Some New Properties for Capacitated Lot-Sizing Problem with Bounded Inventory and Stockouts

Some new properties for capacitated lot-sizing problem with bounded inventory and stockouts

Xiao LIU 1,2 , Feng CHU 1 , Chengbin CHU 1 , Chengen WANG 2 1. Laboratorie d’optimisation des systemes industriels, Universite de technologie de Troyes,12 rue Marie Curie – BP 2060, 10010 Troyes Cedex, France 2. Shenyang Institute of Automation, Chinese Academy of Sciences,China

Abstract: We formulate the single-item inventory capacitated lot size model with lost sales. Unsatisfied demand cannot be backlogged, which means lost sales. Costs are assumed to be time variant. Some new properties are obtained in an optimal solution and a dynamic programming algorithm is developed to solve the problem in O(T 2 ) time.

Key-Words: lot sizing; Dynamatic programming; stockouts; inventory/production

1. Introduction cases. The proposed solution methods are mainly The Single Item Lot Sizing Problem (SILSP) is a based on dynamic programming (such as Kirca 1990; planning problem in which there is time-varying Chen et al.1994; Shaw and Wagelmans 1998), demand for a single production over T periods. The branch and bound (such as Baker et al. 1978), or a objective is to determine those periods where combination of the two (such as Chung et al. 1994). production will take place and the quantities that For many process industries, some production have to be produced. Wagner and Whitin (1958) processes in practical applications (such as oil introduced the first uncapacitated single item lot- drum/barrels), production capacity is high enough, sizing model without backlogging. The demand in but the reorder quantities are restricted by inventory each period must be entirely delivered on time by capacity instead of production capacity. Typically, production or/and by inventory from previous such processes are dedicated to a single product at a periods. They proposed an O(T 2 ) dynamic time, so this kind of production planning is single item lot-sizing problem with bounded inventory. programming algorithm with constant set up cost and Such manufacturing processes can be found in many linear production and holding cost. The above process industries including paper manufacturing, O(T logT) problem was shown to be solvable in time food processing, petrochemical and pharmaceutical by Aggarwal and Park (1993), Federgruen and Tzur industries. Since these products are usually (1991), and Wagelmans et at (1992). Since then, commodities, short term demand is known. In our there has been extensive research by allowing surprise, there are few literatures with inventory backlogging and with more general cost functions. capacity. Love (1973) presented an O(T 3 ) dynamic Capacitated lot-sizing model without backlogging programming algorithm with concave production and problem is very difficult. Florian et al. (1980) and holding costs for the inventory capacity lot-sizing Bitran and Yanasse (1982) showed that the general problem with backlogging. Gutierrez et al.(2002) capacitated single item lot-sizing problem is NP- 3 4 proposed an O(T ) dynamic programming algorithm hard, Florian and Klein (1971) proposed an O(T ) with constant inventory capacity and concave dynamic programming algorithm with constant production and holding costs. This is a special case production capacity and concave production and of the model studied by Love (1973) and the holding cost. Hoesel and Wagelmans (1996) algorithm was 30 times faster than Love’s procedure. O(T 3 ) improved the above algorithm into time In reality, if a company procures large amounts to complexity by assuming constant capacity, concave avoid setup cost, holding cost may be very high production costs and linear holding costs. Swoveland when inventory will have to be carried for many (1975) examined piecewise concave production and periods, or backlogging is not permitted for holding/backlogging costs, and developed an perishable products. The company will decide algorithm similar to that of Florian and Klein. whether or not to satisfy the demand in a period. Polynomial algorithms also exist for many other Unsatified demand means lost sales (lost revenue). A production capacity lot-sizing problems with special part or whole demand of a period must be lost when

1 backlogging is not permitted or non profitable. 2. The mathematical model Sandbothe and Thompson (1990) considered a The following notations will be used in the paper: special case of lost sales (stockouts). A part or whole T number of periods indexed from 1 to T demand of a period must be lost when the inventory K t setup cost of production/procurement in period t is zero. They proved necessary conditions for an pt optimal solution with non time-varying linear unit production/procurement cost (nonincreasing over time) in period t production and holding costs and an O(T 3 ) algorithm ht unit inventory holding cost in period t when production capacity is constant, and an O(2T ) s lost sale cost per unit in period t algorithm for the model with time-varying t d t production capacity for the first k 1 period with a t demand in period computable k . Furthermore, in 1993 they extended St inventory capacity (nondecreasing) at the their research with limitations on production capacity beginning of period t T and storage capacity in O(2 ) time complexity and X t production/procurement quantity in period t gave some new results. Based on the results of I t inventory level at the end of period t Sandbothe and Thompson, Deniz Aksen et al.(2003) Lt amount of unmet demand in period t considered an uncapacitated lot-sizing problem with yt a binary variable indicating whether period t is a lost sales. Setup costs are constant. Production and 1 if X t  0 holding costs are time-varying. They proposed an production period; yt   0 otherwise O(T 2 ) dynamic programming algorithm when lost  X L I sale costs are time-varying. There may be here a sale where t , t are decision variables and t is state loss in a period even if the inventory level is strictly variable. We make the following assumptions: positive, and the period is a conservation period. This 1) Any unmet demand in its period is case is possible when loss cost is very lower at one considered lost; s  p of the periods, loss of demands in positive inventory 2) t t , without loss of generality, sale loss period can be profitable to the total system cost. cost is larger than production cost, t 1,....,T ;

In this paper, a single item lot sizing problem with 3) d t  St which means that demands are less bounded inventory and lost sale is considered. Since than the storage capacity at the same period and are the rule of meet all demands may lead to excessive withdrawn at the ends of periods.; costs, it is cheaper to permit lost sales. In order to get 4) I 0  IT  0 which assumes that the initial maximum profit, we consider a new lost sales model inventory level and the holding level at the end of with time-varying costs during T period the planning period are 0. horizon, and lost sales are allowed in some periods. T I This situation has been considered by few authors, The inventory level t is given by t but it always happens in practical applications. We I  I  (X  d  L ) t 0  j j j assume that unsatisfied demands are lost sales and j 1 that demands cannot be backlogged. For speciality of (1) some constant techniques and production processes in processs industries, we focus our attention on the We assume that the production capacity is special case in which the production costs are non- unlimited. The single item inventory capacitated lot increasing. Inventory capacity is non-decreasing. We sizing model with lost sales can be formulated as follows. present and prove several properities in optimal T solutions. Only stockout periods are existence when Minimize (K t yt  pt X t  ht I t  st Lt ) (2) 2  lost sale cost is non-increasing. An O(T ) algorithm t 1 is developed to solve this problem. subject to This paper is organized as follows: in section 2, It 1  X t  (dt  Lt )  It t 1,....,T (3) we define the notation, and the lost sales model will be developed. In section 3, several properties of an Lt  d t t 1,....,T (4) optimal solution are proved. Furthermore, a new I  X  S t 1,....,T forward recursive dynamic programming algorithm t 1 t t (5) is presented to solve lost sales problem in O(T 2 ) in I 0  I T  0 section 4. Finally, a short conclusion and future (6) research directions are discussed in section 5. X t  S t yt t 1,....,T (7)

2 X t  0, I t  0, Lt  0 , yt {0,1} t 1,....,T (8) (1) Directed arcs (M,P) and (M,L) which ship flows at zero cost and have upper bounds of From (3) and (5), we obtain   ; (2) Directed arcs (P, t ) with lower bounds of 0  I  S  d  L t 1,....,T t t t t (9) X t  1,....,T min t for , which have zero The constraints (3) provide the material balance shipping cost for zero arc flow and a cost of equations which determine the inventory levels from K t  pt  X t if the arc’s flow X t  0 ; the previous decisions. The constraints (4) require (3) Directed arcs (L, t ) with upper bounds of that any lost demand in that period cannot exceed the d t for t  1,....,T , which ship flows at a cost demand of that period. The storage capacity is accomplished by (5). Without loss of generality, the of lost sales st per unit flow; constraint (6) assumes that both the inventory level at (4) Directed arcs ( t , t 1 ) with upper bounds the beginning of the first period and the inventory of S t for t  1,....,T , which ship flows at a level at the end of the last period are zero and cost of holding ht per unit flow; backorders are prohibited. The next constraint in (7) is the binary restriction that tracks any production or T T d X d procurement activity in that period. Finally, (5) Directed arcs with flows t ,  t ,  t , constraint (8) requires the reorder quantities and the t1 t1 inventory levels to be nonnegative. P We formulate our lot sizing problem as a concave network flow problem. It is similar to Sandbothe and T  X t Thompson (1990). First, we begin by making several t 1 X X X X definitions: 1 2 3 T

X  0 t T Definition 1. If t , the period is a production  dt t 1 I I I I period. M 1 1 2 2 3 3 …T- T

I t  0 t d d d 1 d Definition 2. If , the period is a holding 1 2 3 T period.

Definition 3. If I t  0 and Lt  0 , the period t is a T L  L L T t L L 3 stockout period. t1 1 2

Definition 4. If I t  0 and Lt  0 , the period t is a conservation period. L X Definition 5. Define mint ' as the minimum ' production quantity if t is a production period. In Fig 1. Lot sizing problem with stockouts as a network flow problem our model, only when Kt '  pt '  X t '  st ' X t ' i.e. T (s  p )  X  K t ' t ' t ' t ' profit will be obtained by setting L  t have zero unit costs. X  K / s  p t1 up a production. So mint '  t ' t ' t '  , where X  is defined to be the smallest integer larger than or All costs are concave and all constraints incur equal to X . lot-sizing problem are linear. So, our network is j concave cost network. D  d Definition 6. Define i, j  k , which denotes the k i accumulated demand from period i to period j . 3. Properties of optimal solutions The network representation is given in Fig.1. M In this section, some properties are stated and proved is the master supply node containing the total as follows. demand of T periods. P and L are two trans-shipment nodes, which will respectively trans-ship all the Theorem 1. In an optimal solution, if X t  0 , then X  X satisfied demands in each period via production and t mint . all the unsatisfied demands in each period via lost sales. Nodes 1, 2, … T are the period demand nodes Proof: Assume that there is an optimal solution X  X X showing the actual satisfied demand of each period. t mint , according to the definition of mint , we We can observe that costs and bounds on all arcs of have K t  pt  X t  st  X t . This means it is cheaper network.

3 to lose X t units than to produce them, so it is a Proof: Assume that there is a period i , Li  0 , where ' contradiction. t  i  t . It implies the period inventory level I i  0 .

We have Ii  Li  0 , which is a contradiction with Theorem 2. In an optimal solution, Lt  I t  0 for all t . theorem 2.  X ' Proof: Assume that in an optimal solution, Lemma 2. If t ',t is a quantity of production from t ' ' Lt  It  0 for some t . There is a path from node M to t , for each t and t , where 1  t  t  T . In an X  0 X  min{S , D } to node t 1 via nodes L and t at cost st  ht ; optimal solution, if t ',t , then t ',t t ' t ',t Another path via node L using arc (L,t 1) has cost . st1 such that the lower cost solution can be obtained. Proof: By contradiction, let X t',t be a quantity such 0  X  D  S Since by assumption lost sale cost st is non- that t ',t t ',t 1 t ' . One can know that stockouts will be occurred before t , which increasing, thus we have st  ht  st 1 , it is a D  X  S  d contradiction, so the original solution was not contradicts lemma 1. Now if t ',t 1 t ',t t ' t ',t , optimal.  given that dt  I t1  0 , by theorem 3, X t  0 and

According to this theorem, one can know that if hence It  It1  dt  Lt , and by theorem 2, I  0 t all the demands should be met, i.e. no lost Lt It  Lt (It1  dt  Lt )  0 . Therefore, either Lt  0 , sales occur. If and only if the inventory level is less which yields I to be negative (infeasible), or than the demand of that period, lost sales occur. t Therefore, a lost sale problem becomes a stockout Lt  dt  It1 , which represents a feasible decision. case, in other words, there are no conservation According, it is clear that the cost can be reduced by periods. increasing production arc (P,t ' ) till production S D  X  S  D Theorem 3. In an optimal solution, It1  X t  0 for quantity is maximum t' with t ',t 1 t ',t t ' t ',t . t all t . p  h  s L  d  I Only when t '  k t , and hence t t t1 . Proof. Assume that there is a period t with k t' On the other hand, if Dt ',t1  X t',t  Dt ',t  St ' , it can be It1  X t  0 . Then costs can be reduced by t p  h  s increasing production in period t with I t1 and easily proved that if t '  k t then the cost of k t ' decreasing production in period ' with this amount, t X  D ' original plan is reduced by ordering t',t t',t in where t is the previous production period. Then ' period t , and It  Lt  0 . Otherwise, if holding costs of arc (t ' ,t 1) are reduced and Dt ',t  X t',t  S t' then it will have a solution at t 1 p production costs are reduced, because t is non- periods holding positive inventory, which contradicts X  I  S increasing over time. Owing to t t 1 t by our theorem 3. model, a lower cost feasible can be obtained till Lemma 3. In an optimal solution, if It '1  It '  0 for I t1  0 . This contradicts the initial assumptions.  ' X  0 L  d According to this theorem, the original lost sales any t , we have either t' and t ' t ' ; or problem can be decomposed into consecutive X t '  dt ' and Lt'  0 . subproblems g(t ' ,t) , where t ' is a production period, Proof: Since I t '1  I t '  0 , we have the cost of period and t is the next first zero holding period with ' K  p X  s L L  d  X ' t is t ' t ' t' t ' t ' , where t ' t ' t ' . If 1  t  t  T , i.e. It'  It  0 and for each Ii  0 , ' ' K t '  pt ' X t '  st ' X t ' , then minimum cost will be got at i  t ,t 1,...t 1. The problem is minimizing the ' upper bound of production arc (P,t ) with X t'  dt' total production, holding and lost sale cost from ' to t L  0 ' and t ' ; otherwise maximum can be obtained at t . In what follows, we show how to minimize g(t ,t) ' upper bound of lost sale arc (L,t ) with Lt'  dt' and . For this purpose, we establish some properties. X t'  0 if K t '  pt ' X t '  st ' X t ' . Lemma 1. In an optimal solution, if there is a period From this lemma, we can get minimun costs i ' L  0 ' ' , t  i  t , i , for the production period t , if a either K t '  pt ' d t ' or st ' d t ' when t is not the first zero stockout occurs , we have i  t , i.e. a stockout occurs holding period. at the last period of that subproblem.

4 4. Algorithm 5. Conclusions Using the above properties, the number of decision In this paper, we consider the extension of the states per period is reduced. We now show that our standard (Wagner-Whitin) lot-sizing problem. It is model can be solved with a forward recursive assumed that inventory is bounded in each period. dynamic programming algorithm in polynomial time. We assume that backlogging is prohibited, and any By definition, t ' is a production period and t is unsatisfied demand is lost. The objective is to the first zero inventory period, 1  t '  t  T , the total minimize the total cost including setup costs, costs denoted as g(t ' ,t) . production costs, holding costs and lost sale costs. t 1 We discussed the special case of lost sales problem g(t ' , t)  K  p X  h (X  D  D ) t' t ' t ',t  k t ',t 1,k 1,t '1 when production cost and lost sale cost are non- k t' increasing, some new properties have been provided  s (D  D  X ) t 1,t 1,t '1 t ',t in an optimal solution. These new properties translate (10) lost sales problem into stockouts case. Bases on these new properties, we have demostrated that the For the sake of simplification, we use the 2 following notation: proposed algorithm runs in O(T ) . Some numerical results demonstrate that the approach proposed for k H   h lost sale problem is efficient and applicable. Future k t t  1 research will focus on conservation cases. (11) k M   h D References: k t 1,t (12) t  1 [1] Wagner HM, Whitin T. Dynamic version of the In fact (10) can be transformed as follows. economic lot size model. Management Science 1958; 5:89-96. ' g(t ,t)  K t '  pt ' X t ',t  (X t ',t  D1,t '1 ) [2] Florian M, Klein M. Deterministic production (13) planning with concave costs and capacity (H t 1  H t '1  st )  M t1  M t '1  st D1,t constraints. Management Science 1971; 18(1): Therefore, all g(t ' ,t) ’s can be computed O(T 2 ) . 12-20. If f (t) denotes the minimal total production, [3] Swoveland C.A deterministic multi-period holding and lost sales costs over period from 0 to t production planning model with piecewise and concave production and holding-backorder costs. Management Science 1975; 21(9): 1007- f (0)  0 (14) 13.

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