Eindhoven University of Technology
MASTER
A repairable spare parts management strategy for the Royal Netherlands Air Force
Wisse, C.
Award date: 2017
Link to publication
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Hoogerheide, August 2017
A Repairable Spare Parts Management Strategy for the Royal Netherlands Air Force
By C. Wisse
BSc Industrial Engineering Student identity number 0808994
In partial fulfillment of the requirements for the degree of
Master of Science In Operations Management and Logistics
Supervisors : Dr. T. Tan, TU/e, OPAC Dr. Ir. J.J. Arts, TU/e, OPAC Prof. Dr. Ir. G.J.J.A.N. van Houtum, TU/e, OPAC Second Lieutenant Ing. L.W.F. Koops, Royal Netherlands Air Force
TUE. Department of Industrial Engineering and Innovation Sciences Series Master Theses Operations Management and Logistics
Subject headings: spare parts; static scheduling; dynamic scheduling; discrete event simulation; inventory control
It should be noted that the numbers used in this report are fictitious and serve only for illustrative purposes due to confidentiality reasons.
I
ABSTRACT This research has been conducted in collaboration with the RNLAF. Stakeholders at the RNLAF indicate that orders at the component repair shop are currently scheduled inefficient, which causes longer lead times and thus higher inventory levels. The objective of this research is to identify scheduling rules and to study their effect in repairable inventory systems. Furthermore, we aim to optimize the base stock levels based on the optimal scheduling rule, i.e. we jointly optimize the base stock levels and the real-time scheduling decisions. Therefore, we modeled a M/M/1 queueing system that evaluates the aggregate mean number of backorders for a given base stock vector and scheduling rule. Then, for each constraint on the aggregate mean number of backorders, optimized base stock levels follow from Lagrangian relaxation and Newsvendor equations. We conclude that the optimal scheduling rule over all cases has the following main characteristics:
- Takes into account the stochasticity of the arrival process. - Takes into account the average processing time of a SKU. - Takes into account the real-time stock levels of SKU’s.
We show that the Myopic Allocation rule meets all these characteristics and outperforms the other scheduling rules in almost all cases.
II
EXECUTIVE SUMMARY This document contains the results of a master thesis project carried out at the Royal Netherlands Air Force. Most assets used by the RNLAF are expensive and consist of technologically advanced systems. A high availability of these assets is desired, since the assets are used to protect the country and during military missions. Since the development of these assets is very costly and time consuming, repairing an asset is preferred instead of purchasing an entire new one. To achieve a high availability of the assets, orders are placed at 981 SQN, which is the component repair shop of the RNLAF.
The lead time of a component is highly dependent of the route through the repair shop, the capacity of operators, machines, test equipment and the availability of lower level SRU’s. Therefore, to optimize the lead time of a component, the capacity of the repair shop needs to be used optimally and the repair process needs to be controlled as good as possible. The repairable spare part inventory and scheduling methodology that is currently used at the RNLAF is mainly based on the experiences of employees instead of on specified control rules. According to stakeholders, orders are currently scheduled inefficient due to this method. Therefore, the objective of this research is to identify scheduling rules and to study their effect in repairable inventory systems. Furthermore, we aim to optimize the base stock levels based on the optimal scheduling rule, i.e. we jointly optimize the base stock levels and the real-time scheduling decisions. We design a conceptual integrated model which aims to minimize investment costs subject to a constraint on the aggregate mean number of backorders.
This thesis contributes both to science and to industry. Despite the fact that Adan et al. (2009), Tiemessen and van Houtum (2010), and Tiemessen and van Houtum (2013) suggest that research needs to be done to jointly optimize dynamic priority rules and base stock levels, it has not been investigated yet. Besides, the already existing method of jointly optimizing static priority rules and base stock levels is extended the possibility of having different processing times for each SKU. Finally, we design scheduling rules especially for repair shops with high material and capacity uncertainty, which are evaluated in our model. This thesis also contributes to industry, since highly expensive spare parts and high material and capacity complexity are common in the aviation industry. Therefore, the contributions of this research also apply for other parties in this industry.
MODEL FORMULATION
We consider a M/M/1 queueing system with a stock point that keeps multiple repairable spare parts on stock to serve a set of technical systems in a certain region. The stock point keeps repairable SKU’s and we have repair by replacement.
To formulate the model, we made the following assumptions:
- All parts are critical - For each SKU, the number of part failures in the installed base occurs according to a Poisson process with a constant rate - We model the repair facility as a single exponential server - Processing times of all SKU’s are exponentially distributed and mutually independent - Priorities are preemptive III
The objective of the model is to minimize the total investment in spare parts subject to a constraint on the aggregate mean number of backorders. To solve this non-linear integer programming problem, we evaluate the aggregate mean number of backorders for a given base stock vector and scheduling rule. For static scheduling rules, we build an analytic model to evaluate the aggregate mean number of backorders. We use the exact method of Sleptchenko et al. (2015) to find the steady state joint queue length distribution under a given assignment of SKU’s to unique priority classes. Then, optimized base stock levels for each backorder constraint follow from Lagrangian functions and Newsvendor equations. For dynamic rules we build a simulation model to evaluate the aggregate mean number of backorders, due to the high computational complexity of applying an analytical model. The simulation model is used to find the average distribution of the joint queue length under a specific dynamic scheduling rule and specific base stock levels. Then, through an iterative process, optimized base stock levels for each backorder constraint follow from this distribution, Lagrangian relaxation and Newsvendor equations.
SCHEDULING RULES
We evaluate three static scheduling rules. The first heuristic schedules the SKU with the highest costs first (Heuristic 1). Heuristic 2 schedules the order with shortest processing times first. Finally, Heuristic 3 schedules the SKU with the highest product of costs and processing rate first.
Furthermore, we tested six dynamic scheduling rules, starting with two rules that performed best in literature (Tiemessen and van Houtum, 2013). One that aims to maximize the expected backorder reduction (the MA-Rule) and one that aims to equalize the time until the next backorder (the MERT- Rule). Since the repair shop of the RNLAF differs from the repair shops discussed in these studies, we choose to test a four scheduling rules that are designed especially for the repair shop of the RNLAF. The first rule is comes closest to the currently used methodology of the RNLAF and is added to serve as a base line. This rule consist of three priority classes, with in the first class SKU’s with an active backorder, in the second class SKU’s with a net stock level of zero and in the third class SKU’s with a positive inventory level (the CPS-Rule). Ties are broken FCFS. The second rule breaks ties in the priority classes as in the CPS-Rule based on prices of the SKU’s (the MPS1-Rule) and the third rule breaks ties based on the deviation from the base stock level (the MPS2-Rule). Finally, the MA-Rule is adapted to take into account the prices of a spare part (the MMA-Rule).
CASE STUDY
To model the repair shop at the RNLAF as a repair shop with a single location with one workstation and one stock point, the repair shop at the RNLAF is simulated. This simulation is used to decide on the processing times of an order through the repair shop, which are used as input of both M/M/1 models. This case study shows that the costs of inventory can be reduced with at least 54,8% by only taking into account the prices of SKU’s and real-time inventory levels (the CPS-Rule) and with at least 54,6% by taking into account the stochasticity of the arrival rates, the processing rates and the real-time inventory levels (the MA-Rule).
IV
CONCLUSIONS
From our numerical analysis we conclude that the performance of rules that only take into account the prices of SKU’s and real-time inventory levels are not stable when input parameters are varied and have only high performances when prices of SKU’s are extremely varying. Furthermore, we find that the MA- Rule optimizes the performance of a repair shop over almost all tested cases and the performance of this rule are stable when input parameters vary. We conclude that the optimal scheduling rule over all cases has the following main characteristics:
- Takes into account the stochasticity of the arrival process. - Takes into account the average processing time of a SKU. - Takes into account the real-time stock levels of SKU’s.
The MA-Rule and the MMA-Rule are the only scheduling rules tested that meets all these characteristics.
RECOMMENDATIONS
We recommend to the RNLAF to check to which extent the sample used in the case study is representative for the whole data set. When the prices of spare parts in the whole data set are more or equally varying as the prices in the case study, we recommend to implement a rule that takes into account these prices in their planning and scheduling methodology. But when the prices of spare parts in the whole data are less varying than the prices in the case study, we recommend to the RNLAF to apply a rule that takes into account the three identified characteristics.
V
PREFACE This thesis is the result of a graduation project that has been conducted at the RNLAF and serves as a completion of the Master Operations Management & Logistics at Eindhoven University of Technology. I have enjoyed spending the past months in such an interesting organization as the RNLAF.
Many people helped me with this research. In particular I would like to thank my first university supervisor Tarkan Tan. Even though he was on sabbatical leave on the other side of the world during the first months of my project, he always took time to elaborately respond my mails and to support me with critical notes and good advices. Furthermore, I really appreciated his enthusiasm, flexibility, positive approach, and the lots of good advices he gave to me. I also want to thank Joachim Arts for his great commitment. His challenging questions and critical notes helped me improving my thesis. I really appreciated his patience and enthusiasm.
Furthermore, I would like to thank my company supervisor, Leon Loops, for his supervision during my internship. I really appreciated the freedom and trust he gave me to carry out my project, while still keeping me focused on the situation and challenges at 981 SQN. Furthermore, I want to thank Tienka van Campenhout for her interest, involvement, and enthusiasm. I would also like to thank my other colleagues at the operations office of 981 SQN for sharing their passion for the organization with me and the small field trips we made on the air base.
Finally, I want to thank all people close to me for the support and for listening to my endless stories about non-understandable problems and great experiences.
Christianne Wisse
Middelburg, August 2017
VI
TABLE OF CONTENTS
ABSTRACT ...... II
EXECUTIVE SUMMARY ...... III
PREFACE ...... VI
LIST OF ABBREVIATIONS ...... X
LIST OF SYMBOLS ...... XI
LIST OF FIGURES ...... XIII
LIST OF TABLES ...... XIV
1. INTRODUCTION ...... 2
1.1 Logistics Centre Woensdrecht ...... 2
1.2 981 Squadron ...... 3
1.2.1 Material flow ...... 3
1.2.2 Market and demand characteristics ...... 4
1.2.3 Products ...... 5
1.2.4 Control structure ...... 5
1.3 Typology of the repair shop ...... 6
1.4 Environment...... 7
2. RESEARCH DESIGN ...... 8
2.1 Relevant challenges at the RNLAF ...... 8
2.2 Research objective ...... 9
2.3 Research questions ...... 9
2.4 Project approach ...... 9
2.5 Scope ...... 10
2.6 State of the art review ...... 10
2.7 Contribution to science and industry ...... 12
VII
2.7.1 Contribution to science ...... 12
2.7.2 Contribution to industry ...... 12
3. CURRENT SITUATION ...... 14
3.1 Currently used inventory rules ...... 14
3.2 Currently used scheduling rules ...... 14
4. INTEGRATED MODEL ...... 16
4.1 System description ...... 16
4.2 Improved inventory model ...... 18
4.2.1 Determination of mean number of backorders for static scheduling rules ...... 18
4.2.2 Optimization of the base stock levels for static rules ...... 22
4.2.3 Determination of mean number of backorders for dynamic scheduling rules ...... 23
4.2.4 Optimization of the base stock levels for dynamic rules ...... 24
4.3 Scheduling rules ...... 24
4.3.1 Rules from literature ...... 24
4.3.2 Rules designed for the RNLAF ...... 27
5. CASE STUDY: RNLAF ...... 30
5.1 Application of the proposed model ...... 30
5.2 Data sample ...... 30
5.3 Assumptions ...... 31
5.4 Comparison of the scheduling rule ...... 31
5.5 Simulation ...... 32
5.5.1 Simulation 1 ...... 32
5.5.2 Simulation 2 ...... 32
5.6 Results ...... 32
5.7 Model validation and verification ...... 35
VIII
5.7.1 Simulation 1 ...... 35
5.7.2 Simulation 2 ...... 35
6. NUMERICAL ANALYSIS ...... 37
6.1 Input parameters ...... 37
6.2 Results ...... 38
7. IMPLEMENTATION ...... 42
8. CONCLUSIONS AND RECOMMENDATIONS ...... 44
8.1 Conclusions ...... 44
8.2 Limitations and recommendations ...... 45
8.2.1 Limitations and recommendations for the RNLAF ...... 46
8.2.2 Limitation and recommendations for further research ...... 46
BIBLIOGRAPHY ...... 48
APPENDIX ...... 50
Appendix A: Organization chart of the Ministry of Defence ...... 50
Appendix B: Busy periods ...... 51
Appendix C: Lay-out simulation dynamic scheduling (Simulation 2) ...... 52
Appendix D: an example of the application of the proposed dynamic scheduling rules ...... 53
Appendix E: Data sample case study ...... 54
Appendix F: Results case study ...... 58
IX
LIST OF ABBREVIATIONS
APS Advances Planning System AVIO Avionics CPS Current Priority Setting DMO Defence Materiel Organisation FCFS First Comes, First Served GUI Graphical User Interface KMSL Royal Netherlands Air Force Military School IMO Inspections, Metal and Surface Treatment LCW Logistics Centre Woensdrecht LRU Line Replaceable Unit MA Myopic Allocation MERT Modified Equalization of Repair Times MMA Modified Myopic Allocation MM Engines and Modules MOB Main Operating Base MPS Modified Priority Setting NDO Non Destructive Maintenance PRIO Priority PM Program Management PS Priority Setting PVE Product Responsible Unit RFU Ready for Use RNLAF Royal Netherlands Air Force SRU Shop Replaceable Unit SQN Squadron TCO Total Costs of Ownership TU/e Eindhoven University of Technology WIP Work in Process WMS Weapon Technique & Mechanical Systems
X
LIST OF SYMBOLS
푁 Set of SKU’s 푡 Time
푆푛 Base stock level of SKU 푛 ∈ 푁
푂퐻푛(푡) Physical stock on hand of SKU 푛 ∈ 푁 at time 푡
퐼푅푛(푡) Number of parts in repair of SKU 푛 ∈ 푁 at time 푡
퐵푂푛(푡) Number of backorders of SKU 푛 ∈ 푁 at time 푡
휆푛 Arrival rate of SKU 푛 ∈ 푁
휇푛 Processing rate of SKU 푛 ∈ 푁 휌 Utilization rate 휋 Scheduling rule
퐸퐵푂푛(푆푛, 휋) Mean number of backordered demand of SKU 푛, depending on the scheduling rule 휋 and the basestock level 푆푛. 푀 Set of priority classes 푿 Priority matrix
푥푛,푚 The (푛, 푚)th element of matrix 푿 with 푛 ∈ 푁 and 푚 ∈ 푀
푚푛 The priority class 푚 to which SKU 푛 is assigned with 푛 ∈ 푁 and 푚 ∈ 푀
푛푚 The SKU 푛 which is assigned to priority class 푚 with 푛 ∈ 푁 and 푚 ∈ 푀 푚 푝푗 The probability that the number of parts in repair of priority class 푚 ∈ 푀 is equal to 푗 ∈ ℕ0 at an arbitrary moment
휌푚 The workload of class 푚 ∈ 푀
푞푛 The number of class-푛 customers with 푛 ∈ 푁
(푚푛) 풊 (푖푚푛, 푖푚푛−1, … , 푖1) with 푛 ∈ 푁 and 푚 ∈ 푀
(푚푛) 풒 (푞푚푛, 푞푚푛−1, … , 푞1) with 푛 ∈ 푁 and 푚 ∈ 푀
(푚푛) 풋 (푗푚푛, 푗푚푛−1, … , 푗1) with 푛 ∈ 푁 and 푚 ∈ 푀 (푚 ) ퟎ 푛 The zero vector of length 푚푛 with 푛 ∈ 푁 and 푚 ∈ 푀
(푚푛) 풆풌 A zero vector of length 푚푛 with a 1 at position 푚푛 + 1 − 푘 with 푛 ∈ 푁 and 푚 ∈ 푀 (푚 ) 푀 푔 (푚 ) 푛 푘;풊 푛 The probability that, when starting in state (0, … ,0, 푞푚푛+1 − 1, 풒 ) + 풆풌 , the
first passage to class-(푚푛 + 1) level 푞푚푛+1 − 1 happens in state (0, … ,0, 푞푚푛+1 − (푚푛) (푚푛) 1, 풒 + 풊 ) with 푚푛 = 푀 − 1, … ,1; 푘 ≥ 푚푛 + 1; 푛 ∈ 푁
(푚푛) (푀) 푓푘;풊(푚푛) The probability that, when starting in state (0, … ,0, 풒 ) + 풆풌 , the first
passage to class-푚푛 levels less than or equal to 푞푚푛 + 푖푚푛, happens in state XI
(푚푛) (푚푛) (0, … ,0, 풒 + 풊 ) with 푚푛 = 푀 − 1, . . . ,1; 푘 ≥ 푚푛 + 1; 푛 ∈ 푁
퐵푘 The Laplace-Stieltjes transform (LST) of the processing time of a class-푘 customer
퐵 푃푘 The LST of a high priority busy period initiated by a class-푘 customer ( ) The LST of a high-priority busy period (class-( and higher), with 퐵 푃푚푛+1,…,푀 푠 푚푛 + 1) 푛 ∈ 푁 and 푚 ∈ 푀 푃(풒(푀)) Equilibrium probabilities, which is the probability that the state of the system is 풒(푀) 푺 The vector of base stock levels 푺(푥) The vector of base stock levels of iteration x
푇푛 A non-negative stochastic variable that denotes the look-ahead-time of SKU 푛 ∈ 푁
퐷푛(푇푛) The demand of SKU 푛 during time period 푇푛 풙 The state of a system
Δ푔푛(푥푛) The expected backorder reduction at time 푡 = 푇푛 that results from the decision to send one item of SKU 푛 to repair at time 푡 = 0
훿푛 훿푛 = 1 when SKU 푛 ∈ 푁 has an active backorder (i.e. 푥푛 < 0) and 훿푛 = 0 otherwise 푄 The batch run size
퐹푇푛 Flow time of SKU 푛 ∈ 푁
퐴퐹푇푛 Adapted flow time of SKU 푛 ∈ 푁
풇풍풐풘 Low vector of factors
풇풉풊품풉 High vector of factors
풇풗풂풓풊풂풃풍풆 Highly variable vector of factors
XII
LIST OF FIGURES Figure 1: Organizational chart of LCW ...... 3
Figure 2: Material flow of LCW ...... 4
Figure 3: Control structure 981 SQN ...... 5
Figure 4: Repair shop typology (derived from Driessen et al. (2013) ...... 7
Figure 5: Project approach ...... 10
Figure 6: Priority queuing system of 981 SQN ...... 15
Figure 7: Spare parts supply system ...... 16
Figure 8: Optimal solution under different backorder constraints ...... 33
Figure 9: Zoomed in: optimal solution under different backorder constraints ...... 33
Figure 10: Average reduction in costs (%) compared to the CPS-Rule ...... 41
Figure 11: Organization Chart of the Ministry of Defence ...... 50
Figure 12: The busy- and idle periods in a system with one priority class ...... 51
Figure 13: Layout of Simulation 2 ...... 52
XIII
LIST OF TABLES Table 1: Average reduction in costs (%) compared to the CPS-Rule for each scheduling rule under a specific backorder constraint ...... 34
Table 2: Parameter values for the test bed ...... 38
Table 3: Average reduction in costs (%) compared to the CPS-Rule over all scenarios ...... 38
Table 4: Average reduction in costs (%) compared to the CPS-Rule ...... 40
Table 5: Values of the evaluation function for a specific example ...... 53
Table 6: Input case study: arrival rates and purchasing prices ...... 54
Table 7: Input case study: capacity of the workstations in hours per working day of eight hours ...... 54
Table 8: Input case study: distribution of the processing time at the workstations for each SKU ...... 55
Table 9: Input case study: probability on a routing for each SKU ...... 55
Table 10: Input case study: routing matrix ...... 56
Table 11: Results of the case study for Heuristic 1 ...... 58
Table 12: Results of the case study for Heuristic 2 ...... 58
Table 13: Results of the case study for Heuristic 3 ...... 59
Table 14: Results of the case study for the MA-Rule ...... 59
Table 15: Results of the case study for the MERT-Rule ...... 59
Table 16: Results of the case study for the CPS-Rule ...... 59
Table 17: Results of the case study for the MPS1-Rule ...... 60
Table 18: Results of the case study for the MPS2-Rule ...... 60
Table 19: Results of the case study for the MMA-Rule ...... 60
XIV
1
1. INTRODUCTION This document contains the results of a master thesis project carried out at the Royal Netherlands Air Force. In this section the research environment is introduced.
1.1 LOGISTICS CENTRE WOENSDRECHT The Ministry of Defence consists of the Executive Staff, the Defence Materiel Organisation (DMO), the Support Command and the four armed forces: the Navy, Army, Air Force and Marechaussee. DMO and the Support Command provide support to all the armed forces of the Netherlands by providing products and services. The Minister of Defence has the overall lead, while the Chief of Defence controls the armed forces operationally. The organizational chart of the Ministry of Defence can be found in Appendix A. The Royal Netherlands Air Force (RNLAF) lends support in combating unrest and provides disaster relief on a global basis. In the Netherlands, the RNLAF ensures security from the air. In order to do this, it has highly-qualified personnel, aircraft, helicopters and other weapon systems at its disposal. The bases of the RNLAF are dispersed throughout the Netherlands. Woensdrecht Air Base is the Main Support Base of the RNLAF. This base houses two units, which are responsible for training, education, meteorological support, logistics and maintenance, namely the Royal Netherlands Air Force Military School (KMSL) and Logistics Centre Woensdrecht (LCW). LCW is responsible for maintenance on the F-16, teaching aircraft and helicopters, such as Apache, Chinook and NH90. They also ensure that all materials are ordered, repaired and delivered on time. LCW consists of two units: Program Management (PM) and Maintenance and Logistics. PM is responsible for the availability of the flying weapon systems and Maintenance and Logistics is responsible for maintenance and repair. Maintenance and Logistics consists of five underlying units, which are all responsible for specific tasks. The organizational chart of LCW is visualized in Figure 1.
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Logistics Centre Woensdrecht
Program Maintenance Management & Logistics
980 Squadron: 981 982 Squadron: PVE Aircraft & Squadron: Technology & 983 Squadron: Supporting Helicopter Component Mission Logistics Systems Maintenance Maintenance Support
Bureau Production Engineering Operations Office
PVE PVE Weapon Inspections, PVE Engines Technique & Metal & PVE Avionics and Modules Mechanical Surface Systems Treatment
Figure 1: Organizational chart of LCW
1.2 981 SQUADRON In this section we will describe the processes and characteristics of 981 Squadron.
1.2.1 MATERIAL FLOW The flying weapon systems used by the RNLAF operate globally. These systems require an inspection when the system has flown a predefined number of flight hours or upon failure. This inspection takes place at the hangar at an air base. This air base can be one of the Main Operating Bases (MOB’s), but for major maintenance operations the flying weapon systems are mainly transferred to Main Support Base Woensdrecht, 980 Squadron (SQN). A MOB is the home base of a flying weapon system, which can be Volkel Air Base, Leeuwarden Air Base, Gilze-Rijen Air Base or Eindhoven Air Base. In the hangar, an inspection is carried out which reveals which Line Replaceable Units (LRU’s) have failed. These failed LRU’s are disassembled from the flying weapon system and replaced by the same type of Ready For Use (RFU) LRU out of the spare part inventory, i.e. repair by replace. Generally, the failed LRU’s are sent to the warehouse for failed LRU’s. However, sporadically the failed LRU’s are sent directly to the component repair shop, 981 SQN.
PM is responsible for the availability of the flying weapon systems. One of their main tasks is to take control of the spare parts inventory. To maintain sufficient inventory levels, PM needs a reliable material planning. To realize this material planning, PM places orders at 981 SQN to repair LRU’s and Shop Replaceable Units (SRU’s). Besides PM, other customers of 981 SQN are 980 SQN and the MOB’s. Note
3
that PM is not only a customer of 981 SQN. Since they take control of the spare parts inventory, they are also the supplier of RFU LRU’s, SRU’s and lower level SRU’s.
When an order is accepted by 981 SQN, the failed LRU is transferred to one of the following main workstations: Product Responsible Unit (PVE) Weapon Technique & Mechanical Systems (WMS), PVE Avionics (AVIO) or the main workstation of a public private partner: PVE Engines and Modules (MM). At these main workstations, the failed LRU is disassembled and the underlying SRU’s are visually inspected and/or tested. Thereafter, failed parts can be sent to the supporting main workstation PVE Inspections, Metal and Surface Treatment (IMO), where detailed inspection and the repair process take place. PVE IMO consists of eleven workstations, each with their own specialty.
The route through the repair shop is dependent on the type and status of the failed component. When an operation at a workstation is completed, the component is sent immediately to the next workstation. Since it is possible that there is a queue at a workstation, components are placed in the queue of this workstation. When a component has passed all the workstations on the routing, it is sent back to a main workstation. At this main workstation the component is assembled again and sent to the warehouse for RFU LRU’s or directly to the hangar. Occasionally it happens that a component cannot be repaired anymore. These parts are scraped. Note that the lead time of a component is highly dependent on the route through the repair shop, the capacity of operators, machines and test equipment, and the availability of lower level SRU’s.
The material flow described above is visualized in Figure 2.
Figure 2: Material flow of LCW
1.2.2 MARKET AND DEMAND CHARACTERISTICS 981 SQN performs preventive maintenance as well as corrective maintenance. Preventive maintenance takes place when a flying weapon system has flown a certain number of flight hours. After this threshold,
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an inspection is performed according to regulations. Generally, flight hours are planned in advance, and therefore these maintenance activities can be planned. Corrective maintenance takes place upon failure. This generally happens unexpectedly, since it is hard to predict the failures of many components. This makes it challenging plan these maintenance activities. Furthermore, since the flying weapon systems are expensive and the importance of availability is high, the customers expect that 981 SQN can satisfy order request very quickly. In general the demand at 981 SQN can be characterized as uncertain.
1.2.3 PRODUCTS 981 SQN maintains LRU’s and SRU’s of all of the ten different flying weapon systems used by the RNLAF. These weapon systems consist of many different LRU’s with a high value and these LRU’s consist of different SRU’s, which possibly consists of lower level SRU’s. This results in a highly complex product structure of about 1,500 different types of parts per year, which are maintained at 981 SQN. The total number of orders processed by 981 SQN is about 10,000 per year.
1.2.4 CONTROL STRUCTURE The control structure of 981 SQN is as follows. Since the material planning is in control of PM, the control structure of 981 SQN is affected by the capacity planning and resource allocation. The customer sends an order request to the operations office of 981 SQN. This order can be accepted or rejected. When an order is accepted it is sent to engineering, where the job is prepared. Otherwise, the order is sent back to the customer. When the job is prepared, it is released to the operational workstations based on the planning of the order. In these workstations the component is (dis)assembled, inspected and repaired. When the order is finished and the part is RFU again, it is sent to the warehouse for RFU parts. This control structure of 981 SQN is visualized in Figure 3. Note that a line represents a material stream, while a dotted line represents an information stream.
Figure 3: Control structure 981 SQN
5
1.3 TYPOLOGY OF THE REPAIR SHOP Driessen et al. (2013) present a model for the typology of repair shops, which is shown in Figure 4. Their typology introduces two dimensions: capacity complexity and material uncertainty. Capacity complexity refers to the requirement of specialized skills or repair men to complete a repair. According to Driessen et al. (2013), the characteristics of capacity complexity can be quantified by:
- The number of LRU types that are treated; - The average size of the set of operations that is potentially and regularly required for treating a LRU;
Material uncertainty is the extent to which repair jobs for the same item require different sets of materials. The characteristics of material uncertainty can be quantified by:
- The average size of the set of materials that is potentially and regularly required for the repair of a LRU; - The average number of materials required per repair job;
The repair shop typology of Driessen et al. (2013) consists of four types with the following characteristics:
- Type I o Homogenous repair capacity; o Small set of SRU’s that has to be kept on stock; o Narrow product mix; o Low diversity in needed material for the same LRU; - Type II o Homogenous repair capacity; o Large set of SRU’s that has to be kept on stock; o Narrow product mix; o High diversity in needed material for the same LRU; - Type III o Heterogeneous repair capacity; o Small set of SRU’s that has to be kept on stock; o Wide product mix; o Low diversity in needed material for the same LRU; - Type IV o Heterogeneous repair capacity; o Large set of SRU’s that has to be kept on stock; o Wide product mix; o High diversity in needed material for the same LRU; 981 SQN maintains a large number of different parts, so 981 SQN has a wide product mix. A wide diversity of resources and specialized skills are needed to repair these parts, which indicates a
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heterogeneous repair capacity. The wide product mix and the heterogeneous repair capacity indicate that 981 SQN has a high capacity complexity. Furthermore, for maintaining the parts also a wide diversity of high valued materials is needed, so a large set of SRU’s has to be kept on stock. In addition, since the maintained LRU’s have a complex structure, failure of different SRU’s can lead to failure of a LRU. This leads to a high diversity in needed material for the same LRU. The large set of SRU’s that has to be kept on stock and the high diversity in needed material for the same LRU indicate a high material uncertainty. Since 981 SQN has a high capacity complexity and high material uncertainty, it can be classified as a Type IV repair shop. Additionally, we note that the SKU’s repaired by the repair shop of 981 SQN have high variations in costs.
Figure 4: Repair shop typology (derived from Driessen et al. (2013)
1.4 ENVIRONMENT This research is part of a broader research project of the RNLAF and Eindhoven University of Technology. The aim of the broader project is to increase the availability of flying weapon systems and to reduce total maintenance costs by creating a plan tool for 981 SQN. This research is the second master project in line and is followed by a PDEng project. This research and the research of van de Vondervoort (2017) serve both as input for the PDEng project. This PDEng project, conducted by Mark van Someren, aims to result in a plan tool for 981 SQN.
Vondervoort (2017) executed a detailed problem analysis to determine all causes that are resulting in a low delivery reliability of the component repair shop. Since multiple causes are related to the current control structure, a control structure redesign was developed which included multiple decision functions of the component repair shop. This redesign enables the 981 SQN to provide more reliable due dates. Furthermore, Vondervoort recommends including an algorithm in the plan tool for 981 SQN, which minimizes the flow times of orders through the repair shop.
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2. RESEARCH DESIGN In this chapter the design of this research in introduced. This chapter starts with an introduction on the challenges 981 SQN is facing (Section 2.1). After that, we introduce the research objective in Section 2.2. In Section 2.3, we state the research questions and in Section 2.4 the research approach is introduced. Furthermore, is Section 2.5 we formulate the scope of this research. Finally, in Section 2.6 the state of art in literature is elaborated and in Section 2.7 we describe the contribution of this research to science and industry.
2.1 RELEVANT CHALLENGES AT THE RNLAF The assets used by the RNLAF are expensive and consist of technologically advanced systems. A high availability of these assets is desired, since the assets are used to protect the country and during military missions. So, having less assets available result in a lower safety level for the country. Since the development of these assets is very costly and time consuming, repairing an asset is preferred instead of purchasing an entire new one.
The availability of the assets used by the RNLAF is highly related to the availability of RFU LRU’s and SRU’s. Since PM is responsible for the availability of RFU LRU’s and SRU’s, PM needs a reliable material planning to achieve a high availability of the assets. To realize this material planning, PM places orders at 981 SQN for LRU’s and SRU’s. Most of the problems experienced at PM are due to the currently low delivery reliability of 981 SQN. The low delivery reliability makes it challenging for PM to create a reliable material planning. So, the low delivery reliability at the component repair shop has a direct effect on the availability of the assets of the RNLAF.
The lead time of a component is highly dependent of the route through the repair shop, the capacity of operators, machines, test equipment and the availability of lower level SRU’s. So, to optimize the lead time of a component, the capacity of the repair shop needs to be used optimally and the repair process needs to be controlled as good as possible. The planning and scheduling of repair orders is currently mainly based on the experience of employees instead of on specified control rules. According to stakeholders within SQN 981 and PM, orders are currently scheduled inefficient due to this method. Furthermore, it is challenging to create insight in the expected flow times, which leads to a low delivery reliability. This low delivery reliability and inefficient scheduling methodology makes it hard for PM to obtain a reliable material planning. As a result, PM currently chooses to outsource orders to the industry instead of repairing them at 981 SQN more often. The main advantages of outsourcing to industry are the reliable processing times. However, two major drawbacks of outsourcing compared to insourcing them at 981 SQN, are the long processing times and high costs.
In summary, 981 SQN is experiencing a low delivery reliability and suboptimal flow times. This is presumably caused by an inefficient use of resources and by a lack of insight into the status of the orders and processes in the repair shop.
Non-optimal flow times and an inefficient use of resources are common challenges in repair shops of Type IV. Therefore, the conclusions drawn in this resource apply to all repair shops of Type IV.
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2.2 RESEARCH OBJECTIVE The objective of this research is to identify scheduling rules and to study their effect in repairable inventory systems. Furthermore, we aim to optimize the base stock levels based on the optimal scheduling rule, i.e. we jointly optimize the base stock levels and the real-time scheduling decisions. We design a conceptual integrated model which aims to minimize investment costs subject to a constraint on the aggregate mean number of backorders.
2.3 RESEARCH QUESTIONS The research objective above is formulated into the following main research question:
‘’How can the spare parts management methodology of a repair shop be improved to increase the relevant performances?’’
To answer the main research question, we formulate three sub-questions:
- ‘’What is the currently used spare parts management methodology at 981 Squadron and what are the relevant limitations of this methodology?’’
- ‘’How can base stock levels and real-time scheduling decisions be jointly optimized?’’
- ‘’What spare part management methodology optimizes the performances of a repair shop and what are their performances and limitations?’’
2.4 PROJECT APPROACH To answer the research questions we pass multiple phases: problem definition, modeling, analysis and conclusions.
The first phase, problem definition, serves as a preparation for this project. The most important deliverable of this phase is the research proposal. The second phase is modeling, which starts with a description of the current situation of the RNLAF. Thereafter, three sub-activities take place:
- To execute a case study, data from SQN 981 is required. The data gathered from SAP is not immediately applicable, therefore data preparation is required. From the acquired data we select a sample for the case study.
- The current situation of 981 SQN is modeled to serve as a base line model. This is necessary to find the potential value of the proposed scheduling rules.
- To identify the state of art in research on scheduling rules in repair shops, there is search for literature. This literature study aims to identify possible scheduling rules and optimization methods that are applicable in this research. After that, we formulate and design the model.
The third phase is the analyzation phase. In this phase the model created in the modeling phase is executed and evaluated to answer the research questions. Finally, in the conclusion phase, we give an
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answer to the research questions and give recommendations as well for the RNLAF as for industry. This process is visualized in Figure 5.
Research approach
Problem definition Modeling Analyzation Conclusions
Identify possible Run the model Evaluation and Conduct Search for scheduling rules for a sensitivity interviews literature and descibe numerical analysis integrated model Design the analysis proposed integrated model Describe the Model the Conclusions Research current current and Proposal situation of situation of recommendati 981 SQN 981 SQN ons
Read existing Run the model Obtain and Select case research on for a case Evaluation prepare data study 981 SQN study
Figure 5: Project approach
2.5 SCOPE One objective of this research is to increase the currently low delivery reliability at 981 Squadron. The order acceptance of orders handled at 981 Squadron takes place when the order request arrives at the Squadron and the delivery reliability is measured when the component leaves the Squadron. Since both the order acceptance and the determination of the delivery reliability take place in 981 Squadron, the focus of this research is on the entire squadron. Furthermore, we aim to jointly optimize the base stock levels at PM and the real-time scheduling decisions. Therefore, PM is added to the scope of this research.
Research of van de Vondervoort (2017) suggests that capacity at the workstations is currently used inefficient due to the used scheduling rules. Therefore, we aim to use the existing capacity more efficiently.
In this research we focus on all orders. This regards both repair jobs associated with corrective maintenance and preventive maintenance.
2.6 STATE OF THE ART REVIEW The RNLAF desires a high availability of the flying weapon systems. According to Driessen et al. (2015), obtaining a high availability of the capital assets against acceptable supply chain costs is of increasing importance, especially when companies are facing decreasing budgets. To obtain this high availability, the RNLAF removes parts either upon failure of to prevent future failure. The removed parts are repaired at 981 SQN. According to Guide and Srivastava (1997) and Duffua and Raouf (2015) the main difference between a repair shop and a traditional assembly or job shop is that in a repair shop the timing, quantity and variability of jobs are harder to predict. Additionally, a repair shop has to deal with more varying levels of complexity of products (Driessen et al., 2013). 10
Much research has been done on traditional assembly or job shop and repair shop scheduling rules. Since the behavior of a job shop differs from the behavior of a repair shop, we choose to focus only on literature in which repair environments are considered. According to Guide and Srivastava (1997) and Duffua and Raouf (2015) the main difference between a repair shop and a traditional assembly or job shop is that in a repair shop the timing, quantity and variability of jobs are harder to predict. Additionally, a repair shop has to deal with more varying levels of complexity of products (Driessen et al., 2013).
Hausman and Scudder (1982) divide scheduling rules in three different streams. The first stream uses static priority rules to schedule jobs, in which items are assigned to priority classes, independent of the status of the shop and the real-time inventory status. The second stream uses dynamic rules that take into account the progress of jobs through the system. Finally, the third stream uses dynamic rules that take into account the real-time inventory status.
Berry and Rao (1975), Hausman and Scudder (1982) and Scudder (1984) study scheduling rules out of the three identified streams in systems with a limited number of SKU’s via simulation. All of these studies assume that processing times are deterministic. Berry and Rao (1975) present a count-intuitive result, namely that including information on the shop and inventory status does not improve the performance of the shop significantly. Graves (1977) notes that this inferior cost performance of the dynamic scheduling rules can be partly, if not completely, explained by the excessive safety stock levels used when the real-time inventory status is taken into account. This is counter intuitive, since it is expected that safety stock levels decrease when this information is taken into account. The studies of Hausman and Scudder (1982) and Scudder (1984) confirm the conclusion of Graves (1977). They show that rules that include current inventory status outperform the other scheduling rules and that these rules can further be improved by taking into account the real-time shop status. However, optimization of the repair priorities via simulation is time consuming and it is practically impossible to analyze systems with many SKU’s, due to the curse of dimensionality.
Sleptchenko et al. (2005) present an analytical model for multi-echelon, multi-indenture inventory control. They develop heuristics for jointly optimization of spare parts stocks and the assignment of repair priorities. They find that assigning SKU’s to priority classes performs better than the First Comes First Served (FCFS) scheduling rule. However, their model is limited to two priority classes (high and low) and is only applicable on a limited number of SKU’s, due to the complexity of the model.
A model similar to the model of Sleptchenko et al. (2005) is the model of Adan et al. (2009). They formulate an analytical model and develop heuristics for jointly optimization of spare parts stock and scheduling rules. They show that under a given assignment of repair priorities, optimal spare part stocks follow from Newsvendor equations, and then the optimal repair priorities can be found heuristically. They find that the total costs under static priorities based on holding costs and processing time are more than 40% lower than under the FCFS policy. However, Adan et al. assume that processing times are equal for each priority class and like the model of Sleptchenko et al. (2005), the model of Adan et al. (2009) is limited to static priority rules.
In 2010, Tiemessen and van Houtum propose a new solution method to optimize base stock levels and scheduling decisions including dynamic scheduling rules. They use the method of jointly optimizing the base stock levels and static priorities of Adan et al. (2009) to obtain the base stock levels. Then, they 11
replace the static priority rule by a dynamic scheduling rule. So, in this research, the optimized base stock levels do not depend on the proposed dynamic scheduling rules (i.e. the base stock levels and dynamic scheduling rules are not optimized simultaneously). However, they show that even when base stock levels are based on static scheduling rules, dynamic scheduling rules often reduce total cost further by more than 10%. They suggest that further improvements can be possible when the base stock levels for the proposed dynamic scheduling rules are optimized simultaneously with the scheduling rules.
So, the studies of Sleptchenko et al. (2005) and Adan et al. (2009) concludes that cost can be reduced by jointly optimizing static priority rules and base stock levels. Furthermore, earlier research shows that dynamic scheduling rules outperform static priorities rules (e.g. Hausman and Scudder, 1982; Scudder, 1984; Tiemessen and van Houtum, 2010). These two conclusions indicate that joint optimization of dynamic priority rules and base stock levels can lead to further cost reductions. Despite the fact that Adan et al. (2009), Tiemessen and van Houtum (2010), and Tiemessen and van Houtum (2013) suggest that research needs to be done to this subject, it has not been investigated yet.
2.7 CONTRIBUTION TO SCIENCE AND INDUSTRY This master thesis project aims to contribute to both science and industry. In Section 2.7.1 the intended contribution to science is explained and in 2.7.2 the contribution to industry.
2.7.1 CONTRIBUTION TO SCIENCE We aim to contribute to the research gap identified in Section 2.6, by creating a model that jointly optimize the base stock levels and dynamic priority rules. Furthermore, Adan et al. (2009) already developed a method for jointly optimizing base stock levels and static priority rules. However, they assumed that mean processing times of SKU’s in each priority class are equal. Since in practice many repair shops are characterized by large differences in processing times of SKU’s, we include the possibility of having different processing times for each SKU in our method for jointly optimizing base stock levels and static priority rules.
From the large set of scheduling rules tested in literature, we have selected rules that performed best in previous studies to evaluate in this research (see e.g. Adan et al., 2009; Guide et al., 2000; Tiemessen and van Houtum, 2013). Since a type IV repair shop, such as the repair shop of the RNLAF, differs from the repair shops discussed in these studies, we choose to test a number of scheduling rules that are designed especially for type IV repair shops. By doing this we aim to find the optimal scheduling rule for this type of repair shop.
2.7.2 CONTRIBUTION TO INDUSTRY In particular, this research contributes to the aviation industry in the following ways. Since most of the spare parts in the aviation industry are expensive, a reduction in spare part inventory is beneficial. This reduction in spare part inventory of highly expensive parts can be achieved by integrating decisions on priority rules and base stock levels. Furthermore, this method brings new insights for decision making at different levels. It has impact on strategic decision making, since the model gives advices for spare parts investments. Furthermore, the model gives insights in total processing times and optimal stock levels, which supports estimations regarding the total costs of ownership (TCO). The model also supports
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operational decisions, such as when to repair a spare part in-house and when it is profitable to outsource the repair based on the optimal scheduling rule. Note that these contributions are formulated from the perspective of the RNLAF, but these contributions also apply for other parties in the aviation industry or the capital good industry.
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3. CURRENT SITUATION In this chapter we describe the current environment of 981 SQN. First, in Section 3.1, we introduce the currently used inventory rules. Then, we introduce the currently used scheduling methodology in Section 3.2.
3.1 CURRENTLY USED INVENTORY RULES At this moment, the spare part inventory decisions are mainly based on the experiences of the employees at PM, instead of on a pre-specified spare part management methodology. Based on the experiences of these employees and expected flight hours of the weapon systems, the optimized base stock levels are determined. When the current inventory level drops below the base stock level, PM places an order at 981 SQN or at an external company. Also when PM expects that the inventory level shall drop below the base stock levels, they anticipate on a possible shortage and they already place an order. For example, this occurs when a planned overhaul or a planned mission takes place.
3.2 CURRENTLY USED SCHEDULING RULES Currently, planning and scheduling of repair orders is mainly based on the experience of employees instead of on a formal dispatching methodology. However, in general the currently applied dispatching methodology comes closest to a Priority Setting (PS) policy, in which jobs are sequenced according to priority. This priority is set independently of the status of 981 SQN and is only dependent on the inventory status at PM. The job with the highest priority is sequenced first. The methodology used to schedule ties comes closest to a First Comes First Served (FCFS) policy, in which the jobs are sequenced according to their arrival time. The job which arrives first is sequenced first and vice versa.
In practice, some orders are batched in order to process as efficient as possible. However, batching is not based on an unambiguous rule, but is based on the experience and common sense of the planner. Besides that, no fixed pattern can be found in batching orders. Therefore batching is left out of scope.
The scheduling methodology as it is applied by 981 SQN, where each priority level has its own virtual queue, is visualized in Figure 6.
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Figure 6: Priority queuing system of 981 SQN
To sort orders based on their priority, the RNLAF describes three different priority classes. The priority class of a particular order is determined by the customer, so that can be PM as well as 980 SQN or MOB’s. The different priority classes are:
- Priority 1 (PRIO1); There is no stock available of this component and the planned delivery date is later then the planned requirement date.
- Priority 2 (PRIO2); There is no stock available of this component, but the planned delivery date is earlier then the planned requirement date.
- Priority 3 (PRIO3); There is stock available of this component.
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4. INTEGRATED MODEL In this chapter we determine the proposed integrated model, which aims to jointly optimize base stock levels and real-time scheduling decisions. In Section 4.1 the spare part supply system on which our model can be applied is described. After that, in Section 4.2 we formulate the model. Finally, in Section 4.3 we describe the evaluated scheduling rules.
4.1 SYSTEM DESCRIPTION We consider a system with a single location with one repair facility and one stock point. This stock point keeps multiple repairable spare parts on stock to serve a set of technical systems in a certain region. This set of technical systems is also called the installed base. The stock point keeps repairable SKU’s. When a part in a technical system fails, a demand for a RFU part of the same SKU is sent to the stock point and the failed part is sent to the repair facility. When the RFU SKU is on stock at the stock point the demand can be immediately fulfilled, otherwise the demand is backordered and fulfilled as soon as possible a RFU SKU becomes available at the stock point, i.e. we have repair by replacement. While demand is backordered, the technical system is down until a RFU SKU becomes available, since we assume that all SKU’s are critical. Figure 7 gives a graphical representation of the system.
Figure 7: Spare parts supply system
We assume an infinite time horizon, since the lifetime of the technical systems is assumed to be relatively long (e.g., 20-30 years). The scheduling policies at the repair facility are analyzed via a steady-state analysis.
The set of SKU’s is denoted by 푁 and the number of SKU’s is denoted by |푁|. The SKU’s are assumed to be numbered 푛 = 1,2, … , |푁|. Let 푆푛 be the number of SKU’s 푛 that are on stock at the stockpoint at time 푡 = 0. We assume that all parts can always be repaired and no additional parts are bought after 푡 = 0. So, 16
the inventory position of SKU 푛 stays equal to 푆푛. At any time instant 푡 ≥ 0, the following equation holds:
푂퐻푛(푡) + 퐼푅푛(푡) − 퐵푂푛(푡) = 푆푛 ∀푛 ∈ 푁 (1)
Where 푂퐻푛(푡), 퐼푅푛(푡) and 퐵푂푛(푡) are the physical stock on hand, the number of parts in repair, and the number of backorders of SKU 푛 at time instant 푡 respectively. 푆푛 is also denoted as the circulation stock or base stock level of SKU 푛.
We assume that for each SKU 푛, the number of part failures in the installed base occurs according to a
Poisson process with a constant rate 휆푛 (≥ 0). To ease notation, we introduce the total arrival rate |푁| 휆 = ∑푖=푛 휆푛. The assumption of a Poisson process is justified when lifetimes of parts are exponential or when the merged stream of failure processes of individual technical systems is close to Poisson, which happens when the set of technical systems in the installed base is sufficiently large. Furthermore, note that the assumption of a Poisson failure process with constant rates is common in spare parts literature (see Sherbrooke, 2004).
We model the repair facility as a single exponential server. This assumption is made to limit the computational complexity of the exact solutions. However, in practice it is also possible that a server can repair several failed parts simultaneously. Still, the single server assumption captures the feature that several parts compete for the same repair capacity. We assume that the processing times of all SKU’s are 1 exponentially distributed with mean and mutually independent and that the utilization rate 휌 = 휇푛 |푁| 휆푛 ∑푛=1 < 1, to obtain a stable system. 휇푛
Dispatching of jobs in the repair shop occurs according to a scheduling rule 휋. Since the model aims to optimize base stock levels and scheduling rules simultaneously, the optimized base stock levels for a system depends on scheduling rule 휋.
We assume that repair of a failed part can be interrupted when a part with a higher priority enters the repair facility, i.e. priorities are preemptive.
The repair facility has a steady-state behavior that is similar to the behavior of a priority queueing system. The assumptions made are chosen such that evaluation of the steady-state distributions of the number of repair jobs of various types is possible. Although these assumptions do not always reflect reality, they represent the situation that different SKU’s compete for the same repair capacity. Besides that, note that these assumptions are common in the field of capacitated inventory systems (see e.g. Hopp and Spearman, 2000; Sleptchenko et al., 2005; Tiemessen and van Houtum, 2010).
푎 The price of SKU 푛 is equal to 푐푛 (≥ 0). We are interested in the investment in spare parts at the beginning of the time horizon. The objective is to minimize the total investment in spare parts subject to a constraint on the aggregate mean number of backorders, 퐸퐵푂(푺, 휋). The investment in spare parts 푛 is 푎 denoted by 퐶푛(푆푛) = 푐푛 푆푛 and the total investment in spare parts is given by:
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푎 ∀푛 ∈ 푁 (2) 퐶(푺) = ∑ 퐶푛(푆푛) = ∑ 푐푛 푆푛 푛∈푁 푛∈푁
With 푺 = {푆1, … , 푆|푁|}. The aggregate mean number of backorders is denoted by:
∀푛 ∈ 푁 (3) 퐸퐵푂(푺, 휋) = ∑ 퐸퐵푂푛(푆푛, 휋) 푛∈푁
The target level for the aggregate mean number of backorders is equal to 퐸퐵푂표푏푗, and is also named as the backorder constraint. In conclusion, our optimization problem can be formulated as follows:
min 퐶(푺) 푺 = {푆1, … , 푆|푁|} 푆 ≥ 0 Subject to: 푛 (4)
퐸퐵푂(푺, 휋) ≤ 퐸퐵푂표푏푗
4.2 IMPROVED INVENTORY MODEL
To solve the non-linear integer programming problem described in Equation 4, we evaluate 퐸퐵푂푛(푆푛, 휋) for given base stock vector 푺 and scheduling rule 휋. We build an analytical model to evaluate
퐸퐵푂푛(푆푛, 휋) for static priority rules. We made this decision since analytical models provide a perfectly clear relationship between the input parameters and the output. Furthermore, we build a simulation model
to evaluate 퐸퐵푂푛(푆푛, 휋) for dynamic scheduling rules. We made this decision since the computational complexity of applying an analytical model is extremely high (except for sufficiently small problem instances). Therefore we cannot use an analytical model cannot and simulation is a good alternative.
We discuss the evaluation of 퐸퐵푂푛(푆푛, 휋) in Section 4.2.1 for static scheduling rules and in Section 4.2.2. we explain the evaluation of optimized base stock levels for static rules. We describe the evaluation of expected backorders and optimized base stock levels for dynamic rules in Section 4.2.3 and Section 4.2.4, respectively.
4.2.1 DETERMINATION OF MEAN NUMBER OF BACKORDERS FOR STATIC SCHEDULING RULES The determination of the mean number of backorders for static scheduling rules is heavily based on the method of Sleptchenko et al. (2015). The method of Sleptchenko et al. takes into account different mean processing times for each priority class, in contrast to the method of Adan et al. (2009).
We assume that each SKU is assigned to a different priority class 푚 ∈ 푀, so there exist |푁| priority classes (|푁| = |푀|). The higher the number of the class the higher their priority is. Matrix 푿 is of size |푁| × |푀| and describes the assignment of SKU’s to priority classes. The (푛, 푚)th element of this matrix
is 푥푛,푚:
1 푖푓 푆퐾푈 푛 푖푠 푎푠푠푖푔푛푒푑 푡표 푝푟푖표푟푖푡푦 푐푙푎푠푠 푚 (5) 푥 = { ∀푛 ∈ 푁, 푚 ∈ 푀 푛,푚 0 표푡ℎ푒푟푤푖푠푒
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Since each SKU is assigned to a different priority class ∑푛∈푁 푥푛,푚 = ∑푚∈푀 푥푛,푚 = 1. The assignment of SKU’s to priority classes is described by scheduling rule 휋. The notation 푚푛 denotes the priority class 푚 to which SKU 푛 is assigned and 푛푚 denotes SKU 푛 which is assigned to priority class 푚.
To determine the mean number of backorders per SKU (퐸퐵푂푛(푆푛, 휋)), first the steady-state distributions for the number of parts that are in repair per priority class are be determined. The probability that the number of parts in repair of priority class 푚 ∈ 푀 is equal to 푗 ∈ ℕ0 at an arbitrary moment is denoted by 푚 푝푗 (휋). When analyzing priority class 푚, all classes with a lower priority can be ignored, since preemption is assumed.
The workload of class 푚 ∈ 푀, 휌푚, is equal to:
|푁| 푥푛,푚휆푛 ∀푚 ∈ 푀 (6) 휌푚 = ∑ 푛=1 휇푛 and the total workload, 휌, is equal to:
|푀| ∀푚 ∈ 푀 (7) 휌 = ∑ 휌푚 푚=1
Priority class 푚 = |푀| can be modelled as a classical 푀/푀/1 queueing system, since no higher classes |푀| exist. The probabilities 푝푗 (휋), 푗 ∈ ℕ0 is equal to:
|푀| 푗 푗 ∈ ℕ (8) 푝푗 (휋) = (1 − 휌|푀|)휌|푀| 0
For priority classes 푚 < |푀|, the server is working on higher priority classes during certain time intervals. So, these priority classes behave as an M/M/1 preemptive priority system and can be described by a multi-dimensional Markov process. Sleptchenko et al. (2015) created formulas to solve the equilibrium probabilities for these systems with an arbitrary number of customer classes. They use the matrix analytic method for M/G/1 queueing systems to exactly and recursively calculate the distribution of the parts in the system (i.e. the equilibrium probabilities). First, we introduce some notation: 푞푛 is the (푚) (푚) (푚) number of class-푛 customers, 풊 = (푖푚, 푖푚−1, … , 푖1), 풒 = (푞푚, 푞푚−1, … , 푞1), 풋 is the vector (푚) (푚) index of length 푚, ퟎ is the zero vector of length m and 풆풌 defines a zero vector of length 푚 with a 1 at position 푚 + 1 − 푘.
Following the formulas of Sleptchenko et al., two types of first passage probabilities are calculated. The first type is the first passage probability 푔푘;풊(푚), 푤푖푡ℎ 푚 = |푀| − 1, … ,1; 푘 ≥ 푚 + 1. This first passage (푚) |푀| probability is defined as the probability that, when starting in state (0, … ,0, 푞푚+1 − 1, 풒 ) + 풆풌 , the (푚) (푚) first passage to class-(푚 + 1) level 푞푚+1 − 1 happens in state (0, … ,0, 푞푚+1 − 1, 풒 + 풊 ). In other words, the probability that 푖(푚) customers arrive during a busy period of priority classes higher than 푚, starting with the arrival of class a 푘 customer. More details on busy periods can be found in Appendix B.
The second type is the first passage probability 푓푘;풊(푚) , 푤푖푡ℎ 푚 = |푀| − 1, . . . ,1; 푘 ≥ 푚 + 1. This first (푚) (푀) passage probability is the probability that, when starting in state (0, … ,0, 풒 ) + 풆풌 , the first passage (푚) (푚) to class-푚 levels less than or equal to 푞푚 + 푖푚, happens in state (0, … ,0, 풒 + 풊 ). In other words, 19
the probability that at the end of a busy period of a class-푘 customer, there have been at least 푖푚−1 arrivals during this busy period, and when the server brings this number down to 푖푚−1, there have been at least 푖푚−2 arrivals from the start of the busy period of class-푛 customers, and when the server brings down this number to 푖푚−2, there have been at least 푖푚−3 arrivals from the start of the busy period of class-푘 customers. This goes on until the server brings down the number of customer to 푖1. See Sleptchenko et al. for further details. Here, only the formulas to determine the first passage probabilities and the distribution of the number of SKU’s in repair per priority class are stated.
The first passage probability 푔푘;풊(푚) can be calculated with the following formulas:
|푀| ∀푚 ( ) ( ) ( ) (9) 휇푛푘 − (휆 + 휇푛푘 )푔푘;ퟎ 푚 + ∑ 휆푛푥푔푥;ퟎ 푚 푔푘;ퟎ 푚 = 0 푥=푚+1 ∈ {|푀| − 1, … ,1}; 푚 ∀푘 ≥ 푚 + 1 −(휆 + 휇푛 )푔 (푚) + ∑ 휆푛 푔 (푚) (푚) ∀푚 푘 푘;풊 푥 푘;풊 −풆 푥=1 푥 (푚) ∈ {|푀| − 1, … ,1}; 푀 풊 (10) (푚) (푚) (푚) ∀푘 ≥ 푚 + 1; + ∑ 휆푛푥 ∑ 푔푥;풋 푔푘;풊 −풋 = 0 푥=푚+1 풋(푚)=ퟎ(푚) 풊(푚) > ퟎ(푚)
Equation 8 can be used to recursively calculate all 푔푘;풊(푚) where 푔푘;ퟎ(푚) is the minimum non-negative solution of Equation 7. To solve Equation 7, the Laplace-Stieltjes transform (LST) of the processing time of a class-푘 customer, 퐵푘, is introduced. Furthermore, 퐵 푃푘 is the LST of a high priority busy period initiated by a class-푘 customer. Then, 푔푘;ퟎ(푚) can be calculated as follows (for more details, see Kleinrock (1975)):
푚 ( ) 푔푘;ퟎ 푚 = 퐵 푃푘 (∑ 휆푛푥) 푥=1
푚 푀 푚
= 퐵푘 (∑ 휆푛푥 + ∑ 휆푛푥 (1 − 퐵 푃푚+1,…,푀 (∑ 휆푛푥))) ∀푚 (11) 푥=1 푥=푚+1 푥=1 ∈ {|푀| − 1, … ,1};
∀푘 ≥ 푚 + 1 휇푛푘 = ∑푚 ∑|푀| ∑푚 휇푛푘 + 푥=1 휆푛푥 + 푥=푚+1 휆푛푥 (1 − 퐵 푃푚+1,…,|푀|( 푥=1 휆푛푥)) where 퐵 푃푚+1,…,푀(푠) is the LST of a high-priority busy period (class-푚 + 1 and higher), which is equal to the LST of the busy period on a 푀/퐻푀−푚/1 queue with class-푚 + 1, … , 푀 customers:
|푀| 휆푛푥 퐵 푃푚+1,…,푀(푠) = ∑ |푀| 푥=푚+1 ∑ 휆 푙=푚+1 푛푙 ∀푚 (12) 휇푛 ∙ 푥 ∈ {|푀| − 1, … ,1}; 휇 + 푠 + ∑|푀| 휆 (1 − 퐵 푃 (푠)) 푛푥 푙=푚+1 푛푙 푚+1,…,|푀| 푠 ≥ 0
The first passage probability 푓푘;풊(푚) can be recursively calculated with the following formulas: 20
푚 ∀푚 −(휆 + 휇 )푓 (푚) + ∑ 휆 푓 (푚) 푛푘 푘;풊 푛푥 푘;풊(푚)−풆 푥=1 푥 ∈ {|푀| − 1, … ,1}; (13) (푚−1) |푀| 푖푚−1 풊 ∀푘 ≥ 푚 + 1 + ∑ 휆푛 (∑ ∑ 푔 (푚−1) 푓 (푚) (푚) 푥 (푚−1) (푚−1) 푥;푗푚,풋 푘;풊 −풋 (푚−1) (푚−1) 푥=푚+1 푗푚=0 풋 =ퟎ 풊 ≥ ퟎ ; 풊(푚−1) 푖푚 > 0 + ∑ 푓푥;푖 ,풋(푚−1)푔푘;풊(푚−1)−풋(푚−1)) = 0 풋(푚−1)=ퟎ(푚−1) 푚 ∀푚
∈ {|푀| − 1, … ,1}; 푓푘;0,풊(푚−1) = 푔푘;풊(푚−1) ∀푘 ≥ 푚 + 1 (14) 풊(푚−1) ≥ ퟎ(푚−1)
The first passage probabilities 푓푘;풊(푚) for the case 푚 = 1 are computed as follows:
푖푚−1 푚 = 1; 푓 = 1 − ∑ 푔 푘;푖푚 푘;푖푚 (15) 푗푚=0 푘 ≥ 2; 푖푚 > 0
Now, the equilibrium probabilities, 푃(풒(|푀|)) can be determined, which is the probability that the state of the system is 풒(|푀|):
(|푀|−푚) (푚−1) 푃(ퟎ , 푞푚 + 1, 풒 ) 1 = 휇푛푚
풒(푚) |푀| ∗ (∑ 푃(ퟎ(|푀|−푚), 풒(푚) − 풊(푚)) ∑ 휆 푛푥 ∀푚 ∈ {푀, … ,1}; 풊(푚)=ퟎ(푚) 푥=푚+1 (16) 풒(푚) ≥ ퟎ(푚) ∗ (푓푥;풊(푚) − 푔푥;풊(푚) ) 풒(푚−1) (|푀|−푚) (푚−1) + ∑ 푃(ퟎ , 푞푚, 풒 풊(푚−1)=ퟎ(푚−1)
(푚−1) ( ) − 풊 ) 휆푛푚푔푚;풊 푚−1 )
푃(풒(|푀|)) can be solved recursively starting with 푃(ퟎ(|푀|)) = 1 − 휌.
From 푃(풒(|푀|)), the probability that the number of parts in repair of priority class 푚 is equal to 푗 at an 푚 arbitrary moment, 푝푗 (휋), can be calculated:
∞ ∞ 푚 (|푀|−푚) (푚−1) ∀푚 ∈ 푀; 푝푗 (휋) = ∑ ∑ 푃(풒 , 푗, 풒 ) (17) 풒(|푀|−푚)=ퟎ(|푀|−푚) 풒(푚−1)=ퟎ(푚−1) 풒(푚) ≥ ퟎ(푚)
Finally, the mean number of backorders per SKU 푛 can be calculated:
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∞ 푚푛 ∀푆푛 ∈ ℕ0; (18) 퐸퐵푂푛(푆푛, 흅) = ∑ (푗 − 푆푛)푝푗 (휋) 푗=푆푛+1 ∀푚푛 ∈ 푀 ∀푛 ∈ 푁 푚 To compute the 퐸퐵푂푛(푆푛, 흅) truncation is applied at a sufficiently high level. So, calculating 푝푗 (휋) stops when the cumulative probability exceeds 1 − 휀, where 휀 is in the order of 10−6 − 10−8, since the probabilities for higher 푗 are assumed to be zero.
4.2.2 OPTIMIZATION OF THE BASE STOCK LEVELS FOR STATIC RULES To create the optimal solutions of Equation 4, dependent on the 퐸퐵푂풐풃풋 , we are using Lagrangian relaxation as described by van Houtum and Rustenburg (2015). A general description of Lagrangian relaxation can be found in Appendix B of Porteus (2002). The Lagrangian function for Equation 4 is defined as:
푎 표푏푗 푆푛 ≥ 0 (19) 퐿(푺, 흅, 흀) = ∑ 푐푛 푆푛 + 휆 (∑ 퐸퐵푂푛(푆푛, 휋) − 퐸퐵푂 ) 푛∈푁 푛∈푁 휆 ≥ 0 Where 휆 is the Lagrange multiplier. Note that Equation 4 is a linear combination of objectives and constraints, and is therefore separable (see Porteus (2002) for a definition of separable problems). A character of separable problems is that their Lagrangian function is also separable. Therefore, the Lagrangian function can be defined as:
표푏푗 푆푛 ≥ 0 퐿(푺, 흅, 흀) = ∑ 퐿푛(푆푛, 휆) − 휆퐸퐵푂 푛∈푁 휆 ≥ 0 (20) Where
푎 퐿푛(푆푛, 휆) = 푐푛 푆푛 + 휆퐸퐵푂푛(푆푛, 휋) 푆푛 ≥ 0 휆 ≥ 0 Note that there is only one 휆, since Equation 4 has only one constraint and that there are |푁| different decentralized Lagrangian functions, one for each SKU. For each value of 휆, the base stock level that minimizes the decentralized Lagrangian function 퐿푛(푆푛, 휆) can be found. Below we show that these decentralized Lagrangian functions can be solved along the same lines as a Newsvendor problem.
푎 The decentralized Lagrangian function is given by 푓푛(푆푛) = 푐푛 푆푛 + 휆퐸퐵푂푛(푆푛, 휋), 푆푛 ∈ ℕ0. 푓푛(푆푛) is 2 decreasing for 푆푛 > 0 if Δ푓푛(푆푛) ≔ 푓푛(푆푛 + 1) − 푓푛(푆푛) ≤ 0 and convex for 푆푛 > 0 if Δ 푓푛(푆푛) ≔ Δ푓푛(푆푛 + 1) − Δ푓푛(푆푛) ≥ 0. By using Equation 18:
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푎 푎 Δ푓푛(푆푛) = 푐푛 (푆푛 + 1) + 휆퐸퐵푂푛(푆푛 + 1, 휋) − 푐푛 (푆푛)
− 휆퐸퐵푂푛(푆푛, 휋) 푎 = 푐푛 + 휆 ∀푆푛 ∈ ℕ0 (21) ∗ [−(푆푛 + 1) + 푆푛
푆푛+1 푆푛 푚푛 푚푛 + ∑ (푠푛 + 1 − 푗)푝푗 − ∑ (푠푛 − 푗)푝푗 ] 푗=0 푗=0 푆푛 푎 푚푛 = −(휆 − 푐푛 ) + 휆 ∑ (푠푛 − 푗)푝푗 푗=0
So, Δ푓푛(푆푛) is increasing on its whole domain and therefore 푓푛(푆푛) convex. Hence, 푓푛(푆푛) is minimized at the first point where Δ푓푛(푆푛) ≥ 0, i.e. at the smallest 푆푛 ∈ ℕ0 for which:
푎 푆푛 휆 − 푐 푚푛 푛 (22) ∑ 푝푗 (휋) ≥ ∀푛 ∈ 푁 푗=0 휆
Note that this ratio is different from the well-known Newsvendor ratio, since in our problem investment ∗ costs are also paid for parts in repair. The resulting base stock level is denoted as 푆푛(휋, 휆). We can do this for all 푛 ∈ 푁, and the returning base stock vector is denoted as 푺∗(휋, 휆).
According to Everett (1963), the Lagrangian relaxation method gives us optimal solutions of Equation 4 for specific values of 퐸퐵푂표푏푗. Everett claims that if, for a given 휆 ≥ 0, 푺(휋, 휆) minimizes 퐿(푺, 휋, 휆), then 푺(휋, 휆) is optimal for Equation 4 for every 퐸퐵푂표푏푗 that satisfies 퐸퐵푂표푏푗 ≥ 퐸퐵푂(푺(휋, 휆), 휋) and 휆 ∗ (퐸퐵푂(푺(휋, 휆), 휋) − 퐸퐵푂표푏푗) = 0. So, starting with 휆 = 0, the resulting base stock level vector 푺(휋, 휆) is optimal for Equation 4 for every 퐸퐵푂표푏푗 ≥ 퐸퐵푂(푺(휋, 휆), 휋). For each 휆 > 0, the solution 푺(휋, 휆) is optimal for Equation 4 for 퐸퐵푂표푏푗 = 퐸퐵푂(푺(휋, 휆), 휋).
So, by finding 푺∗(휋, 휆), we can find the optimal solutions of Equation 4 for specific values of 퐸퐵푂표푏푗, for which we can calculate the corresponding 퐶(푺) and 퐸퐵푂(푺, 휋).
4.2.3 DETERMINATION OF MEAN NUMBER OF BACKORDERS FOR DYNAMIC SCHEDULING RULES For dynamic scheduling rules, we can calculate the mean number of backorders per SKU 푛 with the following formula:
∞ 푚푛 (23) 퐸퐵푂푛(푆푛, 휋) = ∑ (푗 − 푆푛)푝푗 (푺, 휋) ∀푆푛 ∈ ℕ0 푗=푆푛+1
푚푛 In which the probability that 푗 parts of SKU 푛 are present in the repair facility, 푝푗 (푺, 휋), depends on the 푚푛 vector that contains all the base stock levels for all 푛 ∈ 푁, 푺, and scheduling rule 휋. 푝푗 (푺, 휋) depends on 푺, since dynamic scheduling rules take into account the base stock levels of all SKU’s 푛 in order to 푚푛 schedule jobs. Due to computational complexity, we are not able to find 푝푗 (푺, 휋) and the mean number
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of backorders for dynamic scheduling rules analytically, except for sufficiently small problem instances. So we use simulation to evaluate the dynamic scheduling rules.
4.2.4 OPTIMIZATION OF THE BASE STOCK LEVELS FOR DYNAMIC RULES Also in case of dynamic scheduling rules, optimized base stock levels under a certain backorder constraint can be found by using Newsvendor equations. However, in case of dynamic scheduling rules the optimized base stock level of SKU 푛 is the smallest 푆푛 ∈ ℕ0 for which:
푎 푆푛 휆 − 푐 푚푛 푛 ∀푛 ∈ 푁 (24) ∑ 푝푗 (푺, 휋) ≥ 푗=0 휆
Note that Equation 23 is recursive, since it uses the vector that contains the base stock levels of all SKU’s to obtain the optimized base stock level for SKU 푛. Optimized base stock levels can be found by an iterative process that contains the following steps:
Step 1: Start with 푆푛 = 0 ∀푛 ∈ 푁 and 휆 = 0
Step 2: 푚푛 Derive 푝푗 (푺, 휋) for all 푛 ∈ 푁 as explained in Section 4.2.3.
Find the optimized base stock levels for all 푛 ∈ 푁 by using simulation and Equation 24.
Update 푺
Increase 휆 with 1
Repeat Step 2 until the total number of backorders is smaller than the backorder constraint
∗ Step 3: The optimized base stock levels 푆푛(흅) ∀푛 ∈ 푁 are equal to 푺
The layout of this simulation can be found in Appendix C.
4.3 SCHEDULING RULES From the large set of scheduling rules tested in literature (see e.g. Adan et al., 2009; Guide et al., 2000; Tiemessen and van Houtum, 2013), we have selected rules that perform best in previous studies to evaluate in this research. These rules are discussed in Section 4.3.1. Since the repair shop of the RNLAF differs from the repair shops discussed in these studies, we choose to test a number of scheduling rules that are designed especially for the repair shop of the RNLAF. These scheduling rules are not evaluated in literature before and are discussed in Section 4.3.2.
4.3.1 RULES FROM LITERATURE Adan et al. (2009) test heuristics that takes into account the costs and processing times of the SKU’s. They show that costs savings of 40%-60% can be obtained by using a priority setting based on costs or processing times instead of the FCFS rule. Furthermore, Guide et al. (2000) find that scheduling rules that
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take into account processing times outperform other static scheduling rules. Based on the findings of Adan et al. and Guide et al. we choose to evaluate a static priority rule based on costs and a static priority rule based on processing times.
The dynamic scheduling rules selected for this study meet the following criteria: (i) the rule must be applicable to the situation described in Section 4.1 and (ii) the rule must take into account the real time stock levels. Scheduling rules that are taking into account the routing through the shop are left out of scope.
The study of Tiemessen and van Houtum (2013) concludes that the optimal scheduling rule has the following main characters: (i) in case of backorders, the optimal scheduling rule selects the SKU with the shortest average processing time, (ii) the optimal scheduling rule should also see the benefits of short processing times in case of no backorders, and (iii) the optimal scheduling rule takes the stochasticity of the failure process of a part into account. A rule that meets these characters is the Myopic-Allocation Rule proposed by Tiemessen and van Houtum, which schedules the job with the highest backorder reduction per invested time unit first (Section 4.3.1.2.1). Hax and Meal (1975) describe a scheduling rule for a Make-To-Stock (MTS) environment that maximizes the time until the next backorder. In 2010, Tiemessen and van Houtum modify this scheduling rule to apply it on repairable inventory systems. We include this rule, known as the Modified Equalization of Runout Time Rule, in the model (Section 4.3.1.2.2). An example of the behavior of the dynamic scheduling rules is given in Appendix D.
4.3.1.1 STATIC SCHEDULING RULES
4.3.1.1.1 Priority rule based on investment costs Multiple studies suggest that SKU’s with higher costs should be assigned to higher priority classes, since higher priorities lead to lower base stock levels and this significantly lower investment costs (see e.g. Adan et al., 2009 & Buzacott and Shanthikumar, 1993). At the same time, the base stock levels for less expensive SKU’s increase, but that have only a limited effect on the investment costs. For this heuristic, 푎 푎 푎 which is referred to as Heuristic 1, the SKU’s are ordered, such that 푐1 > 푐2 > ⋯ > 푐|푁|. When two SKU’s have the same price, the SKU’s are assigned to two different priority classes in arbitrary order. Note that ties are broken FCFS.
4.3.1.1.2 Priority rule based on processing rates In the study of Guide et al. (2000) the SKU with the shortest processing time has the highest priority. For this heuristic, which is referred to as Heuristic 2, the SKU’s are ordered, such that 휇1 > 휇2 > ⋯ > 휇|푁|. When two SKU’s have the same processing time, the SKU’s are assigned to two different priority classes in arbitrary order. Note that ties are broken FCFS.
4.3.1.2 DYNAMIC SCHEDULING RULES
4.3.1.2.1 Myopic Allocation Rule (MA-Rule) Pena Perez and Zipkin (1997) proposed the Myopic Allocation Rule. The purpose of the MA-Rule is to give the highest priority to the SKU with the highest expected backorder reduction per invested time unit. To calculate the expected backorder reduction per invested time unit, the following notation is used. Let
푇푛 be a non-negative stochastic variable that denotes the look-ahead-time and 퐷푛(푇푛) be the demand of 25
SKU 푛 during time period 푇푛. The cumulative distribution function of 퐷푛(푇푛) is given by 퐹푛(푥), so 퐹푛(푥) = Pr {퐷푛(푇푛) ≤ 푥}. The state of a system is described by 풙 = (푥1, … , 푥푛), the N-dimensional vector of net stock levels. Then, the expected number of backorders of SKU 푛 under stock level 푥푛 at time 푇푛 is equal to:
+ 푔푛(푥푛) = 퐸[퐷푛(푇푛) − 푥푛] ∀푛 ∈ 푁 (25)
The expected backorder reduction at time 푡 = 푇푛 that results from the decision to send one item of SKU 푛 to repair at time 푡 = 0, Δ푔푛(푥푛), can be calculated:
Δ푔푛(푥푛) = 푔푛(푥푛) − 푔푛(푥푛 + 1) ∀푛 ∈ 푁 (26) + + = 퐸[퐷푛(푇푛) − 푥푛] − 퐸[퐷푛(푇푛) − 푥푛 − 1] = 1 − 퐹푛(푥푛)
Veatch and Wein (1996), Pena Perez and Zipkin (1997), and Liang et al. (2013) propose three values for the look-ahead-time 푇푛: (i) the processing time of SKU 푛, (ii) the sojourn time of SKU 푛 while the repair facility is devoted only to SKU 푛, and (iii) the processing time of SKU 푛. Tiemessen and van Houtum (2013) find no differences in performances between the MA-rule that takes into account the sojourn time and the MA-rule that takes into account the processing time. Based on these findings we decide to take the processing time of SKU 푛 as look-ahead-time.
We assume that demand is Poisson and the processing times are exponentially distributed. Therefore, 휆푛 퐷푛(푇푛) is geometrically distributed with parameter 푝푛 = . We use the notation 풙 ≥ 0 to indicate 휆푛+휇푛 that all stock levels in 풙 are non-negative. Further, 훿푛 = 1 when SKU 푛 has an active backorder (i.e. 푥푛 < 0) and 훿푛 = 0 otherwise. Then the function that evaluate the priority for each SKU, also named as the evaluation function, for the MA-rule, becomes:
푥푛+1 휆푛 ∀푛 ∈ 푁 (27) 푀퐴 휇푛 ( ) 푖푓 풙 ≥ 0 퐹 (풙, 푛) = 휇푛Δ푔푛(푥푛) = { 휆 + 휇 푛 푛 표푡ℎ푒푟푤푖푠푒 휇푛훿푛
The SKU that maximizes the evaluation function 퐹푀퐴 gets the highest priority and ties are broken FCFS.
4.3.1.2.2 Modified Equalization of Runout Time Rule (MERT-Rule) The Equalization of Runout Time rule (ERT-rule) aims to maximize the expected time until the next runout and is proposed by Hax and Meal (1975). This rule gives the highest priority to the SKU with the shortest expected time until the next runout. This ensures that all SKU’s run out at approximately the same time. To calculate the expected time until the next backorder, we use the following notation. Let 푄 be the batch run size and 푞푛 be the production quantity for SKU 푛. The ERT-rule calculates 푞푛 for an arbitrary 푄 with the aim to equalize the expected runout times and the individual production quantities run up to 푄. The production quantities that achieve this are calculated by:
푁 푄 + ∑푖=1 푥푖 ∀푛 ∈ 푁 (28) 푞푛 = −푥푛 + 휆푛 푁 ∑푖=1 휆푖
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To apply the ERT-rule on a repairable inventory system, Tiemessen and van Houtum (2013), create the MERT-rule. If there are no active backorders and the total inventory is larger than zero, the production quantities are calculated for a small batch run size 푄 = 휖 using Equation 27. The SKU with the highest production quantity is scheduled first. However, when there are active backorders, the SKU with the shortest average processing time is scheduled first and when the total inventory equals zero, the SKU with the shortest inter-arrival time is scheduled first. Thus, the evaluation function for the MERT-rule becomes:
∑푁 푥 푁 (29) −푥 + 휆 푖=1 푖 푖푓 풙 ≥ 0 푎푛푑 ∑ 푥 > 0 푛 푛 ∑푁 푖 퐹푀퐸푅푇(풙, 푛) = 푖=1 휆푖 푖=1 휆푛 푖푓 풙 = 0
{ 휇푛훿푛 표푡ℎ푒푟푤푖푠푒
The SKU that maximizes the evaluation function 퐹푀퐸푅푇 gets the highest priority and ties are broken FCFS.
4.3.2 RULES DESIGNED FOR THE RNLAF We include a static scheduling rule that takes into account both costs and processing times, to test the combined effect of Heuristic 1 and 2. This rule assigns SKU’s with a higher product of price and processing rate to higher priority classes (Section 4.3.2.1.1).
The dynamic priority rules designed for the RNLAF meet the same criteria as the dynamic scheduling rules that are selected from literature. Currently, the RNLAF uses a Priority Setting (PS) policy. To evaluate how this scheduling rule performs in our integrated model, we include this dynamic scheduling rule (Section 4.3.2.2.1). Since this rule only split the SKU’s into three different priority classes, it might be beneficial to make a further distinguish within these classes. One option is to distinguish within these classes based on the price of a SKU (Section 4.3.2.2.2). We are considering this distinction, since one of the main characteristics of the SKU’s repaired in the repair shop of the RNLAF is their variability in price. Another option to make a distinction between these classes is to take the deviation from the base stock level into account (Section 4.3.2.2.3). We are considering this option, since it further reduces the total number of expected backorders.
Furthermore, the MA-Rule performed best in previous studies (e.g. Tiemessen and van Houtum, 2010). However, this rule does not take into account the price of a SKU. Since our objective function take into account these prices, we suggest to multiply the evaluation function of the MA-Rule with the price of the concerning SKU (see Section 4.3.2.2.4).
4.3.2.1 STATIC SCHEDULING RULES
4.3.2.1.1 Priority rule based on the product of investment costs and processing rates 푎 푎 This heuristic, which is referred to as Heuristic 3, orders the SKU’s such that 푐1 ∗ 휇1 > 푐2 ∗ 휇2 > ⋯ > 푎 푐|푁| ∗ 휇|푁|. When the product of the price and processing rate of two SKU’s is equal, the SKU’s are assigned to two different priority classes in arbitrary order. Note that ties are broken FCFS.
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4.3.2.2 DYNAMIC SCHEDULING RULES
4.3.2.2.1 Current priority setting rule (CPS-rule) The priority setting rule as is applied by 981 SQN, where each priority level has its own virtual queue, is based on the current inventory levels (see Section 3.2).
When the state of a system is described by 풙 = (푥1, … , 푥푛), a SKU is assigned to priority classes according to the following evaluation function:
1 푖푓 푥푛 < 0 ∀푛 ∈ 푁 (30) 퐶푃푆 퐹 (풙, 푛) = {2 푖푓 푥푛 = 0 3 푖푓 푥푛 > 0
The SKU that minimizes the evaluation function 퐹퐶푃푆 gets the highest priority. Note that this is different from the method described in Section 4.3, since in that method the highest priority class gets the highest priority. Furthermore, ties are broken FCFS.
4.3.2.2.2 Modified priority setting rule I (MPS1-rule) The MPS1-Rule aims to further distinguish the priority classes defined in Section 4.3.2.2.1 (the CPS- rule). The difference between the CPS-Rule and the MPSI-Rule, is that the MPSI-Rule breaks ties according to the price of each SKU. The SKU with the highest price is scheduled first. Thus, the evaluation function for the MPS1-Rule is equal to the evaluation function of the CPS-Rule (Equation 30). But, the MPS1-rule breaks ties according to the following evaluations function: 푀푃푆퐼 푎 퐹 (푛) = 푐푛 ∀푛 ∈ 푁 (31)
The SKU that maximizes the evaluation function 퐹푀푃푆퐼 gets the highest priority and if ties still occur, they are broken FCFS.
4.3.2.2.3 Modified priority setting rule II (MPS2-rule) The MPSII-rule is also a modification of the CPS-Rule. The difference between the CPS-Rule and the MPS2-Rule, is that the MPS2-rule breaks ties according to the difference between the real time inventory level and the base stock level. The SKU with the largest deviation from the base stock level is scheduled first. Thus, the evaluation function for the MPS2-rule is equal to the evaluation function of the CPS-rule (Equation 30). But, the MPS2-rule breaks ties according to the following evaluations function:
푀푃푆퐼퐼 퐹 (푛) = 푆푛 − 푥푛 ∀푛 ∈ 푁 (32)
The SKU that maximizes the evaluation function 퐹푀푃푆퐼퐼 gets the highest priority and if ties still occur, they are broken FCFS.
4.3.2.2.4 Modified Myopic Allocation Rule (MMA-Rule) The MA-Rule aims to give the highest priority to the SKU with the highest backorder reduction per invested time unit. However, the MMA-Rule takes into account the price of a SKU by multiplying the backorder reduction per invested time unit with the price of a SKU. Thus, the evaluation function for the MMA-rule becomes:
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푥푛+1 ∀푛 ∈ 푁 (33) 푎 휆푛 푀푀퐴 푐푛 ∗ 휇푛 ( ) 푖푓 풙 ≥ 0 퐹 (풙, 푛) = 휇푛Δ푔푛(푥푛) = { 휆푛 + 휇푛 푎 표푡ℎ푒푟푤푖푠푒 푐푛 ∗ 휇푛훿푛
The SKU that maximizes the evaluation function 퐹푀푀퐴 gets the highest priority and ties are broken FCFS.
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5. CASE STUDY: RNLAF In this section we describe a case study in which the model and scheduling rules as described in Section 4 are applied. This case study makes use of real life data of the RNLAF and serves as an example from practice. In Section 5.1 we explain how the proposed model is applied to the case study. Next, in Section 5.2 we describe how the data sample is selected. Furthermore, in Section 5.3 we sum up the made assumptions. Then, in Section 5.4 we explain how the scheduling rules are compared to the current situation. In Section 5.5 we describe how the simulations are set up. Then in Section 5.6 we present the results of the case study and finally, in Section 5.7, the simulation models are validated.
5.1 APPLICATION OF THE PROPOSED MODEL The set-up of the repair shop at the RNLAF is different from the repair shop described in Section 4.1, since the repair shop at the RNLAF consists of multiple workstations and variable routings, while the repair shop described in Section 4.1 has a single location with one workstation and one stockpoint. To model the repair shop at the RNLAF as a repair shop with a single location with one workstation and one stockpoint, the repair shop at the RNLAF is simulated. Real life data of the RNLAF on arrival rates, routings, processing times, costs and capacity are used for this simulation. The outcome of this simulation is for each priority rule the estimated flowtime of a part through the shop, the mean utilization of each workstation and the mean number of SKU’s handled at each workstation. We use the mean utilization of each workstation and the mean number of SKU’s handled at each workstation to calculate the weighted average utilization of the whole repair shop. Furthermore we use the weighted average utilization, the arrival rates of the SKU’s and the ratio of the estimated flow times of parts through the shop to calculate the adapted flow times through the shop. In other words, we choose the adapted flow times, 퐴퐹푇, such 푁 that the weighted average utilization is equal to ∑푥=푛 휆푥퐴퐹푇푥 and that the ratio between the estimated 퐴퐹푇 퐴퐹푇 flow times though the shop, 퐹푇, is equal to the ratio between the adapted flow times: 1 = 2 = ⋯ = 퐹푇1 퐹푇2 퐴퐹푇 1 푁. As a result, we use as the mean processing time of SKU 푛 in the single server model as 퐹푇푁 퐴퐹푇푛 described in Section 4.1. Note that the mean processing times of each SKU can be different for each priority rule.
Note that we use in total two simulations for the case study: one to get the mean processing times used in the single server model, denoted as Simulation 1, and one to obtain the expected number of backorders for dynamic scheduling rules as explained in Section 4.2.3, denoted as Simulation 2.
5.2 DATA SAMPLE To create a data set for the case study, we use real data of incoming orders at 981 SQN of the last 15 years. From these orders we use the following data: arrival date, part number, routing and actual processing time at each workstation on the routing.
After having several discussions with multiple stakeholders of 981 SQN we choose to use the following criteria to create the data set used in the case study:
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- We include only SKU’s that have a routing which starts at the workstation WMS Systems the sample. This criterion is set up since the RNLAF is most interested in a solution for these parts. - We include only SKU’s that appear 100 times or more in the available data set in the sample. This criterion is set up since it is most interesting to find a solution for parts that are frequently used. Furthermore, the more data on processing times, arrival dates and possible routings are available, the more accurate the predictions shall be. - We remove routings that appear less than 10 times in the sample created by criteria 1 and 2.
According to these criteria we create a sample that consist of 10 SKU’s with in total 50 possible routings which passes 8 different workstations. Note that this sample is not representative for the whole data set, since parts with a low demand are excluded.
We use data of incoming orders at 981 SQN of the last 15 years to obtain the arrival rates, probabilities of a possible routing, and the distribution of the time spend at each workstation for each SKU. Furthermore, we use historical data on the available capacity of the workstations evaluated in the case study. This data is used as input for Simulation 1. The total input used for the case study can be found in Appendix E.
5.3 ASSUMPTIONS According to several stakeholders inside 981 SQN the data on routings and processing times is not reliable. This is mostly since a prognosis is made of the routing through the repair shop and the processing times at each workstation before a component enters the shop. When an employee decides to change the routing of a component during repair, this is not always recorded in the ERP system. Furthermore, after performing a task, each workstation is obligated to record the actual processing time in the ERP system. However, often the expected processing time is copied as the actual processing time, while the actual processing time is different in practice. These two indications suggest that there are some mistakes in the data set. After discussions with the stakeholders we assume that the data is right, since it cannot be traced which routings and actual processing times are correct and which are not.
At workstations IMO Chemical Cleaning and IMO Non Destructive Maintenance (NDO) it is possible to batch orders and to process them parallel. Unfortunately, there is no data on batch sizes available. Based on interviews with technicians working at these workstations, we decided to set the average batch size at IMO Chemical Cleaning and IMO NDO on 4 and 2 respectively. Furthermore, at all workstations orders are batched sometimes to process them serially. Since there is no data available on the time gain from serial batching, we ignore serial batching.
5.4 COMPARISON OF THE SCHEDULING RULE At this moment, the spare part inventory decisions are mainly based on the experience of the employees at PM, instead of on a pre-specified spare part management methodology. Besides that, there are no records available on the base stock levels of the sample during the last 15 years. This makes it impossible to compare the integrated model with the current unintegrated situation. Therefore, we use the CPS-Rule and the optimized base stock levels according to the CPS-Rule as the base line situation. Based on the results of Adan et al. (2009) and Tiemessen and van Houtum (2010), we assume that the total costs of jointly optimizing base stock levels and priority rules, are lower than or equal to the total costs of separately optimizing the base stock levels and priority rules for each priority rule tested. According to the 31
assumption, when in the proposed model an arbitrary priority rule performs better than the CPS-Rule, this arbitrary rule performs better than the current situation.
5.5 SIMULATION In this section we describe how the simulations as described in Section 4.2.3 and Section 5.1 are set up.
5.5.1 SIMULATION 1 Simulation 1 is performed in the discrete event simulation software package Arena Enterprise Suite Academic.
To reduce the influence of variety, we choose to run the model for 780 days of 8 hours (after a warm-up period of 260 days of 8 hours). A year consists of 52 weeks and each week consists of five workdays (52*5=260 days a year). Therefore we execute the model for 3 years after a warming up period of 1 year. Furthermore, for each priority rule we conduct 200 replications.
5.5.2 SIMULATION 2 Simulation 2 is performed in Java Eclipse. To decide on the number of replications, we apply the non- overlapping batch means method. For more details on this method we refer to Steiger and Wilson (2001). For each run, we simulate the same 550,000 part failures. We divide each run is divided into 11 sub-runs of 50,000 failure events. The first sub-run is used as warming up period and for the other 10 sub-runs we calculate the average number of backorders and the average annual costs. From these 10 samples the sample mean and standard error of the average annual costs are calculated. When the standard error is lower than 1% of the sample mean, we terminate the simulation. Otherwise, we simulate another 10 sub- runs and merge them into 10 new samples by merging sub-run 1 with 2, 3 with 4,…, and 9 with 10. We repeat this method until the standard error is lower than 1% of the sample mean.
We simulate the same part failures for each tested priority rule. However, note that in the case study the processing rates differ for each priority rule, therefore the processing times for each (similar) part failure differs for each priority rule.
5.6 RESULTS In Table 1 we present the average reduction in costs (%) compared to the CPS-Rule for each scheduling rule under a specific backorder constraint. Furthermore, in Figure 8, the optimal investments costs for each scheduling rule are plotted versus the expected total backorders. The markers in the figure represent these optimal solutions, which are connected with a smooth line. Note that due to the discrete base stock levels, the optimal solutions are discrete. Therefore, only the solutions visualized by a marker are possible. Furthermore, due to the discrete base stock levels, the deviation from the 퐸퐵푂표푏푗 is different for each scenario and scheduling rule. This partially explains the fluctuating percentages in Table 1. As can be seen from Figure 8 and Table 1, the differences in performance of Heuristic 1, Heuristic 3, the MA- Rule, the MERT-Rule, the MPS1-Rule and the MMA-Rule are relatively small. Therefore, we zoom in on the investment costs under different backorder constraints in Figure 9. We refer to Appendix F for more details on the optimal solutions
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4
3,5 Heuristic 1 3 Heuristic 2
2,5 Heuristic3
2 MA-Rule
EBO(S) MERT-Rule 1,5 CPS-Rule 1 MPS1-Rule
0,5 MPS2-Rule MMA-Rule 0 0 200 400 600 800 1000 C(S) (x €1000)
Figure 8: Optimal solution under different backorder constraints
2 1,8 1,6 Heuristic 1 1,4 Heuristic 2 Heuristic3
1,2
1 MA-Rule EBO(S) 0,8 MERT-Rule CPS-Rule 0,6 MPS1-Rule 0,4 MPS2-Rule 0,2 MMA-Rule 0 10 30 50 70 90 110 130 C(S) (x €1000)
Figure 9: Zoomed in: optimal solution under different backorder constraints
Table 1 shows that, for each evaluated backorder constraint, the costs of inventory can be reduced with at least 54,8% compared to the CPS-Rule. As can be seen from Table 1, Figure 8 and Figure 9, the MPS1- Rule is the best performing scheduling rule under all backordering constraints, followed by Heuristic 1,
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Heuristic 3, the MA-Rule, the MERT-Rule and the MMA-Rule. An explanation for the strong performances of Heuristic 1, Heuristic 3, the MPS1-Rule, and the MMA-Rule is that these are the only rules that take into account the price on a spare part. By scheduling SKU’s with high prices first, lower inventory levels are needed for these SKU’s. Since prices of SKU’s in this case study vary from €2134,87 to €258.824, −, it is beneficial to reduce inventory levels of SKU’s with high prices. A possible reason for the strong performance of the MA-Rule and the MERT-Rule is that these are the only scheduling rules that take into account both the real time inventory levels, the expected arrival rates and the expected processing times of the SKU’s. Furthermore, the MA-Rule performs slightly better than the MERT-Rule. This can be explained, since the MA-Rule takes into account the stochasticity of the part failure process, while the MERT-Rule does not. Next, the MPS2-Rule performs only slightly better than the CPS-Rule (see Figure 8 and Figure 9). A possible reason for this result is that in this case study the more expensive SKU’s have the lowest arrival rates (see Appendix E). Therefore the probability that less expensive SKU’s are having a bigger deviation from the base stock levels is higher and the probability is higher that less expensive SKU’s are scheduled first. Furthermore, from Table 1, Figure 8 and Figure 9 we see that Heuristic 2 is by far the worst performing scheduling rule in this case study. This is partially caused by the fact that in this case study the SKU’s with the highest prices have the longest processing times and are therefore scheduled latest at each work station (see Appendix E). As a result, higher inventory levels are needed for the parts with the longest processing times and highest prices. This causes high investment costs and therefore a bad performance. Interestingly to mention is that having relatively long routings and processing times at the workstations for high valued SKU’s is typical for the RNLAF and thus not a coincidence in this case study.
Table 1: Average reduction in costs (%) compared to the CPS-Rule for each scheduling rule under a specific backorder constraint
푬푩푶풐풃풋 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule 4 100,0% 100,0% 100,0% 100,0% 100,0% 100,0% 25,9% 100,0% 3,5 100,0% 33,1% 100,0% 100,0% 100,0% 100,0% 15,3% 100,0% 3 85,8% -611,4% 79,5% 85,8% 85,8% 85,8% -14,3% 85,8% 2,5 89,9% -810,0% 87,8% 89,9% 89,9% 89,9% 52,2% 89,9% 2 82,6% -687,9% 78,9% 86,6% 86,6% 87,2% 48,1% 86,6% 1,75 84,0% -794,1% 80,6% 84,0% 84,0% 84,0% 3,2% 84,0% 1,5 81,8% -922,4% 78,0% 84,6% 84,6% 84,6% 5,7% 84,6% 1,25 84,4% -645,4% 79,5% 86,4% 84,0% 86,4% 29,2% 84,0% 1 81,2% -726,0% 75,5% 80,6% 80,6% 82,9% -1,5% 80,6% 0,8 79,8% -733,6% 75,1% 77,6% 77,6% 82,9% 18,5% 80,9% 0,6 61,7% -639,7% 60,2% 66,5% 65,1% 79,9% 14,7% 63,7% 0,4 75,3% -452,3% 69,6% 78,0% 77,2% 78,9% 17,9% 77,2% 0,2 63,9% -499,0% 54,5% 66,8% 66,1% 66,8% 14,9% 65,9% 0,1 52,9% -388,3% 52,1% 54,6% 54,2% 54,8% -12,6% 53,8% Average 78,72% -555,50% 76,52% 81,53% 81,12% 83,15% 15,51% 81,21%
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5.7 MODEL VALIDATION AND VERIFICATION To check if we can indeed apply the simulation models to the situation at the RNLAF, we validate and verify the simulation models. Eddy et al. (2012) describe multiple types of validation that are commonly used in literature. Two of these types are face validity, which we use to validate Simulation 1 (see Section 5.7.1) and cross validation, which we use to validate Simulation 2 (see Section 5.7.2).
5.7.1 SIMULATION 1 Face validity is the extent to which a model corresponds to the real world, as judged by experts. We interviewed employees of the operations office and the workstations in the repair shop to denote which workstations are a bottleneck according to their experience. All interviewed employees indicated that the workstation WMS Systems (Workstation 8) has the highest utilization. The output of Simulation 1 confirms these statements.
Verification examines the extent to which the mathematical calculations are performed correctly and are consistent with the models’ specifications. We verify the simulation models by using extreme input values. For all evaluated scheduling rules, we test six scenarios of extreme values:
- Extremely high arrival rates - Extremely low arrival rates - Extremely high capacity at the workstations - Extremely low capacity at the workstations - Extremely high processing rates - Extremely low processing rates
When we use extremely high arrival rates, extremely low capacity at the work stations, or extremely low processing rates as input values, the utilization of one or more of the workstations exceeds one and the weighted average of the utilization over all workstation is close to one. Furthermore, when we use extremely low arrival rate, extremely high capacity at the workstation or extremely high processing rates as input values the utilizations at the workstations decrease to close to zero. Therefore, according to these findings it is verified that Simulation 1 behaves according to expectations.
5.7.2 SIMULATION 2 Cross validation compares the output of the model to the output of models that addresses the same problem. Therefore, we run Simulation 2 with Heuristic 1, Heuristic 2 and Heuristic 3 as scheduling rule. As expected, the difference between the output of these simulations and the output of the analytical model with Heuristic 1, Heuristic 2 and Heuristic 3 as scheduling rule is smaller than 휀, where 휀 is 10−3.
To verify the Simulation 2 we test the model with extreme input values, as is done by verifying Simulation 1. For all evaluated scheduling rules, we test four scenarios of extreme values:
- Extremely high arrival rates - Extremely low arrival rates - Extremely high processing rates - Extremely low processing rates 35
As expected, the expected work in process becomes extremely high when we use extremely high arrival rates or extremely low processing rates as input values. Furthermore, the expected work in process becomes extremely low when we use extremely low arrival rates or extremely high processing rates as input.
Therefore, according to the findings from verification and validation, we conclude that Simulation 2 behaves according to expectations.
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6. NUMERICAL ANALYSIS In this section we provide further insight on the performance of the scheduling rules in our integrated model. The case study discussed in Section 5 already gives insights for a predefined sample of SKU’s of the RNLAF. However, there exist many types of spare parts with different combinations of input parameters. In this section, we discuss the performance of the integrated model for multiple combinations of input parameters. This section starts with a description of the input parameters (Section 6.1) and after that we present the results in Section 6.2.
6.1 INPUT PARAMETERS To examine the performance of the integrated model, we use a factorial design for our test bed that consists of a wide range of problem instances of real-life size. To create our test bed, we choose to consider different scenarios for the following input parameters: the number of SKU’s, the utilization, the mean arrival rate, the mean processing rate, and price of a SKU. We choose to consider 5, 10 and 50 SKU’s in our test bed. Furthermore, we consider three scenarios for the arrival rate. Note that the arrival rate can vary for different SKU’s within a scenario. Therefore, the arrival rate in our test bed is equal to a percentage of a fixed value. We choose to consider a scenario with low, high and highly varying arrival rates. For all of these scenarios, |푁| values are picked, which are uniformly distributed on the range [1.0,1.1] for low arrival rates, [20.0,20.1] for high arrival rates and on the range [1.0,20.1] for highly varying arrival rates. These values are given by 풇푙표푤, 풇ℎ푖𝑔ℎ, and 풇푣푎푟푖푎푏푙푒, respectively and are named as the vector of factors. Note that |푁| is equal to the number of SKU’s considered in the scenario. To obtain the arrival rates for each SKU in each scenario the fixed arrival rate value, which is equal to 10, is multiplied by a random chosen value out of 풇푙표푤 for the scenario with low arrival rates, 풇ℎ푖𝑔ℎ for the scenario with high arrival rates and 풇푣푎푟푖푎푏푙푒 for the scenario with highly varying arrival rates. Note that a value from the arrival rate vector can only be chosen once, so there are no SKU’s in a sample with the same arrival rate. Next, to obtain a stable system, we choose the mean processing time, based on two scenarios for the utilization of the system, namely 0.6 and 0.95, respectively. Then, we consider two scenarios to calculate the mean processing rate, namely low varying processing rates and highly varying processing rates. We obtain these low varying and highly varying processing rates by a random chosen value out of 풇푙표푤 and 풇푣푎푟푖푎푏푙푒 respectively. Then the processing times are multiplied by a fixed value that is different for low varying and highly varying processing rates. We choose this fixed value based on the utilization and arrival rates in the evaluated scenario. Finally, we consider three scenarios for the prices of SKU’s, namely low, high and highly varying prices. These prices are obtained by the same method as the arrival rates, with the same vector of factors and with a fixed price value equal to 1000.
In total, we test 3 ∗ 3 ∗ 2 ∗ 2 ∗ 3 = 108 unique scenarios. For these 108 unique scenarios the optimal total costs are calculated for multiple backorders constraints, namely {10; 9; 8; 7; 6; 5; 4; 3.5; 3; 2.5; 2, 1.5; 1.25; 1}
Furthermore we test all scenarios five times, to reduce the effects on output of randomly chosen input values out of 풇. So, in total we test 108 ∗ 5 = 540 instances. Furthermore, we use the same randomly picked values out of 풇푙표푤, 풇ℎ푖𝑔ℎ, and 풇푣푎푟푖푎푏푙푒, for each scenario, i.e. we assign for each scenario which is in the same replication the same input values to the same SKU. 37
To decide on the number of replications we apply the non-overlapping batch means method, which is described in Section 5.5.2.
The values of the input parameters that we use for the test bed are presented in Table 2. Table 2: Parameter values for the test bed Variable Description Values |푵| Number of SKU’s {5; 10; 50} 풇풍풐풘 Low vector of factors A vector of length |푁| with values that are equally distributed on range [1.0,1.1] 풇풉풊품풉 High vector of factors A vector of length |푁| with values that are equally distributed on range [20.0,20.1] 풇풗풂풓풊풂풃풍풆 Highly varying vector A vector of length |푁| with values that are equally distributed on of factors range [1.0,20.1]
풄풏 Price {1000 ∗ 푅퐴푁퐷(풇푙표푤); 1000 ∗ 푅퐴푁퐷(풇ℎ푖𝑔ℎ); 1000 ∗ ∗ 푅퐴푁퐷(풇푣푎푟푖푎푏푙푒)} Arrival rate ∗ 흀풏 {10 ∗ 푅퐴푁퐷(풇푙표푤); 10 ∗ 푅퐴푁퐷(풇ℎ푖𝑔ℎ); 10 ∗ 푅퐴푁퐷(풇푣푎푟푖푎푏푙푒)} 흆 Utilization rate {0.6; 0.95} ∗∗ 흁풏 Mean processing rate {푥 ∗ 푅퐴푁퐷(풇푙표푤); 푥 ∗ 푅퐴푁퐷(풇푣푎푟푖푎푏푙푒) } 푬푩푶풐풃풋 Backorder constraint {10; 9; 8; 7; 6; 5; 4; 3.5; 3; 2.5; 2, 1.5; 1.25; 1}
*RAND(x) denotes a random number out of array x **In which x depends on the arrival rate of the SKU and the utilization of the scenario
6.2 RESULTS In, Table 3 we show the average costs reduction compared to the CPS-Rule over all 540 problem instances. We see that over all cases all static and dynamic rules tested except Heuristic 1and the MPS1- Rule outperform the CPS-Rule. According to literature, dynamic scheduling rules that take into account the real time stock levels outperform dynamic scheduling rules in most cases (Hausman and Scudder, 1982; Scudder, 1984; Tiemessen and van Houtum, 2013; Liang et al., 2013). Therefore, it seems counterintuitive that the static scheduling rules Heuristic 2 and Heuristic 3 outperform the dynamic CPS- Rule. However, these studies do not included a dynamic scheduling rule similar to the CPS-Rule.
An interesting observation is that over all tested cases the MA-Rule, the MERT-Rule, and the MMA-Rule are the best performing rules, with a reduction in investment costs of 44,2%, 42,3%, and 41,7%, respectively. An explanation for the strong performances of these rules is that these are the rules that take into account characteristics that should be included according to literature, namely real time stock levels, processing times, and the failure process (e.g. Hausman and Scudder, 1982; Scudder, 1984; Tiemessen and van Houtum, 2013; Liang et al., 2013).
Table 3: Average reduction in costs (%) compared to the CPS-Rule over all scenarios
Heuristic 1 Heuristic 2 Heuristic 3 MA-Rule MERT-Rule MPS1-Rule MPS2-Rule MMA-Rule -101,4% 8,5% 11,2% 44,2% 42,3% -11,5% 3,4% 41,7%
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To find the exact effect of the input parameters on the performances of the scheduling rules, we analyze how de costs reduction depends on five out of six factors of our test bed, namely the number of SKU’s, the utilization, the mean arrival rate, the mean processing rate, and price of a SKU. These results are shown in Table 4a-e.
Table 4a shows that when we increase the number of SKU’s in the system, the difference in performance with the CPS-Rule increases for all scheduling rules. This can be explained since a low number of SKU’s in the system (|푁| = 3), leads to a higher probability that there are only parts of one SKU in the queue. This causes that the choice of which SKU is put into repair is no longer dependent on the applied scheduling rule. The strength of this effect decreases as the number of SKU’s grows. Table 4b shows that the costs reduction of all scheduling rules is relatively stable when the arrival rate increases or is highly varying between SKU’s. In Table 4c we see that the difference in performance with the CPS-Rule increases for all scheduling rules when the utilization increases. The reason that for a low utilization rate the scheduling rules perform more equal than for a high utilization rate, is that for a low utilization rate the total Work In Process (WIP) is lower and thus, in general, the queue is shorter. Therefore, it happens more often that there is only one SKU in the queue, which causes that the choice of which SKU is put into repair is no longer dependent on the applied scheduling rule. In Table 4d we see that the performances of Heuristic 1, the MPS1-Rule and the MPS2-Rule decrease extremely when processing rates are highly varying. An explanation for these large reductions is that these are the only rules that do not take into account the mean processing times of a SKU by making scheduling decisions. As a result, it is possible that a SKU with a high mean processing time gets the highest priority. Giving higher priorities to SKU’s with high processing times decreases the backorder reduction of the server per time unit. The size of this reduction depends on the variation in processing rate amongst the SKU’s. Since backorders of all SKU have the same weight, it is more beneficial to first schedule a SKU with a short processing time. For this reason, the performance of rules that do take into account processing time while scheduling orders increases when processing rates are highly varying. Finally, in Table 4e we see that when prices of SKU’s are more varying, the relative performances of Heuristic 1 and the MPS1-Rule compared to the CPS-Rule increases extremely. This can be explained, since Heuristic 1 and the MPS1-Rule take into account the price of a SKU by making scheduling decisions. Higher priorities for expensive SKU’s lead to lower base stock levels for these SKU’s. When the difference in price amongst SKU’s is higher, the effect on total investment costs shall also be higher.
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Table 4: Average reduction in costs (%) compared to the CPS-Rule (a) Costs reduction as function of 푵 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule 3 -89,3% 7,5% 10,3% 40,1% 38,3% -10,4% 3,1% 38,5% 10 -106,6% 8,9% 11,6% 45,4% 43,9% -12,0% 3,6% 42,6% 50 -108,2% 9,1% 11,6% 47,0% 44,7% -12,2% 3,7% 43,9%
(b) Costs reduction as function of 흀 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule Low -102,2% 8,2% 10,9% 42,8% 41,1% -11,2% 2,6% 40,6% High -93,9% 7,0% 11,2% 43,9% 41,6% -10,5% 2,9% 42,8% Highly varying -108,0% 10,3% 11,4% 46,0% 44,2% -12,9% 4,7% 41,6%
(c) Costs reduction as function of 흆 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule 0.6 -62,3% 2,0% 5,2% 38,0% 36,3% -7,9% 0,5% 35,3% 0.95 -140,4% 14,9% 17,1% 50,4% 48,3% -15,2% 6,3% 48,1%
(d) Costs reduction as function of 흁 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule Low varying -6,0% -58,6% 0,9% 9,4% 8,9% -1,2% 11,0% 7,9% Highly varying -196,7% 75,5% 21,5% 79,0% 75,7% -21,8% -4,1% 75,4%
(e) Costs reduction as function of 풄 Heuristic Heuristic Heuristic MA-Rule MERT- MPS1- MPS2- MMA- 1 2 3 Rule Rule Rule Rule Low -152,1% 37,2% 11,4% 43,1% 41,4% -24,4% 3,2% 41,5% High -145,7% 37,5% 11,3% 43,4% 41,6% -22,7% 3,2% 41,0% Highly -6,3% -49,3% 10,8% 46,1% 43,9% 12,5% 4,0% 42,6% varying
Furthermore, we analyze how the costs reduction depends on the sixth factor in our test bed, namely the backorder constraint. This result is shown in Figure 10. An interesting observation is that the performance differences almost disappear when the backorder constraint is high. An explanation for this observation is that the higher the backorder constraint, the more scenarios there are in which it is not necessary to invest in spare parts for all scheduling rules. Therefore, for higher backorder constraints, less investment decisions are made, which reduces the differences in performance of the scheduling rules. On the other hand, the lower the backorder constraint, the more investment decisions are made, which increases the differences in performance of the scheduling rules.
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150% -
100% Heuristic 1 Heuristic 2
50% Heuristic 3
MA-Rule Rule MERT-Rule 0% 0 2 4 6 8 10 12 MPS1-Rule MPS2-Rule -50%
MMA-Rule Improvementcosts in compared thetoCPS
-100% EBOobj
Figure 10: Average reduction in costs (%) compared to the CPS-Rule
Overall, note from Table 4 and Figure 10 that the MA-Rule consistently outperforms the other scheduling rules. Therefore we summarize how the structure of the MA-Rule differs from the other evaluated scheduling rules. Below we discuss the main findings:
- The MA-Rule takes into account the stochasticity of the arrival process. This research shows rules that take into account the stochasticity of the arrival rates (the MA-Rule and the MMA- Rule) perform equally or better than other scheduling rules that does not take into account this stochasticity. - The MA-Rule takes into account the average processing time of a component. This research shows that in case of highly varying processing times for each SKU, rules that take into account the processing times (Heuristic 2, Heuristic 3, the MA-Rule, the MERT-Rule, and the MMA- Rule) outperform all other scheduling rules. - The MA-Rule takes into account real-time inventory levels. This research concludes that rules that take into account real-time inventory levels perform more stable when input parameters change.
Finally, from this research we indicate that when the prices of SKU’s are more varying than tested in this numerical analysis, the performances of Heuristic 1 and the MPS1-Rule are further improved. This indication is based on as well the numerical analyses as the case study performed for the RNLAF.
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7. IMPLEMENTATION In this section we describe the actions that should be taken by the RNLAF to benefit fully from the results of this research.
The case study concludes that changing the currently used scheduling rule helps the RNLAF in their goal of minimizing the flow times in the repair shop and reducing the total investment costs. According to the case study, which is based on 10 selected SKU’s, the costs of inventory can be reduced with at least 54,8% compared to the CPS-Rule. However, while reading this research, it should be taken into account that the CPS-Rule differs from the current situation at the RNLAF, since currently PM does not optimize their base stock levels based on the scheduling rule applied at 981 SQN. Furthermore, the currently used dispatching methodology at 981 SQN is based on the experience of employees. Although the decisions of the employees come closest to the CPS-Rule, the current dispatching methodology cannot be framed in a formal rule.
Nevertheless, we recommend to the RNLAF to check to which extent the sample used in the case study is representative for the whole data set. When the prices of SKU’s in the whole data set are more or equally varying as the prices in the case study, we recommend to implement the MPS1-Rule in their planning and scheduling methodology. But when the prices of SKU’s in the whole data are less varying than the prices in the case study, we recommend to the RNLAF to apply the MA-Rule. We recommend this, since the performances of the MA-Rule are more stable than the performances of the MPS1-Rule when input parameters change. However, currently the RNLAF does not have a planning tool that automatically schedules orders in the system. Instead, orders are assigned to a priority classes and then manually placed in the correct sequence. When an order needs to be rescheduled, the start times of all jobs at all workstations are manually adjusted. Rescheduling an order happens for example when a job at a work station takes longer or shorter than expected or when an order with a higher priority enters the system. The currently used scheduling methodology is based on three different priority classes, which makes it possible for the planners at the operations office to put orders in the correct sequence. However, it is time consuming and almost impossible for the planners to keep the start times of all jobs at all workstations up to date. As a result, it is challenging to get insight the status of the orders and work stations in the repair shop. Therefore, when orders are scheduled based on a more complex scheduling rule, the RNLAF needs an Advanced Planning System (APS). This also helps them to get insight into the status of the orders and work stations in the repair shop.
An APS offers the functionality to support planning or scheduling and is characterized by having at least the following elements (Wiers & the Kok, 2016):
- A Graphical User Interface (GUI) that shows the allocation of tasks to resources in time; - A model of a physical problem that needs to be planned or scheduled. This model allocates capacity demand to supply (e.g. machines, operators) in time; - An immediate transaction propagation engine that calculates the results of planning actions or imported data; Wiers and de Kok identified another typical characteristic of an APS, but which is not regarded as necessary to classify a system as an APS:
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- An algorithm which is used to generate plans and schedules;
Since this research is followed by a PDEng project which aims to result in a plan tool for 981 SQN, this APS is built during that project.
Furthermore, we set up this model as general as possible, so it can be applied on multiple types of repair shops. For this purpose, we assume that each backorder has the same weight. However, at the RNLAF, in practice the weight of a backorder depends on the mission of the weapon systems, which makes it variable in time. Since it is challenging to include this in a model, we recommend the human planner to include this in the planning. This means that the scheduling rule used at the RNLAF supports humans by making the planning decisions instead of replacing the human planner.
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8. CONCLUSIONS AND RECOMMENDATIONS In this chapter we summarize the conclusions of this research (see Section 8.1). However, limitations should be taken into account by interpreting these conclusions. Therefore, these limitations and several recommendations for the RNLAF and for further research are given in Section 8.2.
8.1 CONCLUSIONS The primary objective of this research is to identify scheduling rules and to study their effect in repairable inventory systems. Furthermore, we aim to optimize the base stock levels based on the optimal scheduling rule, i.e. we jointly optimize the base stock levels and the real-time scheduling decisions. This objective is translated into a main research question and three sub-questions, which are formulated in Section 2.3. In this section, we discuss the main conclusions of each research question, starting with the first sub- question:
‘’What is the currently used spare parts management methodology at 981 Squadron and what are the relevant limitations of this methodology?’’
The repairable spare part inventory and scheduling methodology that is currently used at the RNLAF is mainly based on the experiences of the employees, instead of on a pre-specified spare part management methodology. According to these experiences and expected flight hours of the weapon systems the employees of PM decide on the optimized base stock levels. Furthermore, in general the currently applied scheduling methodology comes closest to a Priority Setting policy. The RNLAF uses three priority classes, with in the first class SKU’s with an active backorder, in the second class SKU’s with a net stock level of zero and in the third class SKU’s with a positive inventory level. Note that jobs in the lowest priority class are scheduled first and ties are broken according to the FCFS rule. Since the currently used scheduling rule only takes into account the current stock levels and no other potentially relevant parameters, we expect that other scheduling rules can improve the current performances. Furthermore, it is expected that optimizing the base stock levels according to a scheduling rule improves the performances of the total supply chain.
The next sub-question which we answer in this research is:
‘’How can base stock levels and real-time scheduling decisions be jointly optimized?’’
We develop an analytic model which jointly optimizes base stock levels and static scheduling decisions. This model is heavily based on the methods of Sleptchenko et al. (2015) and Adan et al. (2009). We combine these methods to be applicable to systems with SKU’s with different processing times for each priority class. The analytic model considers an M/M/1 queueing system with preemptive priorities. We use the exact method of Sleptchenko et al. to find the steady state joint queue length distribution under a given assignment of SKU’s to unique priority classes. Then, optimized base stock levels for each backorder constraint follow from Lagrangian relaxation and Newsvendor equations.
It is not possible to create an exact method for dynamic scheduling rules, due to computational complexities. Therefore, we create a simulation model to jointly optimize base stock levels and dynamic scheduling decisions. This simulation model also considers an M/M/1 queueing system with preemptive 44
priorities. The simulation model is used to find the average distribution of the joint queue length under a specific dynamic scheduling rule and specific base stock levels. Then, through an iterative process, optimized base stock levels for each backorder constraint follow from this distribution, Lagrangian relaxation and Newsvendor equations.
Since we set up both models as systems with one server and a single stock point, we are using a second simulation to model the repair shop at the RNLAF as a M/M/1 queueing system. In this simulation the actual repair shop at the RNLAF is modeled and we use this simulation to decide on the weighted average utilization of the workstations and the flow times of an order through the shop. These findings are used to find the processing rates, which are used as input of both M/M/1 models.
The last sub-question which we answer in this research is:
’What spare part management methodology optimizes the performances of a repair shop and what are their performances and limitations?’’
We evaluated scheduling rules that performed best in previous studies. However, the repair shop described in this research differs from the repair shops discussed in these studies. Therefore, we choose to test scheduling rules that we designed especially for this type of repair shop. According to our case study, the best performing scheduling rule for the repair shop of the RNLAF is the MPS1-Rule. This rule reduces for each evaluated scenario at the RNLAF, the costs of inventory compared to the currently used scheduling rule with at least 54,8%. However, from the numerical analysis we conclude that the performance of the MPS1-Rule is not stable when input parameters vary. Furthermore, we conclude that this scheduling rule only has high performances when prices of SKU’s are extremely varying. Next, we find that the MA-Rule optimizes the performance of a repair shop over almost all tested cases. In contrast to the MPS1-Rule, the performances of the MA-Rule are stable when input parameters vary.
This research shows that the optimal scheduling rule has the following main characteristics:
- Takes into account the stochasticity of the arrival process. - Takes into account the average processing time of a SKU. - Takes into account the real-time stock levels of SKU’s.
The MA-Rule and the MMA-Rule are the only scheduling rules tested that meets all these characteristics.
These answers on the sub-questions together give an answer on the main research question.
‘’How can the spare parts management methodology of a repair shop be improved to increase the relevant performances?’’
8.2 LIMITATIONS AND RECOMMENDATIONS While reading this research, some limitations should be taken into account. Therefore we set up recommendations for as well the RNLAF (Section 8.2.1) as for further research (Section 8.2.2) to take away these limitations.
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8.2.1 LIMITATIONS AND RECOMMENDATIONS FOR THE RNLAF We based the input of the case study on historical data of incoming orders at 981 SQN of the last 15 years. From these orders we use the arrival date, part number, routing and actual processing time at each workstation on the routing. However, according to several stakeholders inside 981 SQN, the data on routings and processing times is not reliable. The reason for this is that often, the data on actual processing times and actual routings is not properly stored in the ERP system. Since the planning is based on this data, we recommend to the RNLAF to do further research to the accuracy of the data and the possibilities to improve the accuracy of the data.
Next, the sample we use for the case study consists of 10 SKU’s with in total 50 possible routings which passes 8 different workstations. This sample is not representative for the whole data set, since we excluded parts with a low demand. Therefore we recommend to the RNLAF to check to which extent the sample used in the case study is representative for the whole data set. When the characteristics of the spare parts in the whole data set differs from the scenarios tested in the case study or numerical analysis, we recommend to test the model with input parameters that represents the whole data set of the RNLAF.
Finally, we assume that inventory space and inventory budgets are infinite. However, it should be checked whether the warehouse of PM has enough space to stock the SKU’s suggested by the model.
8.2.2 LIMITATION AND RECOMMENDATIONS FOR FURTHER RESEARCH The model presented in this research gives some interesting insights. However, due to limitations of the model there are opportunities for further research.
We execute Simulation 2 to translate the job shop environment into a system with one repair facility and one stock point, such that it fits the proposed M/M/1 model. Due to the single server, we are not able to test scheduling rules that take into account the status of the work stations. However, according to literature, taking this into account can lead to further costs reductions (e.g. Hausman and Scudder, 1982; Scudder, 1984; Guide et al. 1997). Therefore, we suggest designing a model that jointly optimizing base stock levels and scheduling decisions in a job shop environment. This model makes it possible to test rules that take into account the status of the work stations.
In this research we make the simplifying assumption that all parts are critical. However, in most real life cases, not all parts are critical. Therefore, we recommend investigating the possibility of including non- critical parts in a model that jointly optimizes base stock levels and scheduling decisions.
Furthermore, in this research we leave the benefits of serial batching out of scope. In other words, we assume that the set-up time of changing operations from one SKU to another is incorporated in the processing time. However, in real life, serial batching can be time efficient. Therefore, we suggest doing research on the benefits of including serial batching into the model.
Next, we assume that priorities are preemptive. However, in case of non-preemption priorities, scheduling decisions can be different, since there is a risk that while repairing a less expensive SKU with non- negative inventory, a highly expensive SKU with a backorder arrives at the repair shop. Therefore, it is more risky to repair a less expensive SKU with a high average processing time and no backorders. We
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recommend for further research to investigate the effects of non-preemptive priorities on scheduling decisions and base stock levels.
Finally, in our numerical analysis we analyzed how the costs reductions depend on one of the factors of our test bed. For further research we suggest to do further research on the interaction effects of the factors in our factorial design.
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BIBLIOGRAPHY Adan, I.J.B.F., Sleptchenko, A., & Houtum, van, G.J. (2009). Reducing costs of spare parts supply systems via static priorities. Asia-Pacific Journal of Operational Research, 26(4), 559-585.
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Houtum, van, G.J.J.A.N. & Kranenburg, A.A. (2015). Spare parts inventory control under system availability constraints. International Series in Operations Research & Management Science. New York: Springer.
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Sleptchenko, A.V., Selen, J., Adan, I.J.B.F. & Houtum, van, G.J.J.A.N. (2015). Joint queue length distribution of multi-class, single-server queues with preemptive priorities. Queueing Systems: Theory and Applications, 81(4), 379-395.
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APPENDIX
APPENDIX A: ORGANIZATION CHART OF THE MINISTRY OF DEFENCE In Figure 11 the organization chart of the Ministry of Defence can be found.
Figure 11: Organization Chart of the Ministry of Defence
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APPENDIX B: BUSY PERIODS In this appendix we give a more detailed definition of the busy period of a system.
A queueing system passes through two types of periods: busy periods and idle periods. To give more details on these periods, the notation of Kleinrock (1975) is used. Note that this notation is only used in this appendix. Let 퐶푛 be the n-th custumers in the system, 휏푛 be the arrival time of 퐶푛, 푡푛 be the interarrival time of 퐶푛−1 and 퐶푛, also denoted as (휏푛 − 휏푛−1 ) and 푥푛 be the service time of 퐶푛. Furthermore 푈(푡) is defined as the unfinished work at time 푡, the busy-period durations are denoted by
푌1, 푌2, 푌3, …. and the idle period durations by 퐼1, 퐼2, 퐼3, ….. . In Figure 12 the busy- and idle periods are visualized in a system with one possible priority class. It can be seen that, for example, an arrival of 퐶1 gives an amount of unfinished work of 푥푛. Since the system was idle before its arrival, the arrival of 퐶1 initiates a new busy period 푌1. As long as 푈(푡) > 0, the system is uninterrupted. When 퐶3 is finished, then 푈(푡) = 0, which indicates the start of a new idle period, 푌1. So, only when the state of a system turns from idle to busy, a new busy period is initiated by the first arriving customer of that period.
Figure 12: The busy- and idle periods in a system with one priority class
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APPENDIX C: LAY-OUT SIMULATION DYNAMIC SCHEDULING (SIMULATION 2) In Figure 13 we show the layout of the optimization and repair shop module of the simulation which is used to evaluate the dynamic scheduling rules. The simulation has been programmed using Java Eclipse.
Simulation engine
Optimization module Repair shop module
Initialize the number of arrival events and the Start current time as 0 (nArr=t=0)
Decide on Start input parameters
Find next time that a Find next time that For all priority rules tested SKU enters the shop the server is empty (tArr) (tLeave) Initialize λ= 0
True tArr Repair shop module Place order in Find new tLeave queue Decide on aggregate number of backorders Find new tArr and Is queue optimal base stock empty? levels nArr=nArr+1 False Pick order from Is server idle? True λ=λ+1 queue according True to priority rule False Is preemption True needed? EBO(S) False Calculate costs End Stop e s a h P Figure 13: Layout of Simulation 2 52 APPENDIX D: AN EXAMPLE OF THE APPLICATION OF THE PROPOSED DYNAMIC SCHEDULING RULES In this appendix, we illustrate the behavior of the dynamic scheduling rules on a small example. This example considers two SKU’s with base stock level vector 푺 = (2,4), arrival rate 휆 = (2,10), processing rate 휇 = (10,20), and prices 푐 = (50,30). For a selection of system states, we give the values of the evaluation functions of the evaluated scheduling rules for each SKU and denote which SKU is selected for repair, 훼 . The results are shown in Table 5. Table 5: Values of the evaluation function for a specific example MA-Rule MERT-Rule CPS-Rule MPS1-Rule MPS2-Rule MMA-Rule F(1) F(2) α F(1) F(2) α F(1) F(2) α F(1) F(2) α F(1) F(2) α F(1) F(2) α (-1,-2) 10 20 2 10 20 2 1 1 1,2 50 30 1 3 6 2 500 600 2 (-1,-1) 10 20 2 10 20 2 1 1 1,2 50 30 1 3 5 2 500 600 2 (-1,0) 10 0 1 10 0 1 1 2 1 50 30 1 3 4 1 500 200 1 (0,-1) 0 20 2 0 20 2 2 1 2 50 30 2 2 5 2 500 600 2 (0,0) 1,67 6,67 2 2 10 2 2 2 1,2 50 30 1 2 4 2 83,33 200 2 (0,1) 1,67 2,22 2 0,17 -0,17 1 2 3 1 50 30 1 2 3 1 83,33 66,67 1 (0,2) 1,67 0,74 1 0,33 -0,33 1 2 3 1 50 30 1 2 2 1 83,33 22,22 1 (0,3) 1,67 0,25 1 0,50 -0,50 1 2 3 1 50 30 1 2 1 1 83,33 7,41 1 (1,0) 0,28 6,67 2 -0,83 0,83 2 3 2 2 50 30 2 1 4 2 13,89 200 2 (1,1) 0,28 2,22 2 -0,67 0,67 2 3 3 1,2 50 30 1 1 3 2 13,89 66,67 2 (1,2) 0,28 0,74 2 -0,50 0,50 2 3 3 1,2 50 30 1 1 2 2 13,89 22,22 2 (1,3) 0,28 0,25 1 -0,33 0,33 2 3 3 1,2 50 30 1 1 1 1,2 13,89 7,41 1 (2,0) 0,05 6,67 2 -1,67 1,67 2 3 2 2 50 30 2 0 4 2 2,31 200 2 (2,1) 0,05 2,22 2 -1,50 1,50 2 3 3 1,2 50 30 1 0 3 2 2,31 66,67 2 (2,2) 0,05 0,74 2 -1,33 1,33 2 3 3 1,2 50 30 1 0 2 2 2,31 22,22 2 (2,3) 0,05 0,25 2 -1,17 1,17 2 3 3 1,2 50 30 1 0 1 2 2,31 7,41 2 53 APPENDIX E: DATA SAMPLE CASE STUDY According to these criteria set up in Section 5.2, a sample is created that consist of 10 SKU’s with in total 50 possible routings which passes 8 different workstations. In Table 6 the arrival rate and the purchasing prices of the 10 SKU’s in the sample are presented and in Table 7 the capacity of each workstation measured in hours available per working day of eight hours are visualized. Furthermore, in Table 8 for each SKU the distribution of the processing time at each workstation is given and in Table 9 the probability on a specific routing for each SKU is presented. Finally in Table 10, the routing matrix is visualized, which are the workstation sequences of the routings. Table 6: Input case study: arrival rates and purchasing prices SKU number Arrival rate 흀풏* Purchase price 1 0,101027 € 2.134,87 2 0,339897 € 97.103,11 3 0,372432 € 2.461,64 4 0,100171 € 2.461,64 5 0,862158 € 28.138,80 6 0,184075 € 32.552,67 7 0,090753 € 258.824,00 8 0,189212 € 15.899,00 9 0,097603 € 3.095,00 10 0,093322 € 4.617,48 *measured in arrivals per hour Table 7: Input case study: capacity of the workstations in hours per working day of eight hours Workstation Capacity* 1 0,299887 2 0,022985 3 1,347212 4 0,679015 5 1,480464 6 0,527258 7 0,330007 8 13,15829** *measured in hours available per working day of eight hours **Note that this number of hours exceeds the length of a working day, which indicates that for a part of the day there are two men working at the machine. 54 Table 8: Input case study: distribution of the processing time at the workstations for each SKU ퟏ Distribution of the processing time at workstation * 흁풏 SKU 1 2 3 4 1 0 0 2 + LOGN(2.8, 351) 0 2 20 * BETA(0.601, 5.16) 0 ERLA(0.567, 2) 0 3 0 0 0.25 + GAMM(0.157, 7.45) 8 * BETA(1.51, 21.2) 4 0 0 0.73 + LOGN(0.429, 0.358) EXPO(0.449) 5 LOGN(1.12, 0.971) EXPO(4) EXPO(1.95) WEIB(0.636, 1.01) 6 0 0 0 0 7 0.14 + 4.32 * BETA(0.706, UNIF(0,4) 0 0.608) EXPO(1.05) 8 0 0 0 0 9 0 0 0 GAMM(0.122, 4.29) 10 0.12 + 0.881 * BETA(1.27, 0 0 LOGN(0.699, 0.311) 2.35) SKU 5 6 7 8 1 0.12 + EXPO(0.272) 0.22 + ERLA(0.0358, 2) 0 LOGN(0.465, 0.324) 2 LOGN(0.686, 0.399) ERLA(0.0919, 6) 0 LOGN(1.57, 1.49) 3 WEIB(0.638, 1.62) GAMM(0.124, 3.98) TRIA(0.35, 1.49, 2) GAMM(0.238, 2.5) 4 1.76 * BETA(1.45, 2.78) LOGN(0.574, 0.299) 0 LOGN(0.847, 0.656) 5 LOGN(0.594, 0.526) LOGN(0.706, 0.421) NORM(1.78, 0.417) LOGN(0.739, 0.668) 6 0.12 + LOGN(0.376, 0 0.22) 0 LOGN(0.962, 0.746) 7 0.14 + 3.86 * BETA(0.485, LOGN(1.56, 1.24) 0 0.621) GAMM(1.55, 1.05) 8 LOGN(0.882, 0.446) LOGN(0.517, 0.243) 0.23 + LOGN(0.353, 0.277) LOGN(0.799, 0.551) 9 LOGN(0.814, 0.533) 0 LOGN(0.66, 0.351) LOGN(1.33, 1.27) 10 0.06 + LOGN(0.178, LOGN(0.604, 0.528) 0.193) ERLA(0.0899, 4) LOGN(0.545, 0.238) *Measured in hours Table 9: Input case study: probability on a routing for each SKU Routing SKU 1 SKU 2 SKU 3 SKU 4 SKU 5 SKU 6 SKU 7 SKU 8 SKU 9 SKU 10 1 0,58586 0,01661 0,01923 0,11111 0,04348 0,26540 0,04348 0,07586 0,00000 0,00000 2 0,22222 0,03654 0,04396 0,04444 0,03333 0,08057 0,21739 0,06207 0,68254 0,00000 3 0,05051 0,01993 0,08516 0,07778 0,03768 0,05687 0,17391 0,00690 0,06349 0,00000 4 0,00000 0,27243 0,03571 0,00000 0,07536 0,02844 0,00000 0,45517 0,00000 0,04167 5 0,00000 0,00332 0,00000 0,00000 0,00000 0,00000 0,21739 0,06207 0,00000 0,00000 6 0,00000 0,00332 0,00000 0,00000 0,00000 0,56872 0,00000 0,00000 0,00000 0,00000 7 0,00000 0,01329 0,00000 0,00000 0,00000 0,00000 0,17391 0,03448 0,00000 0,00000 8 0,00000 0,03987 0,00824 0,00000 0,03188 0,00000 0,08696 0,11724 0,00000 0,00000 9 0,14141 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 10 0,00000 0,03654 0,02473 0,00000 0,00000 0,00000 0,00000 0,00000 0,01587 0,00000 11 0,00000 0,02326 0,00000 0,00000 0,00000 0,00000 0,00000 0,03448 0,01587 0,01042 12 0,00000 0,00000 0,04945 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 13 0,00000 0,00000 0,06044 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 14 0,00000 0,14286 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 15 0,00000 0,05980 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 16 0,00000 0,16279 0,02198 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 17 0,00000 0,12957 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 18 0,00000 0,02990 0,00000 0,00000 0,00145 0,00000 0,00000 0,00000 0,00000 0,00000 55 19 0,00000 0,00997 0,00000 0,00000 0,00000 0,00000 0,08696 0,06207 0,00000 0,00000 20 0,00000 0,00000 0,05495 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 21 0,00000 0,00000 0,00549 0,22222 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 22 0,00000 0,00000 0,05495 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 23 0,00000 0,00000 0,32143 0,00000 0,00000 0,00000 0,00000 0,00000 0,20635 0,00000 24 0,00000 0,00000 0,04396 0,00000 0,00000 0,00000 0,00000 0,00000 0,01587 0,00000 25 0,00000 0,00000 0,04670 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 26 0,00000 0,00000 0,03297 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 27 0,00000 0,00000 0,04396 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 28 0,00000 0,00000 0,04670 0,07778 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 29 0,00000 0,00000 0,00000 0,46667 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 30 0,00000 0,00000 0,00000 0,00000 0,01739 0,00000 0,00000 0,00000 0,00000 0,00000 31 0,00000 0,00000 0,00000 0,00000 0,01884 0,00000 0,00000 0,00000 0,00000 0,00000 32 0,00000 0,00000 0,00000 0,00000 0,02464 0,00000 0,00000 0,00000 0,00000 0,00000 33 0,00000 0,00000 0,00000 0,00000 0,02174 0,00000 0,00000 0,00000 0,00000 0,00000 34 0,00000 0,00000 0,00000 0,00000 0,20290 0,00000 0,00000 0,00000 0,00000 0,00000 35 0,00000 0,00000 0,00000 0,00000 0,16377 0,00000 0,00000 0,00000 0,00000 0,00000 36 0,00000 0,00000 0,00000 0,00000 0,04493 0,00000 0,00000 0,00000 0,00000 0,00000 37 0,00000 0,00000 0,00000 0,00000 0,04928 0,00000 0,00000 0,00000 0,00000 0,00000 38 0,00000 0,00000 0,00000 0,00000 0,04493 0,00000 0,00000 0,00000 0,00000 0,00000 39 0,00000 0,00000 0,00000 0,00000 0,05072 0,00000 0,00000 0,00000 0,00000 0,00000 40 0,00000 0,00000 0,00000 0,00000 0,01884 0,00000 0,00000 0,00000 0,00000 0,00000 41 0,00000 0,00000 0,00000 0,00000 0,01449 0,00000 0,00000 0,00000 0,00000 0,00000 42 0,00000 0,00000 0,00000 0,00000 0,02029 0,00000 0,00000 0,00000 0,00000 0,00000 43 0,00000 0,00000 0,00000 0,00000 0,03913 0,00000 0,00000 0,00000 0,00000 0,00000 44 0,00000 0,00000 0,00000 0,00000 0,02899 0,00000 0,00000 0,00000 0,00000 0,00000 45 0,00000 0,00000 0,00000 0,00000 0,01594 0,00000 0,00000 0,00000 0,00000 0,00000 46 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,08966 0,00000 0,00000 47 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,51042 48 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,18750 49 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,12500 50 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,00000 0,12500 Table 10: Input case study: routing matrix Operation Routing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 8 8 2 8 8 8 3 8 4 8 8 8 8 5 8 8 5 8 8 8 6 8 6 8 7 8 8 8 8 8 8 8 8 8 8 8 8 9 8 6 8 8 5 8 10 8 8 6 6 5 8 11 8 5 8 8 8 12 8 8 6 7 5 8 13 8 8 6 6 5 8 4 14 8 8 8 8 6 8 15 8 1 8 8 8 16 8 8 8 6 6 5 8 17 8 8 8 6 6 5 8 8 56 18 8 8 8 6 8 19 8 8 5 8 8 8 8 20 8 8 6 7 5 8 8 4 8 3 8 21 8 6 6 5 5 8 4 8 8 22 8 8 6 6 5 5 8 4 8 8 8 8 23 8 8 6 6 5 8 4 8 8 8 24 8 8 6 6 5 5 8 4 8 8 8 25 8 8 6 6 5 8 8 8 8 8 26 8 8 4 8 8 8 27 8 8 8 6 6 5 8 8 4 8 8 28 8 8 8 6 6 5 8 8 4 8 3 8 29 8 6 6 5 8 4 8 8 30 8 1 1 8 8 3 8 3 8 8 8 31 8 8 2 2 8 5 4 8 8 3 8 3 8 8 8 32 8 8 8 6 7 5 5 8 4 8 33 8 8 3 8 3 8 8 8 34 8 8 8 6 6 5 5 5 8 35 8 8 8 6 6 5 5 8 36 8 4 8 8 8 8 37 8 8 8 6 6 5 5 5 8 8 38 8 8 8 6 6 5 5 5 8 4 8 8 8 4 8 39 8 8 8 6 6 5 5 5 8 4 8 8 8 8 40 8 8 8 6 6 5 5 5 8 8 4 8 8 8 8 41 8 8 8 6 6 5 5 5 4 8 8 8 8 42 8 8 8 6 6 5 5 5 8 5 4 8 8 8 8 43 8 8 8 6 6 5 5 5 8 4 44 8 4 5 8 4 4 45 8 1 1 8 8 8 8 46 8 8 8 8 6 8 8 47 8 4 7 8 3 8 8 48 8 6 7 8 3 8 8 49 8 4 7 3 8 8 50 8 4 7 5 8 3 8 8 57 APPENDIX F: RESULTS CASE STUDY In Table 11 until Table 19 the results of the case study are presented in detail. Table 11: Results of the case study for Heuristic 1 Table 12: Results of the case study for Heuristic 2 Investment Expected Investment Expected Max λ Base stock levels Max λ Base stock levels costs backorders costs backorders 1828 0,0,0,0,0,0,0,0,0,0 € - 3,381 13729 0,0,0,0,0,0,0,0,0,0 € - 3,803 2336 1,0,0,0,0,0,0,0,0,0 € 2.134,87 2,843 20928 0,0,1,0,0,0,0,0,0,0 € 2.461,64 3,651 4681 1,0,1,0,0,0,0,0,0,0 € 4.596,51 2,329 28776 0,0,1,0,0,0,0,0,1,0 € 5.556,64 3,522 6611 2,0,1,0,0,0,0,0,0,0 € 6.731,38 2,016 35603 0,0,1,1,1,0,0,0,1,0 € 8.018,28 3,443 8480 2,0,2,0,0,0,0,0,0,0 € 9.193,02 1,745 83420 1,0,1,1,1,0,0,0,1,0 € 10.153,15 3,387 9138 2,0,2,1,1,0,0,0,0,0 € 11.654,66 1,520 95290 1,1,1,1,1,0,0,0,1,0 € 107.256,26 2,849 14502 3,0,2,1,1,0,0,0,0,0 € 13.789,53 1,330 116182 1,1,2,1,1,0,0,0,1,0 € 109.717,90 2,824 15957 3,0,3,1,1,0,0,0,0,0 € 16.251,17 1,185 129194 1,1,2,1,1,0,0,1,1,0 € 125.616,90 2,703 26346 4,0,3,1,1,0,0,0,0,0 € 18.386,04 1,067 171036 1,1,2,1,1,0,0,1,1,0 € 153.755,70 2,524 28962 5,0,3,1,1,0,0,0,0,0 € 20.520,91 0,992 175899 1,1,2,1,1,0,0,1,2,0 € 156.850,70 2,507 31659 5,0,4,1,1,0,0,0,0,0 € 22.982,55 0,914 224856 1,1,2,1,1,0,1,1,2,0 € 415.674,70 1,911 41451 5,0,4,1,1,0,0,0,1,0 € 26.077,55 0,825 321142 1,2,2,1,1,0,1,1,2,0 € 512.777,81 1,610 44023 6,0,4,1,1,0,0,0,1,0 € 28.212,42 0,776 433670 1,2,2,2,2,0,1,1,2,0 € 515.239,45 1,602 55736 6,0,4,2,2,0,0,0,1,0 € 30.674,06 0,723 466606 1,2,2,2,2,0,2,1,2,0 € 774.063,45 1,228 63247 6,0,5,2,2,0,0,0,1,0 € 33.135,70 0,681 533017 1,3,2,2,2,0,2,1,2,0 € 871.166,56 1,056 93880 7,0,5,2,2,0,0,0,1,0 € 35.270,57 0,648 555589 1,3,2,2,2,1,2,1,2,0 € 903.719,23 0,998 104076 8,0,5,2,2,0,0,0,1,0 € 37.405,44 0,626 558609 2,3,2,2,2,1,2,1,2,0 € 905.854,10 0,995 121783 8,0,6,2,2,0,0,0,1,0 € 39.867,08 0,603 829494 2,3,3,2,2,1,2,1,2,0 € 908.315,74 0,990 141727 8,0,6,2,2,0,0,0,1,0 € 68.005,88 0,415 851517 2,3,3,2,2,1,3,1,2,0 € 1.167.139,74 0,752 151200 9,0,6,2,2,0,0,0,1,0 € 70.140,75 0,400 877157 2,3,3,2,2,1,3,1,2,0 € 1.195.278,54 0,720 168961 9,0,6,2,2,0,0,0,1,1 € 74.758,23 0,371 917355 2,4,3,2,2,1,3,1,2,0 € 1.292.381,65 0,621 190984 9,0,6,2,2,0,0,1,1,1 € 90.657,23 0,285 1090050 2,4,3,2,2,1,3,2,2,0 € 1.308.280,65 0,604 196169 9,0,7,2,2,0,0,1,1,1 € 93.118,87 0,272 1245301 2,4,3,2,2,1,3,2,2,1 € 1.312.898,13 0,600 217880 9,0,7,3,3,0,0,1,1,1 € 95.580,51 0,259 1432074 2,4,3,2,2,1,3,2,3,1 € 1.315.993,13 0,597 319415 10,0,7,3,3,0,0,1,1,1 € 97.715,38 0,250 1570545 2,4,3,2,2,1,4,2,3,1 € 1.574.817,13 0,444 356497 11,0,7,3,3,0,0,1,1,1 € 99.850,25 0,243 2352070 2,5,3,2,2,1,4,2,3,1 € 1.671.920,24 0,386 362976 11,0,8,3,3,0,0,1,1,1 € 102.311,89 0,236 2747924 2,5,3,2,2,1,5,2,3,1 € 1.930.744,24 0,287 471814 11,0,8,3,3,0,0,1,2,1 € 105.406,89 0,228 3006681 2,6,3,2,2,1,5,2,3,1 € 2.027.847,35 0,252 659887 12,0,8,3,3,0,0,1,2,1 € 107.541,76 0,223 3648234 2,6,4,2,2,1,5,2,3,1 € 2.030.308,99 0,252 691396 12,0,9,3,3,0,0,1,2,1 € 110.003,40 0,220 3721776 2,6,4,3,3,1,5,2,3,1 € 2.032.770,63 0,251 774229 12,0,9,3,3,0,0,1,2,1 € 138.142,20 0,180 4693508 2,6,4,3,3,1,6,2,3,1 € 2.291.594,63 0,186 865032 13,0,9,3,3,0,0,1,2,1 € 140.277,07 0,178 4749031 2,6,4,3,3,1,6,2,3,1 € 2.319.733,43 0,180 1065339 13,0,9,4,4,0,0,1,2,1 € 142.738,71 0,175 5809212 2,7,4,3,3,1,6,2,3,1 € 2.416.836,54 0,160 1074545 13,0,9,4,4,1,0,1,2,1 € 175.291,38 0,145 5991140 2,7,4,3,3,1,7,2,3,1 € 2.675.660,54 0,117 1160791 13,1,9,4,4,1,0,1,2,1 € 272.394,49 0,062 7868947 2,7,4,3,3,1,7,3,3,1 € 2.691.559,54 0,115 8051566 3,7,4,3,3,1,7,3,3,1 € 2.693.694,41 0,114 8536652 3,8,4,3,3,1,7,3,3,1 € 2.790.797,52 0,102 8937615 3,8,4,3,3,2,7,3,3,1 € 2.823.350,19 0,099 58 Table 13: Results of the case study for Heuristic 3 Table 14: Results of the case study for the MA-Rule Investment Expected Expected Max λ Base stock levels Max λ Base stock levels Investment costs costs backorders backorders 2632 0,0,0,0,0,0,0,0,0,0 € - 3,332 2009 0,0,0,0,0,0,0,0,0,0 € - 3,257 3209 0,0,0,0,0,0,0,0,1,0 € 3.095,00 2,792 2153 1,0,0,0,0,0,0,0,0,0 € 2.134,87 2,742 5629 0,0,1,0,0,0,0,0,1,0 € 5.556,64 2,358 2154 0,0,1,0,0,0,0,0,0,0 € 2.461,64 2,703 6758 0,0,1,1,1,0,0,0,1,0 € 8.018,28 2,053 2434 0,0,1,0,0,0,0,0,0,0 € 2.461,64 2,695 10000 0,0,1,1,1,0,0,0,2,0 € 11.113,28 1,739 4816 1,0,1,0,0,0,0,0,0,0 € 4.596,51 2,213 13215 0,0,2,1,1,0,0,0,2,0 € 13.574,92 1,542 7388 1,0,2,0,0,0,0,0,0,0 € 7.058,15 1,856 20193 0,0,2,1,1,0,0,0,3,0 € 16.669,92 1,352 8671 2,0,2,0,0,0,0,0,0,0 € 9.193,02 1,616 20453 1,0,2,1,1,0,0,0,3,0 € 18.804,79 1,256 13761 2,0,3,0,0,0,0,0,0,0 € 11.654,66 1,398 24082 1,0,2,2,2,0,0,0,3,0 € 21.266,43 1,149 20690 2,0,4,0,0,0,0,0,0,0 € 14.116,30 1,245 24613 1,0,2,2,2,0,0,0,4,0 € 24.361,43 1,035 25645 2,0,5,0,0,0,0,0,0,0 € 16.577,94 1,138 42205 1,0,3,2,2,0,0,0,4,0 € 26.823,07 0,944 33690 3,0,5,0,0,0,0,0,0,0 € 18.712,81 1,048 55587 1,0,3,2,2,0,0,0,5,0 € 29.918,07 0,876 38333 3,0,6,0,0,0,0,0,0,0 € 21.174,45 0,984 58274 1,0,4,2,2,0,0,0,5,0 € 32.379,71 0,833 48883 3,0,6,0,0,0,0,0,0,1 € 25.791,93 0,864 71253 1,0,4,3,3,0,0,0,5,0 € 34.841,35 0,793 56091 3,0,6,0,0,0,0,0,1,1 € 28.886,93 0,803 117655 1,0,4,3,3,0,0,0,6,0 € 37.936,35 0,751 75149 3,0,7,0,0,0,0,0,1,1 € 31.348,57 0,759 118016 1,0,4,3,3,0,0,0,6,1 € 42.553,83 0,713 80829 3,0,7,0,0,0,0,0,1,1 € 59.487,37 0,501 120302 1,0,4,3,3,0,0,0,6,1 € 70.692,63 0,521 84023 3,0,7,1,1,0,0,0,1,1 € 61.949,01 0,456 124128 1,0,4,3,3,0,0,0,7,1 € 73.787,63 0,496 106909 3,0,8,1,1,0,0,0,1,1 € 64.410,65 0,431 130870 1,0,5,3,3,0,0,0,7,1 € 76.249,27 0,476 140767 4,0,8,1,1,0,0,0,1,1 € 66.545,52 0,400 150770 1,0,5,3,3,0,0,1,7,1 € 92.148,27 0,368 200609 4,0,9,1,1,0,0,0,1,1 € 69.007,16 0,384 202113 1,0,5,4,4,0,0,1,7,1 € 94.609,91 0,352 266841 4,0,10,1,1,0,0,0,1,1 € 71.468,80 0,373 211515 1,0,5,4,4,0,0,1,8,1 € 97.704,91 0,337 286020 4,0,11,1,1,0,0,0,1,1 € 73.930,44 0,363 264592 2,0,5,4,4,0,0,1,8,1 € 99.839,78 0,327 360552 4,0,11,1,1,0,0,0,1,1 € 102.069,24 0,282 336561 2,0,6,4,4,0,0,1,8,1 € 102.301,42 0,318 367521 4,0,11,1,1,0,0,0,1,2 € 106.686,72 0,263 376450 2,0,6,4,4,0,0,1,9,1 € 105.396,42 0,309 476306 4,0,11,1,1,0,0,1,1,2 € 122.585,72 0,208 439427 2,0,6,5,5,0,0,1,9,1 € 107.858,06 0,302 564064 4,0,12,1,1,0,0,1,1,2 € 125.047,36 0,204 556979 2,0,6,5,5,1,0,1,9,1 € 140.410,73 0,233 786032 5,0,12,1,1,0,0,1,1,2 € 127.182,23 0,197 566748 2,0,6,5,5,1,0,1,10,1 € 143.505,73 0,228 874913 5,0,13,1,1,0,0,1,1,2 € 129.643,87 0,195 658311 2,0,7,5,5,1,0,1,10,1 € 145.967,37 0,223 1020681 5,0,13,1,1,0,0,1,2,2 € 132.738,87 0,189 843967 2,0,7,5,5,1,0,1,10,1 € 174.106,17 0,182 1033070 5,0,13,1,1,0,0,1,2,2 € 160.877,67 0,165 950523 2,0,7,5,5,1,0,1,11,1 € 177.201,17 0,179 1192933 5,0,13,1,1,1,0,1,2,2 € 193.430,34 0,127 958604 2,0,7,6,6,1,0,1,11,1 € 179.662,81 0,176 1192934 5,1,13,1,1,1,0,1,2,2 € 262.394,65 0,065 1143903 2,1,7,6,6,1,0,1,11,1 € 276.765,92 0,084 Table 15: Results of the case study for the MERT-Rule Table 16: Results of the case study for the CPS-Rule Expected Investment Expected Max λ Base stock levels Investment costs Max λ Base stock levels backorders costs backorders 1957 0,0,0,0,0,0,0,0,0,0 € - 3,305 2704 0,0,0,0,0,0,0,0,0,0 € - 5,713 2130 1,0,0,0,0,0,0,0,0,0 € 2.134,87 2,783 6863 0,0,1,0,0,0,0,0,0,0 € 2.461,64 5,066 2131 0,0,1,0,0,0,0,0,0,0 € 2.461,64 2,753 10093 0,0,2,0,0,0,0,0,0,0 € 4.923,28 4,503 2386 0,0,1,0,0,0,0,0,0,0 € 2.461,64 2,744 12671 1,0,2,0,0,0,0,0,0,0 € 7.058,15 4,112 4674 1,0,1,0,0,0,0,0,0,0 € 4.596,51 2,258 14320 1,0,2,1,1,0,0,0,0,0 € 9.519,79 3,768 7260 1,0,2,0,0,0,0,0,0,0 € 7.058,15 1,891 16960 1,0,3,1,1,0,0,0,0,0 € 11.981,43 3,385 8376 2,0,2,0,0,0,0,0,0,0 € 9.193,02 1,649 25093 1,0,3,1,1,0,0,0,1,0 € 15.076,43 2,985 13218 2,0,3,0,0,0,0,0,0,0 € 11.654,66 1,427 32367 1,0,4,1,1,0,0,0,1,0 € 17.538,07 2,693 19596 2,0,4,0,0,0,0,0,0,0 € 14.116,30 1,273 38642 1,0,4,1,1,0,0,0,1,0 € 45.676,87 2,240 24377 2,0,5,0,0,0,0,0,0,0 € 16.577,94 1,160 50643 1,0,5,1,1,0,0,0,1,0 € 48.138,51 2,033 31145 3,0,5,0,0,0,0,0,0,0 € 18.712,81 1,069 55881 1,0,5,1,1,0,0,0,1,1 € 52.755,99 1,929 38270 3,0,6,0,0,0,0,0,0,0 € 21.174,45 0,999 74335 1,0,6,1,1,0,0,0,1,1 € 55.217,63 1,769 49441 3,0,6,0,0,0,0,0,0,1 € 25.791,93 0,878 83055 2,0,6,1,1,0,0,0,1,1 € 57.352,50 1,634 51315 3,0,6,0,0,0,0,0,1,1 € 28.886,93 0,819 88777 2,0,7,1,1,0,0,0,1,1 € 59.814,14 1,515 73034 3,0,7,0,0,0,0,0,1,1 € 31.348,57 0,773 104176 2,0,7,1,1,0,0,1,1,1 € 75.713,14 1,318 75137 3,0,8,0,0,0,0,0,1,1 € 33.810,21 0,742 121015 2,0,7,1,1,0,0,1,1,1 € 103.851,94 1,123 80575 3,0,8,0,0,0,0,0,1,1 € 61.949,01 0,489 134912 2,0,7,2,2,0,0,1,1,1 € 106.313,58 1,037 102733 3,0,8,1,1,0,0,0,1,1 € 64.410,65 0,442 160683 2,0,7,2,2,0,0,1,2,1 € 109.408,58 0,928 125295 4,0,8,1,1,0,0,0,1,1 € 66.545,52 0,413 211358 2,0,8,2,2,0,0,1,2,1 € 111.870,22 0,868 170636 4,0,9,1,1,0,0,0,1,1 € 69.007,16 0,393 233002 2,0,8,2,2,0,0,1,2,1 € 140.009,02 0,776 231662 4,0,10,1,1,0,0,0,1,1 € 71.468,80 0,380 279215 2,0,9,2,2,0,0,1,2,1 € 142.470,66 0,740 284576 4,0,11,1,1,0,0,0,1,1 € 73.930,44 0,372 324426 2,0,9,2,2,1,0,1,2,1 € 175.023,33 0,618 322125 4,0,11,1,1,0,0,0,1,1 € 102.069,24 0,288 347995 2,0,10,2,2,1,0,1,2,1 € 177.484,97 0,589 362585 5,0,11,1,1,0,0,0,1,1 € 104.204,11 0,278 416903 2,0,10,2,2,1,0,1,2,1 € 205.623,77 0,529 365156 5,0,11,1,1,0,0,0,1,2 € 108.821,59 0,263 512646 2,1,10,2,2,1,0,1,2,1 € 302.726,88 0,326 489563 5,0,11,1,1,0,0,1,1,2 € 124.720,59 0,206 579736 2,1,11,2,2,1,0,1,2,1 € 305.188,52 0,305 662330 5,0,12,1,1,0,0,1,1,2 € 127.182,23 0,200 616646 2,1,11,2,2,1,0,1,2,1 € 333.327,32 0,274 863902 5,0,13,1,1,0,0,1,1,2 € 129.643,87 0,197 812750 3,1,11,2,2,1,0,1,2,1 € 335.462,19 0,256 890478 5,0,14,1,1,0,0,1,1,2 € 132.105,51 0,196 888864 3,1,11,2,2,1,0,1,2,1 € 363.600,99 0,226 59 1031834 5,0,14,1,1,0,0,1,2,2 € 135.200,51 0,191 1097994 3,1,11,2,2,1,0,1,3,1 € 366.695,99 0,206 1034139 5,0,14,1,1,0,0,1,2,2 € 163.339,31 0,165 1158068 3,1,11,2,2,1,0,2,3,1 € 382.594,99 0,187 1185920 5,0,14,1,1,1,0,1,2,2 € 195.891,98 0,127 1371094 3,1,11,3,3,1,0,2,3,1 € 385.056,63 0,174 1185921 5,1,14,1,1,1,0,1,2,2 € 264.856,29 0,064 1416821 3,1,12,3,3,1,0,2,3,1 € 387.518,27 0,170 1862678 3,1,12,3,3,1,0,2,3,1 € 415.657,07 0,156 1911366 3,1,12,3,3,1,0,2,3,1 € 443.795,87 0,148 2405401 3,1,13,3,3,1,0,2,3,1 € 446.257,51 0,142 2540750 4,1,13,3,3,1,0,2,3,1 € 448.392,38 0,139 2625850 4,1,13,3,3,1,0,2,3,2 € 453.009,86 0,135 2877164 4,1,13,3,3,1,0,2,3,2 € 481.148,66 0,126 3711919 4,2,13,3,3,1,0,2,3,2 € 578.251,77 0,098 Table 17: Results of the case study for the MPS1-Rule Table 18: Results of the case study for the MPS2-Rule Investment Expected Investment Expected Max λ Base stock levels Max λ Base stock levels costs backorders costs backorders 1859 0,0,0,0,0,0,0,0,0,0 € - 3,354 0 0,0,0,0,0,0,0,0,0,0 € - 5,218 2342 1,0,0,0,0,0,0,0,0,0 € 2.134,87 2,819 2814 0,0,1,0,0,0,0,0,0,0 € 2.461,64 4,688 5321 1,0,1,0,0,0,0,0,0,0 € 4.596,51 2,292 4478 1,0,1,0,0,0,0,0,0,0 € 4.596,51 4,229 5892 2,0,1,0,0,0,0,0,0,0 € 6.731,38 1,981 5579 1,0,1,1,1,0,0,0,0,0 € 7.058,15 3,817 10345 2,0,2,0,0,0,0,0,0,0 € 9.193,02 1,663 6868 1,0,1,1,1,0,0,0,1,0 € 10.153,15 3,379 11333 2,0,3,0,0,0,0,0,0,0 € 11.654,66 1,460 13439 1,0,2,1,1,0,0,0,1,0 € 12.614,79 3,059 15517 2,0,3,1,1,0,0,0,0,0 € 14.116,30 1,246 17162 1,0,2,1,1,0,0,0,1,1 € 17.232,27 2,793 20417 3,0,3,1,1,0,0,0,0,0 € 16.251,17 1,093 25016 2,0,2,1,1,0,0,0,1,1 € 19.367,14 2,555 29736 3,0,4,1,1,0,0,0,0,0 € 18.712,81 0,981 31618 2,0,2,2,2,0,0,0,1,1 € 21.828,78 2,321 31722 4,0,4,1,1,0,0,0,0,0 € 20.847,68 0,898 38102 2,0,2,2,2,0,0,0,2,1 € 24.923,78 2,065 38842 4,0,4,1,1,0,0,0,1,0 € 23.942,68 0,785 43808 2,0,3,2,2,0,0,0,2,1 € 27.385,42 1,887 58412 4,0,5,1,1,0,0,0,1,0 € 26.404,32 0,722 53369 2,0,3,2,2,0,0,0,2,1 € 55.524,22 1,569 68890 4,0,6,1,1,0,0,0,1,0 € 28.865,96 0,679 57988 2,0,3,2,2,0,0,1,2,1 € 71.423,22 1,341 99470 5,0,6,1,1,0,0,0,1,0 € 31.000,83 0,642 101819 3,0,3,2,2,0,0,1,2,1 € 73.558,09 1,226 113035 5,0,6,2,2,0,0,0,1,0 € 33.462,47 0,612 110101 3,0,4,2,2,0,0,1,2,1 € 76.019,73 1,136 121888 6,0,6,2,2,0,0,0,1,0 € 35.597,34 0,588 146223 3,0,4,3,3,0,0,1,2,1 € 78.481,37 1,043 148629 6,0,6,2,2,0,0,0,1,0 € 63.736,14 0,385 163308 3,0,4,3,3,1,0,1,2,1 € 111.034,04 0,859 190230 6,0,7,2,2,0,0,0,1,0 € 66.197,78 0,367 182063 3,0,4,3,3,1,0,1,3,1 € 114.129,04 0,776 194831 6,0,7,2,2,0,0,0,1,1 € 70.815,26 0,343 223352 3,0,4,3,3,1,0,1,3,2 € 118.746,52 0,720 224060 6,0,7,2,2,0,0,1,1,1 € 86.714,26 0,263 284750 3,0,5,3,3,1,0,1,3,2 € 121.208,16 0,691 267629 6,0,8,2,2,0,0,1,1,1 € 89.175,90 0,249 306111 3,0,5,3,3,1,0,1,3,2 € 149.346,96 0,610 393879 7,0,8,2,2,0,0,1,1,1 € 91.310,77 0,240 380917 4,0,5,3,3,1,0,1,3,2 € 151.481,83 0,573 418919 8,0,8,2,2,0,0,1,1,1 € 93.445,64 0,233 405819 4,1,5,3,3,1,0,1,3,2 € 248.584,94 0,378 522734 8,0,8,2,2,0,0,1,2,1 € 96.540,64 0,222 448108 4,1,5,3,3,1,0,2,3,2 € 264.483,94 0,342 677501 8,0,9,2,2,0,0,1,2,1 € 99.002,28 0,219 568284 4,1,5,4,4,1,0,2,3,2 € 266.945,58 0,322 823780 8,0,9,2,2,0,0,1,2,1 € 127.141,08 0,177 587613 4,1,5,4,4,1,0,2,4,2 € 270.040,58 0,286 967893 8,0,9,3,3,0,0,1,2,1 € 129.602,72 0,173 756493 4,1,6,4,4,1,0,2,4,2 € 272.502,22 0,276 1057538 9,0,9,3,3,0,0,1,2,1 € 131.737,59 0,171 924006 4,1,6,4,4,1,0,2,4,2 € 300.641,02 0,249 1080251 9,0,9,3,3,1,0,1,2,1 € 164.290,26 0,139 977092 4,1,6,4,4,1,0,2,4,3 € 305.258,50 0,232 1348817 9,1,9,3,3,1,0,1,2,1 € 261.393,37 0,052 1344668 5,1,6,4,4,1,0,2,4,3 € 307.393,37 0,227 1498374 5,1,7,4,4,1,0,2,4,3 € 309.855,01 0,214 1672495 5,1,7,4,4,1,0,3,4,3 € 325.754,01 0,196 1857075 5,1,7,4,4,1,0,3,4,3 € 353.892,81 0,183 2079484 5,1,7,4,4,1,0,3,5,3 € 356.987,81 0,169 2100312 5,1,7,5,5,1,0,3,5,3 € 359.449,45 0,164 2562894 5,1,7,5,5,2,0,3,5,3 € 392.002,12 0,147 3129544 5,1,7,5,5,2,1,3,5,3 € 650.826,12 0,071 Table 19: Results of the case study for the MMA-Rule Max λ Base stock levels Investment costs Expected backorders 1854 0,0,0,0,0,0,0,0,0,0 € - 3,355 2179 1,0,0,0,0,0,0,0,0,0 € 2.134,87 2,820 4769 1,0,1,0,0,0,0,0,0,0 € 4.596,51 2,305 6617 1,0,2,0,0,0,0,0,0,0 € 7.058,15 1,957 8457 2,0,2,0,0,0,0,0,0,0 € 9.193,02 1,690 13069 2,0,3,0,0,0,0,0,0,0 € 11.654,66 1,468 16238 2,0,4,0,0,0,0,0,0,0 € 14.116,30 1,315 19496 2,0,4,1,1,0,0,0,0,0 € 16.577,94 1,137 24115 3,0,4,1,1,0,0,0,0,0 € 18.712,81 1,023 31830 3,0,5,1,1,0,0,0,0,0 € 21.174,45 0,940 37913 3,0,5,1,1,0,0,0,1,0 € 24.269,45 0,828 48667 3,0,6,1,1,0,0,0,1,0 € 26.731,09 0,772 58173 4,0,6,1,1,0,0,0,1,0 € 28.865,96 0,722 81150 4,0,7,1,1,0,0,0,1,0 € 31.327,60 0,681 60 83611 4,0,8,1,1,0,0,0,1,0 € 33.789,24 0,654 122110 4,0,8,2,2,0,0,0,1,0 € 36.250,88 0,616 126582 4,0,8,2,2,0,0,0,1,0 € 64.389,68 0,427 153735 4,0,9,2,2,0,0,0,1,0 € 66.851,32 0,409 184716 5,0,9,2,2,0,0,0,1,0 € 68.986,19 0,391 193857 5,0,9,2,2,0,0,0,1,1 € 73.603,67 0,363 210668 5,0,9,2,2,0,0,1,1,1 € 89.502,67 0,279 281516 5,0,10,2,2,0,0,1,1,1 € 91.964,31 0,267 323316 5,0,11,2,2,0,0,1,1,1 € 94.425,95 0,259 396989 5,0,11,2,2,0,0,1,2,1 € 97.520,95 0,246 510200 5,0,12,2,2,0,0,1,2,1 € 99.982,59 0,239 572366 5,0,13,2,2,0,0,1,2,1 € 102.444,23 0,235 586940 5,0,13,2,2,0,0,1,2,1 € 130.583,03 0,191 807566 6,0,13,2,2,0,0,1,2,1 € 132.717,90 0,186 865406 6,0,14,2,2,0,0,1,2,1 € 135.179,54 0,183 1054831 6,0,14,3,3,0,0,1,2,1 € 137.641,18 0,179 1081874 6,0,14,3,3,1,0,1,2,1 € 170.193,85 0,145 1340401 6,1,14,3,3,1,0,1,2,1 € 267.296,96 0,054 61