020-0500
A New Approach for Master Production Schedule under Theory of Constraints
Davood Golmohammadi a and S. Afshin Mansouri b a College of Management, University of Massachusetts Boston, USA. [email protected] b Brunel Business School, Brunel University West London, UK. [email protected]
POMS 22nd Annual Conference Reno, Nevada, U.S.A. April 29 to May 2, 2011
Abstract: Product mix optimization is one of the main challenges in a production system.
Several algorithms have been developed based on the Theory of constraints (TOC) approach to determine an optimized master production schedule (MPS). Most of these algorithms are evaluated based on simple examples and they may not be very efficient in dealing with real- world operations. We investigate the inefficiency of recent algorithms, and demonstrate some of the fundamental factors that have not been considered in any of the current algorithms.
Finally, we propose a new algorithm under TOC approach to create an efficient MPS.
Key Words: Theory of constraints, Master Production Schedule (MPS)
1. Introduction:
The Theory of constraints (TOC) is a successful operations philosophy, focused on the
concept of marginal attention to the constraints that inhibit the performance of the entire
system (Linhares, 2009). A constraint prevents a system from achieving its goal and may include a machine whose capacity limits the throughput of the entire production process,
a highly specialized operator or scarce tool. The TOC attempts to identify constraints in
the system, and to exploit and elevate them to improve the overall output of the system
(Fawcett 1991). Principles for achieving a continuous improvement process are as
follows:
Identify the system’s constraint(s).
Decide how to exploit the system’s constraint(s).
Subordinate everything else to the above decision.
Elevate the system’s constraint(s).
If in any of the previous steps a constraint is broken, then go back to the first
step.
In Step 1, the constraint or bottleneck is identified. In Step 2, an MPS is developed to maximize the throughput. Step 3 develops a detailed schedule for production, ensuring that the constraint resource can fulfill the MPS schedule established in Step 2. Step 4 encourages continuous improvement to fully utilize the existing capacity and to increase the capacity at the constraint. Step 5 continually reevaluates the entire system to see if there is a new constraint after making the improvements identified in Step 4. If the constraint has changed, then the heuristic starts over again with Step 1 (Fredendall and Lea, 1997).
TOC emphasizes that all activities of a system should be harnessed by the constraint, and the constraint’s capacity should be fully utilized. How to optimize the resource utilization to maximize the enterprise’s throughput within the determined product types is a dilemma for operations managers. It involves the determination of the type and quantity of products which is known as the product mix optimization problem. Product mix optimization is one of the main challenges in a production system with a direct impact on the manufacturing enterprise’s performance, such as profit, work-in-process (WIP), and customer service. The product mix heuristic generates a master production schedule (MPS) which is capable of maximizing the firm’s net profitability.
Several algorithms have been developed based on TOC to determine an optimized MPS.
Some of the algorithms (Goldratt,1990; Patterson,1992; Luebbe and Finch,1992; Lea and
Plenert, 1993; Lee and Plenert, 1996) are only proven to find the optimal solution of a single constraint example. Also, in some problems, the results are inefficient and produce non- optimal or infeasible solutions when certain new product alternatives are available. Posnack
(1994) and Maday (1994) argued that TOC approach should be properly used, and for non- integer solutions, partial products should be allowed to be manufactured in the next planning horizon. The other issue arises when there are multiple constraints; TOC approach might generate an optimal solution (Luebbe and Finch 1992; Lee and Plenert,1993), or an infeasible solution (Plenert, 1993; Fredendall and Lea,1997; Hsu and Chung,1998; Balakrishnan and
Chen, 2000). Other scholars developed improved algorithms (Fredendall and Lea, 1997; Hsu and Chung, 1998; Aryanezhad and Komijan, 2004 and Komijan et al., 2009). However some of the main drawbacks still exist in their method. Mainly, most of these algorithms are validated based on simple examples and may not be very efficient when dealing with large- scale or real-world operations in a job-shop system. Operation scheduling in a job-shop system is generally complicated. Linhares (2009) criticized the current algorithms and claimed that an efficient and optimum heuristic is simply impossible.
In this study, we investigate inefficiency of the recent algorithms, and demonstrate some of the fundamental factors that have not been considered in them. In the next step, we develop an algorithm under TOC approach to create MPS. In this unique method, identification of constraints is explicitly different from the current procedure within all algorithms. We demonstrate that constraints are not just those resources that do not have adequate capacity to meet the demand.
In order to conduct a comprehensive study, we implement the proposed model in a production line inspired from a real case that has a great deal of complexity and similarity to most job-shop systems in the real world. Setup times and different processing times are incurred in this system. More details about the case and its exclusive features will be illustrated later.
2. Research methodology
The MPS planning is developed based on the first two principles of the TOC. In situations where there are complexities, it may be difficult to create an efficient MPS. In the following conditions, complexity may occur (Fox et al., 1986):
One constraint feeds another constraint.
There are a number of setups in the constraints and many parts use them.
There is a significant difference among the production lead times (from the constraint
to the end of the line) for products.
The different parts of one product need to use the constraint.
In our case study, we faced not only the above complexity conditions, but also the following situations:
The difference between available capacity and required capacity for some of the
resources (machines) was not significant (this may create constraint shiftiness). The sequence of operations showed that most of the machines (constraint and non-
constraint) were used by all products.
Processing times and setup times were different for the same operation of different
products.
We consider complex situations in operations such as different parts of one product need to use the constraint, or a number of setups need to be done in the constraints for many products. Without careful consideration to these situations which are common in real-world operations, the accuracy of MPS is questionable. This is one of the crucial drawbacks of the current methods in the literature that we address in our method. We define the key factors which play a vital role to create an accurate MPS, and take them into consideration for the model development. A function is defined and scores are assigned to all resources based on a decision making model. Simply, these scores expose the potential of resources to act as bottlenecks. The highest score shows the primary bottleneck. This is a unique manner in creating MPS. Then, via several steps, the best combination of products is finalized as MPS.
2.1 MPS Design
We propose our unique algorithm in this section. We emphasize that all resources may play the role of constraints; even those that might show excess capacity; i.e. the difference between the available capacity and required capacity to meet the demand is positive. There are n available resources, ri (i=1 to n) is a resource belonging to the resource constraints set with a degree of caution from [0 to ∞]. The higher degree means that the resource can have more effect on the operations as a bottleneck. The highest degree is the primary bottleneck. In this algorithm, we take into account the number of setups and the sequence of operations as well as the capacity requirement to determine bottlenecks. The method’s steps are illustrated in two phases as follows:
Phase 1: Identify the system’s constraint(s):
In this phase, via several steps, the bottlenecks and their priority are determined.
These steps are as follows:
a) Calculate the following ratio for all resources
* (-100)
b) Count the number of products that use ri.
c) Count the number of times a product uses ri to produce one unit of a final product.
Follow this step for all products
d) Add all above scores
e) Normalize them based on the highest score achieved in part d
f) Add the results of parts e and a
g) Divide the results of part f into the absolute value of the summation of the results
in part f. The normalized scores which are greater than 0 are the membership
value of ri for (i=1 to n). The highest membership is the primary bottleneck.
h) Sort the machines in descending order of the ratio obtained in part g.
Phase 2: MPS Design
In this phase, MPS is developed based on the bottlenecks and improved via
several steps. i) Consider the first machine as the main bottleneck (B); develop a feasible MPS
through the following steps which have partly been inspired by the algorithm of
Fredendall and Lea (1997):
i.1) Sequence products in non-decreasing order of Ri ratio which is defined as .
In the event of a tie, give priority to the product with the higher CMi.
i.2) Develop the initial feasible MPS by allocating market demand of products
(Pi) in the above order. Make sure the MPS is feasible at all stages, i.e. there is
always enough capacity on all critical machines identified in step (a).
i.3) Calculate total throughput (TP) of the initial MPS: .
i.4) Verify that the total throughput can be improved by trading-off
production quantities (Pi) between higher and lower ranked products in the
sequence.
i.4.1) Set i=1; examine potential trade-offs between Pi and Pj; j=i+1,…,n.
Accept those trade-offs that improve throughput.
i.4.2) Set i=i+1.
i.4.3) If i i.5) Consider the resultant MPS as a final solution. We will explain the proposed algorithm by an example in the next section. 3. Case Study We applied MPS scheduling logic to an automotive part manufacturer that operates based on a job-shop system. We selected the operations of five products, A, B, C, D and E which have 2000, 5000, 2000, 3000 and 3000 units in demand, respectively, for one month as the period of production. Product B consists of two types of raw materials, B1 and B2. Figure 1 shows the different operations routings of the products. The processing times and capacities are shown in Table 1. Figure 1. Operations Routings of the Products Table 1. The Processing Times and Capacities 3.1 Proposed Algorithm Implementation Phase 1: The results of the first phase of the proposed algorithm are shown in Table 2. This algorithm considers the first machine in the priority list as the primary bottleneck. Table 2. Identifying the Bottlenecks- Results of Phase 1 Implementation Phase 2: The MPS is created and improved in several steps. Table 3 shows a group of critical machines or bottlenecks based on the priority resulting from Phase 1. The Priority Index for products based on machine 4 is shown in Table 4. Table 3. Considered Resources as Possible Bottlenecks Table 4. Priority Index for Products based on Machine 4 Contribution Margin Priority index (PRij) Product (CMi) M4 A 1,500 1500 B 2,200 2200 C 3,000 6000 D 1,700 3400 E 2,800 1400 Total 14500.00 The next step is to create the initial MPS. The results are shown in Table 5. Table 5. Initial MPS Trade-off analysis is performed in the next step after the initial MPS development to determine whether throughput can be improved. The results of rounds 1 and 2 of the trade-off analysis are shown in Tables 6 and 7. Table 6. Trade-off Analysis- Round 1 Table 7. Trade-off Analysis- Round 2 Table 7 shows the best MPS resulting from the proposed algorithm. The next step is to evaluate the proposed model and its capability. 4. Comparison and Validation Ultimately, to prove the capability of the proposed algorithm, the MPS for the above complex job-shop system, inspired from a case in auto industry, is generated using the proposed algorithm as well as a benchmark algorithm. We considered the well-known algorithm of Fredendall and Lea (1997) in the literature as a benchmark. After following the MPS development based on their algorithm, a new MPS is developed and the results are shown in Table 8. Table 8. MPS Development based on Fredendall and Lea Algorithm (1997) The results are then evaluated and compared by a simulation technique. A discrete event simulation model of the proposed model was developed for the purpose of experimentation. Generally, a discrete event simulation model of a system is constructed by defining the events that can occur and then modelling the logic associated with each event to capture the changing status in the system. We used Arena software version 12 to model and analyze the dynamic of the system. To find an optimal or a very satisfactory solution based on the best set of input variables, we used OptQuest optimization software. OptQuest overcomes the above limitation by automatically searching for optimal solutions within Arena simulation models. Without an appropriate tool, finding an optimal solution for a simulation model generally requires that you search in a heuristic or ad hoc fashion. This usually involves running a simulation for an initial set of decision variables, analyzing the results, changing one or more variables, re-running the simulation, and repeating this process until a satisfactory solution is obtained. This process can be very tedious and time-consuming, even for small problems, and it is often not clear how to adjust the variables from one simulation to the next. All MPS were implemented in the production line, and the results of simulation are shown in Table 9. Table 9. Simulation Results and Comparisons MPS Simulation Results Total Throughputs Fredendall Fredendall Proposed Proposed Fredendall and Proposed Products and Lea and Lea Algorithm Algorithm Lea (1997) Algorithm (1997) (1997) C 1375 2,000 850 847 $8,961,400 $9,751,800 D 3000 2,500 1400 1452 B 5000 5,000 1160 1392 A 0 0 0 0 E 3000 2,125 500 600 The results show that the developed algorithm could accurately identify the main bottlenecks and create an efficient MPS. The role of defined factors in modelling was salient. The proposed algorithm could provide more throughputs than the Fredendall and Lea (1997) algorithm. The output and profit of this MPS is very satisfactory. However, the performance of the current algorithm shows more deviation from the expected output of the production line. 5. Conclusion Product mix optimization is one of the main challenges in production management. We defined the key factors which play a vital role in creating an accurate MPS for real-world operations. We demonstrated that the common constraints consideration in a production system based on TOC may not be appropriate for a job-shop system in the real-world. We have included the sequence of operations, number of setups, and number of products using resources for MPS development. Moreover, we considered how products utilize resources in the proposed algorithm. This is one of main contributions of this research, to consider key factors for MPS development. We developed a new and unique approach to identify constraints and create an efficient MPS. Furthermore, the research addresses the complexity of creating an efficient MPS, and creates new avenues for future research. 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