An optimal batch size for a production system under linearly increasing time-varying demand process

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1 Computers & Industrial Engineering 4 (00) 35±4 An optimal batch size for a production system under linearly increasing time-varying demand process Mohd Omar a, *, David K. Smith b a Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia b School of Mathematical Sciences, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, UK Accepted 1October 001 Abstract In this paper we consider a manufacturing system which procures raw materials from suppliers and processes them to make a nished product. The problem is to determine an ordering policy for raw materials and a production policy for the nished product to satisfy a deterministic time-varying demand process. We present an optimal solution procedure, using a dynamic programming approach, and also two heuristic procedures. These procedures are illustrated with numerical examples. q 00 Elsevier Science Ltd. All rights reserved. Keywords: Batch size; Manufacturing system; Time-varying demand; Optimisation; Heuristic; Dynamic programming 1. Introduction In many manufacturing systems, the quantity of raw materials (QR) needed in production is dependent on the production size (QM). Therefore, is often desirable to consider QR and QM simultaneously. This could be done by treating purchasing and production in a single model. Sarker, Musta zul Karim, and Anwarul Haque (1995) have shown that such a joint ordering policy provides lower costs under certain conditions compared with policies which separate ordering and production. Sarker and Khan (1999), Sarker and Parija (1994) and Sarker et al. (1995) developed a variety of models for this system under continuous supply and a constant rate of demand. However this assumption is very restrictive during the growth and decline phases of the product life cycle where demand is either increasing or decreasing with time. In this paper we extend the above models to satisfy a deterministic linearly increasing time-varying * Corresponding author. addresses: mohd@mnt.math.um.edu.my (M. Omar), d.k.smith@ex.ac.uk (D.K. Smith) /0/$ - see front matter q 00 Elsevier Science Ltd. All rights reserved. PII: S (01)

2 36 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 Fig. 1. Plot of stock of nished product against time. demand process. For each batch we determine an economic manufacturing quantity and an order policy for raw material, which together minimize total cost. We consider the case where the supply of raw material and demand for nished product are continuous. We apply a dynamic programming approach that gives an optimal policy. We also develop two heuristic procedures. The rss based on Phelps' approaches (Phelps, 1980) where the period between consecutive orders is constant. The second is an extended Silver±Meal heuristic procedure (Silver, 1979).. Mathematical formulation In this section, a general cost model is developed by considering QM and QR. Usually the following costs are considered: ² raw material ordering cost, ² manufacturing set-up cost, ² raw materials inventory carrying cost, ² nished producnventory carrying cost. To develop the model, the following terminology is used: ² The demand rate of nished product at time n (0, H) isf(t). H is the time horizon after which no demand will be met. ² The nite production rate is P units per unit time and P. f(t) for all t. ² There is a xed manufacturing set-up cost of c p for each production run. ² There is an ordering cost of c j for raw material j. ² There is a carrying inventory cost of h p per unit per unit time for nished goods. ² There is a carrying inventory cost of h j per unit per unit time for raw material j. ² QM i is the production quantity of the (i 1 1)st batch. ² QR ij is the raw material quantity for raw material j of the (i 1 1)st batch. ² n is the total number of batch replenishment (and therefore we de ne t n ˆ H). ² r j is the amount/quantity of raw material j required in producing one unit of a product.

3 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 37 Fig. 1shows the graph of nished product against time for the (i 1 1)st batch. Fig. represents the raw material level during the production period. We assume that during production nished product becomes immediately available to meet the demand process. If the input rate (P) is in nite then the stock of nished producs described by the curves y 1 t for n ; 11 : If the input rate is nite then the stock level is given by y t for n ; t p i and y 1 t for n t p i ; 11 : For an in nite P the time-weighted stockholding for the batch is area A i whereas for the nite P is area C i. The area of B i is the difference between A i and C i. D ij is the time-weighted stockholding for raw material j for the (i 1 1)st batch. From these de nitions, we arrived at y 1 t ˆ Z 1 1 f t dt Zt f t dt ˆ Z 1 1 t f t dt # t # 11 : 1 Zt y t ˆP t f t dt # t # ti p : It follows that Fig.. Plot of stock of raw material against time. A i ˆ Z 1 1 y 1 t dt: 3 B i ˆ Zt p i y 1 t y t Šdt: 4 Therefore the inventory carrying cost of nished products for the (i 1 1)st batch is h p A i B i : Let y 3 (t) be the level of raw material during the production period. For a raw material j, we get Z 1 1 y 3 t ˆr j f t dt P t ; # t # ti p : 5

4 38 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 It follows that the total time-weighted raw material for this batch is given by D ij ˆ Zt p i y 3 t dt: 6 In order to satisfy the demand during ; 11 ; we have Pti p Z 1 1 ˆ f t dt: 7 3. The dynamic programming formulation For simplicity, we only consider one type of raw material j ˆ 1 : De ne the single stage cost (SSC) of meeting all demand from time to time 11 by a single production at time, when stock levels of both the nished product and the raw material are zero, as SSC ; 11 ˆc p 1 h p A i B i 1 c 1 1 h 1 D i1 : 8 Let C n; t be the minimum total cost of meeting all future demand, initially with no stock of nished product and raw material at time t with exactly n batches. The dynamic programming formulation for this problem is C 0; H ˆ0; C 0; t ˆ1 0 # t # H; C n; t ˆ min {SSC t; x 1 C n 1; x }; 9 t#x#h where n ˆ 1; ; 3; ¼; 0, t, H: The solution to Eq. (9) proceeds by calculating values C n; t backwards in time for a nely-graded spectrum of values of n [0,H]. C(n,0) is the optimal total cost for n-batch policy. For a linearly increasing demand rate over the interval (0,H), we have f t ˆa 1 bt a. 0; b. 0; 0 # t # H: 10 Substituting Eq. (10) into Eqs. (1), () and (5) we obtain y 1 t ˆa 11 t 1 b t i11 t ; y t ˆP t a t b t ti y 3 t ˆr 1 a 11 1 b t i11 ti P t ; and so Eqs. (3) and (4) give A i ˆ 11 a 1 b : 11 ;

5 Zti p B i ˆ a 11 1 b t i11 ti P t dt ˆ a 11 ti p M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 39 b 1 t i11 ti We found that D i1 ˆ r 1 B i : It follows that: ti p P tp i ˆ 11 P a 1 b 11 1 : 1 SSC ; 11 ˆc p 1 h p A i B i 1 c 1 1 h 1 D i1 ( t ˆ c p 1 c 1 1 h i11 p a 1 b p a 1 b h!) 13 1r 1 : Thus we can determine the total cost of the optimal n batch policy, C(n,0), together with the corresponding production schedule. We evaluate C(n,0) for different values of n in order to determine the optimal policy. Fig. A1of Appendix A shows the variation of C(n,0) with n for several sets of parameter values. In each case, the function is convex. Is relatively easy to prove by calculus that C(n,0) is convex for situations where the reorder points are equally spaced. This convexity result also applies when the reorder points are varied a small amount from such equal spacing, as is true for smooth functions f(t) in this work. The argumennvolves co-ordinate geometry in 3n dimensions and is outside the scope of this paper. Assuming convexity, n p is the optimal batch size if C n p ; 0 # C n p 1; 0 and C n p ; 0 # C n p 1 1; 0 : However, it would be desirable to use simpler procedures instead of the dynamic programming approach, if the cost penalty of so doing were insigni cant. In Section 4, we consider two heuristic approaches. The total relevant cost (TRC(n)) of the model in n such batches is ( ) n X 1 TRC n ˆn c p 1 c 1 1 h p A i B i iˆ0 n X 1 1 h 1 iˆ0 D i1 : h p Heuristic procedures In this section we extend Phelps' approaches which are similar to the policy T; Q i in Omar (1998). We also extend the Silver±Meal heuristic procedure (Silver, 1979). Again, we assumed j ˆ 1: 4.1. Heuristic 1: T; Q i policy In this policy, batches are made at xed and regular intervals of time T. For n replenishments, we have T ˆ H=n and ˆ ih=n; i ˆ 0; ¼; n:

6 40 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 Using Eqs. (13) and (14) we obtain TRC H1 n ˆn c p 1 c 1 1 h ph n X ( a 1 b 3 3i 1 H 1 n p a 1 b i 1 1 H ) n n 1 iˆ0 1 h 1r 1 H Pn a 1 bh i 1 1 n ˆ n c p 1 c 1 1 h ph ( n an 1 bh 3n 1 1 an " # 6 P a 1 bh b H ) 1n 4n 1 1 h! 1r 1 H Pn an 1 abhn 1 b H n b H : 3 1n 15 where TRC H1 (n) is the TRC under Heuristic 1with n replenishments. The convexity of the function TRC H1 (n) can be proved for f(t) a linear function. By treating TRC H1 (n) as a continuous function of n, following Phelps' approach (Phelps, 1980), then for a linear increasing demand, the second derivative of TRC H1 (n) with respect to n is positive. 4.. Heuristic : extended Silver's heuristic procedure Silver (1979) modi ed the Silver±Meal heuristic to produce an approximation for the case of continuous linear increasing demand over a nite time horizon. He determined each lot size sequentially, one at a time, by minimizing the total cost per unit time. However this procedure needs some adjustment particularly when there is a well-de ned ending poinn the demand pattern. From Eq. (13), the total relevant cost per unit time (TRCUT) for a batch made at time 0 which meets all demand of nished product and raw material up to time T is TRCUT T ˆ 1 ( T c p 1 h pt " a 1 bt 1 3 P a 1 bt # 1 c 1 1 h 1r 1 T a 1 bt ) : P 16 The necessary condition for TRCUT(T ) to be minimum is that d TRCUT T Š=dT ˆ G T ˆ0: If f t ˆ a 1 bt; this gives G T ˆ " ah # p a h p a h 1 r 1 T 1 P " # bh p abh p abh 1 r 1 T 3 1 3h 1r 1 b 3h p b 3 P 8P T 4 c p 1 c 1 : (17)

7 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 41 Table 1 Comparison between optimal method and heuristic procedures Op. method Heuristic 1Heuristic b n OC n SC Pen. n SC Pen When b. 0, the root of Eq. (17) is unique and is a global minimum. Is easier to solve Eq. (17) by numerical methods. 5. Numerical example To demonstrate the effectiveness of these methods, we present four numerical examples. For these examples, demand is linearly increasing from slower to faster where b takes values of 5, 10, 15 and 0. The other parameters are a ˆ 6; b ˆ 10; c p ˆ 300; h p ˆ 15; c 1 ˆ 50; h 1 ˆ 5; r 1 ˆ 1; P ˆ 1000 and H ˆ 5: For example when b ˆ 10; from a dynamic programming method, an optimal number of batches is 4 with the production starting times 0, 1.660,.8978 and The total cost for this optimal policy is Similarly, Heuristic 1gives the same number of production batch. However, the minimum total cost for Heuristic 1 is , which is.07% high than the optimal cost. The best batch production starting times from the Silver approaches are at 0, ,.5460, and with the minimum total coss Table 1 gives the full result of these examples. In Table 1, OC means Optimal Cost, SC means Solution Cost from heuristic procedures and Pen. is the percentage cosncrease. n is a corresponding optimal number of production schedule. From these results, the Heuristic 1is superior then the Heuristic for slower cases. On the other hand, the Heuristic gives very good solution cost when b is equal to 15 and Conclusions In this paper we have extended the constant demand model to the case for a linearly increasing timevarying demand process. We have used a dynamic programming approach to obtain the optimal solution. We also proposed two heuristic solution procedures with relatively low penalty cost with all our examples. Furthermore, these methods are easily adapted to other demand patterns such as linearly and exponentially declining. AppendixA See Figs. A1and A.

8 4 M. Omar, D.K. Smith / Computers & Industrial Engineering 4 (00) 35±4 Fig. A1. Plot of the total cost against number of replenishment. Fig. A. Plot of the total cost against number of replenishment (Heuristic 1). References Omar, M. (1998). Production and inventory modelling for time-varying demand processes, PhD Thesis, Department of Mathematical Statistics and Operational Research, University of Exeter, United Kingdom. Phelps, R. I. (1980). Optimal inventory rule for a linear trend in demand with a constant replenishment period. Journal of the Operational Research Society, 31, 439±44. Sarker, B. R., & Khan, L. R. (1999). An optimal batch size for a production system operating under a xed-quantity, periodic delivery policy. Journal of Computers andindustrial Engineering, 37, 711±730. Sarker, B. R., & Parija, G. R. (1994). An optimal batch size for a production system operating under a xed-quantity, periodic delivery policy. Journal of the Operational Research Society, 45 (8), 891±900. Sarker, M. R. A., Musta zul Karim, A. N., & Anwarul Haque, A. F. M. (1995). An optimal batch size for a production system operating under a continuous supply/demand. International Journal of Industrial Engineering, (3), 189±198. Silver, E. A. (1979). A simple replenishment rule for a linear trend in demand. Journal of the Operational Research Society, 30, 71±75.