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Optimal (s, S) policies with delivery time guarantees for manufacturing systems with early set-up
Computers & Industrial Engineering 40 (2001) 249±257
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Optimal (s, S) policies with delivery time guarantees for manufacturing systems with early set-up Wai Ki Ching Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK
Abstract We consider optimal (s, S) policies with delivery time guarantees for production planning in one-machine manufacturing systems with early set-up. The machine produces one type of product and delivery time guarantee is offered to the customers for each unit of ordered product. The inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. In a (s, S) policy, the machine will shut down when an inventory level of S is attained and once the inventory level drops to s, the machine will re-start. A set-up time is required for the machine. We model the set-up by the exponential distribution. We obtained an analytical form of the steady state probability distribution for the inventory levels derived. The average pro®t of the system can be written in terms of this probability distribution. Hence the optimal (s, S) policy can be obtained by varying different possible values of s and S. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Manufacturing systems; (s, S) policies; Delivery time; Early set-up
1. Introduction In this paper, we study optimal (s, S) policies with product delivery time guarantees for production planning in manufacturing systems with early set-up. We observe that there was a series of shifts in strategic focus in American manufacturing over the past four decades. Firstly the `throughput' was the main concern in 60s because of the increasing demand in manufacturing products. The `cost' became the main concern when there were internal and foreign competitions in 70s. In 80s people focused on `process quality' to improve productivity and gain markets. Now in 90s, `time' is a source of competition. Successful manufacturers are now working to shorten the cycle and delivery time and improve the service level so as to remain competitive. There is a growing trend in using the delivery time guarantee in commercial companies as a marketing strategy to attract customers. For example, the United Parcel Services guarantees the next day delivery by 8:30 a.m. Pizza Hut in Hong Kong offers free pizza if the E-mail address: [email protected] (W.K. Ching). 0360-8352/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 02 4 - 9
250
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
ordered pizza cannot be served within 20 min. Lucky a major supermarket chain in California, use a `three's a crowd' campaign, which guarantees a new checkout counter will be open if there are more than three people waiting in its checkout queue. Wells Fargo Bank offers a `®ve minute maximum wait policy' which offers ®ve dollars to the customer if the customer waits for more than 5 min in line. Numerous papers have studied the relationships between the delivery time, the pricing and the capacity. Li (1992) considered the application of inventory in delivery time competition. Buss, Lawrence and Kropp (1994) studied optimal demand and capacity for the ¯ow time of jobs. Hill and Khosla expressed the demand as a function of delivery time and price. For more related works see for instance Ching (1998), Dewan and Mendelson (1990), Karmarker (1994) and Donohue (1994). However, to our best knowledge, the application of delivery time guarantee as a marketing strategy in manufacturing systems with set-up time seems not yet addressed in literature. In our study, we assume that the delivery time guarantee T and the unit price P are the main factors affecting the demand rate l l P; T of the product. One standard formulation for the relation (Cobb±Douglas) is
l l P; T sP2t1 T 2t2 ;
1
where t 1, t 2 are the price and delivery time guarantee elasticity respectively, and s is a positive constant; see Freeland (1980) and Hill and Khosla (1992) for instance. In the delivery time guarantee strategy, a delivery time guarantee of T on each unit of ordered product is offered to the costumers. In our model, an order can always be served within the delivery time T whenever there are products available in the inventory. Here for simplicity, we consider a one-machine system, extension to multiple machines case is possible. The machine produces one type of product, the inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. Finite backlog of the demands is allowed in the system. A cost of cB is assigned to each unit of backlog and a cost of cI is assigned to each unit of inventory. Using the inventory level as an indicator, our interest is to determine the production strategy such that the average pro®t is maximized. We employ the (s, S) policy as the production (inventory) control. An (s, S) policy is characterized by two integers s and S. The machine keeps on producing until an inventory level of S is reached. Once this inventory level is reached, the production is stopped by shutting down the machine. We allow the inventory level fall to s, (s , S), and re-start the production. The machine requires a set-up time that is assumed to be exponentially distributed. In many manufacturing systems, set-up time is a vital procedure. Moreover, our model can be further extended to multi-item case. In this case, set-up procedure becomes necessary and important. The hedging point policy discussed in Akella and Kumar (1986) is a particular case of the (s, S) policy with S s 1 1: When the optimal hedging policy is a zero-inventory policy (i.e. the hedging point is zero), the policy matches with the just-in-time (JIT) policy. The JIT policies have strongly been favored in real-life production systems for process discipline reasons even when they are not optimal. By using the JIT policy, the TOYOTA company can manage to reduce work-in-process and cycle time, see Monden (1983), pp. 13±34. In this paper, we obtain the analytical form of the steady state probability of the inventory levels under (s, S) production policies. The average pro®t of the system can then be written in terms of this probability distribution, and the optimal values of s and S are obtained for any given unit price P of the product and the guaranteed delivery time T. The remainder of this paper is organized as follows. In Section 2, we present the model of manufacturing system under (s, S) policy. In Section 2.1, we derive a desirable maximum backlog when the guaranteed delivery time T is given. In Section 2.2, we give the balanced equations for the steady state
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
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distribution of the machine-inventory system. Analytic form of this probability distribution is obtained in Section 2.3. In Section 3, the average pro®t for the systems is written in terms of the steady state probability distribution and numerical examples are also given. In Section 4, concluding remarks are given to discuss possible extensions of our model.
2. The manufacturing system In this section, we present a one-machine manufacturing system with product delivery time guarantee. Let us de®ne the following notations: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
m 21, the mean processing time for a demand for one unit of the product, l 21, the mean inter-arrival time for one unit of the product, t 21, the mean set-up time of the machine, O, operating cost for one unit of the product, P (P . O), unit price of the product, T, the guaranteed time of delivery, s , S, the parameters of the (s, S) policy, cI, inventory carrying cost per unit of the product (dollars), cP, penalty cost per unit of the product (dollars), cS, machine set-up cost (dollars), cD, penalty cost per unit of delay product (dollars), m, the maximum allowable backlog, Imax, the maximum allowable inventory.
We employ (s, S) policy as the production control. The processing time for one unit of product by the machine and the inter-arrival time for one unit of demand are assumed to be exponentially distributed. The demand rate of the product depends on the unit price P and the guaranteed delivery time T, and is given by Eq. (1). Backlog of products is allowed in the system. We reject any arrival demand when the inventory level is 2m, i.e. our maximum backlog capacity is m. A penalty cost of cP is assigned to each unit of rejected demand. To determine the optimal values of s and S, we have to set an appropriate guaranteed delivery time. 2.1. Determination of maximum allowable backlog In manufacturing system of limited production capacity, to guarantee customers a delivery time of the product, there should be an upper limit for the maximum allowable backlog m. Here we propose a simple estimation of the maximum allowable backlog m. Suppose we have k backlogged orders, the probability that they can be offered within the guarantee time T should be greater than a pre-determined value g (0 , g , 1). The reliability value (or the service level) g can be 90 or 95% for example. The following proposition gives an inequality relating the quantities g , T and the maximum allowable backlog m. Proposition 1. Given the delivery guarantee time T and the reliability value g , the maximum allowable
252
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
Fig. 1. The Markov Chain of the System
backlog m is the largest positive integer k which satis®es the inequalities: 8 kX 21 j j > T m 2 mk m 2 t j2k mk e2 t2mT > mT > g e $ for m ± t; 1 1 2 > > j! m 2 tk < j0
> > > > > :
mT
1 2 g e
$
kX 21 j0
mTj mk T k e2mT 1 j! k!
2
for m t:
2.2. The inventory levels When the desirable guaranteed delivery time T and the unit price P of the product are decided, the maximum allowable backlog m (Proposition 1) and the demand rate l (cf. Eq. (1)) can also be determined. We then employ the (s, S) policy in the production planning of the manufacturing system. We note that under the (s, S) policy, the inventory levels can take integer values in [2m, S]. When the inventory level reaches S, the machine is shut off. When the machine is shut off, the inventory levels can take integer values in [s, S] and once it falls to s, the machine is turned on and a set-up time is required. During the set-up time, the inventory level can take integer values in [2m, s]. There are (2(S 1 m) 1 1) possible states in the inventory system, see Fig. 1. We order the states as follows. The ®rst (S 2 s) states correspond to the situation that the machine is turned off, and the inventory levels are ordered from S down to (s 1 1). The next (m 1 s 1 1) states correspond to the situation that the machine is in the set-up time, and we order the inventory levels from (S 2 1) down to 2m. The next (m 1 S) states correspond to the situation that the machine is in the operating state, the inventory level takes integer values in [2m, S 2 1]. We order the inventory levels from (S 2 1) down to 2m. We construct the generator matrix for the machine-inventory system. A generator matrix A of a Markov process gives the transition rate [A]ij from state j to state i. Under our
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253
ordering, we obtain the following generator matrix A for the machine-inventory system: 2
l
6 6 2l l 6 6 6 ] ] 6 6 6 6 6 6 6 6 6 6 6 6 6 A6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0
2l
l1t ]
] 2l
t
0
0
m1l 2l
2m
m 1 l 2m ]
2t ] ]
0
2t
]
]
2l
m1l
2m
]
]
]
2l
l1m 2l
3
S .. . .. .
7 7 7 7 7 7 7 7 s 7 7 7 .. 7 7 . 7 7 7 2m 1 1 7 7 7 2m 7 7 7 7 S21 7 7 7 S22 7 7 7 .. 7 . 7 7 7 .. 7 . 7 7 2m 5 2m 1 1 m 2m
The generator matrix A is irreducible, has zero column sum, positive diagonal entries and nonpositive off-diagonal entries. Therefore A has one-dimensional null space; see Varga (1963), p. 30. The steady state probability distribution is the normalized positive null vector of A. 2.3. The steady state distribution of the inventory levels We derive the analytic form of the steady state probability distribution of the inventory levels. We let probability distribution vector p be written as follows: p pS ; pS21 ; ¼p2m ; qS21 ; qS22 ; ¼q2m t : Here qi is steady state probability that the inventory level is i and the machine is operating and pi is steady state probability that the inventory level is i and the machine is shut off or in set-up stage. By letting qS21 1 and direct veri®cation, we have the following proposition. Proposition 2. The steady state probability distribution p of A can be written as follows: p a pS ; pS21 ; ¼p2m ; qS21 ; qS22 ; ¼q2m t
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where
8 > > > > > > > > > > > > > > > > <
pS2i ps2i
m ; l
i 0; ¼; S 2 s 2 1;
mli ; l 1 ti11
i 0; ¼; m 2 s;
qS21 1; > > mi 2 li > > > qS2i i21 ; i 2; ¼; S 2 s; > > m m 2 l > > > > 22 j > > mS2s1i 2 lS2s1i tli21 t l 1 m i21 iX lm > > 2 ; q 2 > : s2i l1t m l 1 t l 1 m l 1 ti mS2s1i21 m 2 l j0
Here a ps 1
PS 2 1
i 1; ¼; m 1 s:
3
i2 m
pi 1 qi is the normalization constant.
3. The average pro®t and numerical examples The average pro®t of the system, C(s, S) can be written as follows: C s; S P 2 O£ S s; S 2 I s; S 2 U s; S 2 P s; S 2 D s; S; where S(s, S), I(s, S), U(s, S), P(s, S) and D(s, S) are the average sales, the average inventory holding cost, the average set-up cost, the average demand rejection penalty cost and average product delay cost, respectively. Here P and O are the unit price and unit operation cost of the product, respectively. In terms of the inventory distribution, the average sales S(s, S), the average inventory cost I(s, S), the average set-up cost U(s, S) and the average demand rejection penalty cost P(s, S) can be written as ! ! SX 21 SX 21 pi 1 qi ; I s; S cI SpS 1 i pi 1 qi ; S s; S l pS 1 i1
u s; S cS
s X i2 m
pi ;
i1
and
P s; S cP l p2m 1 q2m ;
respectively. For the average product delay cost D(s, S), we note that an arrival demand will not be delayed if the inventory level is positive. When a demand arrived, with probability p2k that the backlog is k and the machine is in the set-up state and with probability q2k that the backlog is k and the machine is the production state. In the ®rst case, the probability that the demand will delay is given by hk 1 2
ZT m mtk e2mt 1 2 e2t T2t dt: k! 0
In the second case, the probability that the demand will delay is given by gk 1 2
k ZT m mtk X mTj e2mt dt e2mT : k! j! 0 j0
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255
Table 1 The optimal (m p, s p, S p, pro®t) for different d and r
r
d
0.1 0.3 0.5 0.7 0.9
0.1
0.3
0.5
0.7
0.9
(1,16,17,151.6) (202,16,17,125.0) (236,16,17,109.0) (266,15,16,99.9) (294,15,16,92.6)
(63,19,20,142.1) (1,19,20,120.3) (67,19,20,104.4) (266,19,20,97.6) (294,19,20,91.9)
(60,19,20,134.2) (15,19,20,116.2) (1,19,20,104.6) (17,19,20,97.3) (9,19,20,91.0)
(51,19,20,128.1) (16,19,20,112.8) (7,19,20,102.7) (1,19,20,96.2) (9,19,20,90.6)
(40,19,20,123.2) (14,20,21,111.2) (7,19,20,100.8) (4,19,20,95.1) (1,19,20,89.8)
Thus the average product delay cost is given D s; S cD
m X k0
p2k hk 1 q2k gk :
In what follows, we give some numerical examples. Recall that the demand rate
l sP2t1 T 2t2 : We let the elasticity constants t 1, t 2, s to be 1, 2, 1000 respectively. The unit price P is $120 and the unit operating cost O is $20. The guaranteed delivery time T is one day. The parameters T0 and b of the delivery distribution are to be 0.1 and 10, respectively. The maximum inventory capacity Imax is 100. The inventory cost cI per unit of product is $10. Furthermore, we let the reliability factor to be 90%, the penalty cost cP to be $100. For the set-up cost cU, we set it to be $50 and the unit product delay cost to be $120. In Table 1 we give the optimal pair (s p, S p) and their corresponding average pro®t C (s p, S p) for different values of r l=m and d l=t:
4. Further researches In this paper, we discuss optimal (s, S) policies for production planning with delivery time guarantees. We establish a relationship (Proposition 2) between the desirable maximum allowable backlog m and the guaranteed delivery time T. Analytic form of the average pro®t is also derived. By varying different possible values of s and S, the optimal (s, S) policy can be obtained. Many interesting problems remain open. 1. 2. 3. 4.
We may consider the case when the machine is unreliable. Extension to the case of multi-item is also possible. Furthermore, other distributions (e.g. the Weibull distribution) can be used to model the delivery time. Another possible extension of our model is the following non-linear programming problem. We do not pre-determine the unit price P and the guaranteed delivery time T (Recall that the demand rate is a function of P and T cf. Eq. (1)). Rather, they are part of the decisions. The generalized formulation is
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W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
given as follows: ( max l P 2 O 2 cI SpS 1
P;T;s;S
SX 21 i1
l P 2 O 2 cI iqi 2 cP l q2m 1 p2m 2 cS
s X i2 m
) pi
subject to Eq. (2), s , S # Imax, and 1 # m.
Acknowledgements I would like to thank the two anonymous referees for their helpful comments. Their constructive suggestions improved the readability of this paper. Appendix A A.1. Proof of Proposition 1 Proof. The processing time for one product is exponentially distributed with mean 1/m , therefore the probability density function for the total processing time of k products is given by the Erlangian function fk t m mtk21 e2mt = k 2 1!: The delivery time is modeled by the shifted exponential distribution t e 2t t. Hence we have ZT m mtk21 e2mt 1 2 e2t T2t dt $ g: 0 k 2 1! By making use of the formula kX 21 ZT m mtk21 mTj e2mt dt 1 2 e2mT ; j! 0 k 2 1! j0
we get the inequalities (2). Therefore the maximum allowable backlog m is the largest positive integer k satisfying the above inequality. A A.2. Proof of Proposition 2 Proof. We let qS21 1; we can solve pS2i
m ; l
i 0; ¼; S 2 s 2 1
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by backward substitution. Then by forward substitution, we have qS2i and qs2i
mi 2 li ; mi21 m 2 l
i 2; ¼; S 2 s
j 22 mS2s1i 2 lS2s1i tli21 t l 1 m i21 iX lm S2s1i21 2 ; 2 l1t m l 1 t l 1 m l 1 ti m m 2 l j0
i 1; ¼; m 1 s: The steady state probability is then obtained by the normalization. A References Akella, R., & Kumar, P. (1986). Optimal control of production rate in a failure prone manufacturing systems. IEEE Transaction on Automatic Control, 31, 116±126. Buss, A., Lawrence, S., & Kropp, D. (1994). Volume and capacity interaction in facility design. IIE Transaction, 26, 36±49. Ching, W. (1998). An inventory model for manufacturing systems with delivery time guarantees. Computers and Operational Research, 25, 367±377. Dewan, S., & Mendelson, H. (1990). User delay costs and internal pricing for a service facility. Management Science, 36, 1502±1517. Donohue, K. (1994). The economics of capacity and marketing measure in a simple manufacturing environment. Production and Operations Management, 3, 78±99. Freeland, J. (1980). Coordination strategies for production and marketing in a functionally decentralized ®rm. IIE Transaction, 12, 126±132. Hill, A., & Khosla, I. (1992). Models for optimal leadtime reduction. Production and Operations Management, 1, 185±197. Karmarkar, U. (1994). A robust forecasting techniques for inventory and leadtime management. Journal of Operations Management, 12, 45±54. Li, L. (1992). The role of inventory in delivery-time completion. Management Science, 38, 182±197. Monden, Y. (1983). Toyota production system, Atlanta, GA: Industrial Engineering and Management Press. Varga, R. (1963). Matrix iterative analysis, New Jersey: Prentice-Hall.
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Optimal (s, S) policies with delivery time guarantees for manufacturing systems with early set-up Wai Ki Ching Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK
Abstract We consider optimal (s, S) policies with delivery time guarantees for production planning in one-machine manufacturing systems with early set-up. The machine produces one type of product and delivery time guarantee is offered to the customers for each unit of ordered product. The inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. In a (s, S) policy, the machine will shut down when an inventory level of S is attained and once the inventory level drops to s, the machine will re-start. A set-up time is required for the machine. We model the set-up by the exponential distribution. We obtained an analytical form of the steady state probability distribution for the inventory levels derived. The average pro®t of the system can be written in terms of this probability distribution. Hence the optimal (s, S) policy can be obtained by varying different possible values of s and S. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Manufacturing systems; (s, S) policies; Delivery time; Early set-up
1. Introduction In this paper, we study optimal (s, S) policies with product delivery time guarantees for production planning in manufacturing systems with early set-up. We observe that there was a series of shifts in strategic focus in American manufacturing over the past four decades. Firstly the `throughput' was the main concern in 60s because of the increasing demand in manufacturing products. The `cost' became the main concern when there were internal and foreign competitions in 70s. In 80s people focused on `process quality' to improve productivity and gain markets. Now in 90s, `time' is a source of competition. Successful manufacturers are now working to shorten the cycle and delivery time and improve the service level so as to remain competitive. There is a growing trend in using the delivery time guarantee in commercial companies as a marketing strategy to attract customers. For example, the United Parcel Services guarantees the next day delivery by 8:30 a.m. Pizza Hut in Hong Kong offers free pizza if the E-mail address: [email protected] (W.K. Ching). 0360-8352/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 02 4 - 9
250
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
ordered pizza cannot be served within 20 min. Lucky a major supermarket chain in California, use a `three's a crowd' campaign, which guarantees a new checkout counter will be open if there are more than three people waiting in its checkout queue. Wells Fargo Bank offers a `®ve minute maximum wait policy' which offers ®ve dollars to the customer if the customer waits for more than 5 min in line. Numerous papers have studied the relationships between the delivery time, the pricing and the capacity. Li (1992) considered the application of inventory in delivery time competition. Buss, Lawrence and Kropp (1994) studied optimal demand and capacity for the ¯ow time of jobs. Hill and Khosla expressed the demand as a function of delivery time and price. For more related works see for instance Ching (1998), Dewan and Mendelson (1990), Karmarker (1994) and Donohue (1994). However, to our best knowledge, the application of delivery time guarantee as a marketing strategy in manufacturing systems with set-up time seems not yet addressed in literature. In our study, we assume that the delivery time guarantee T and the unit price P are the main factors affecting the demand rate l l P; T of the product. One standard formulation for the relation (Cobb±Douglas) is
l l P; T sP2t1 T 2t2 ;
1
where t 1, t 2 are the price and delivery time guarantee elasticity respectively, and s is a positive constant; see Freeland (1980) and Hill and Khosla (1992) for instance. In the delivery time guarantee strategy, a delivery time guarantee of T on each unit of ordered product is offered to the costumers. In our model, an order can always be served within the delivery time T whenever there are products available in the inventory. Here for simplicity, we consider a one-machine system, extension to multiple machines case is possible. The machine produces one type of product, the inter-arrival time of the demand and the processing time for one unit of product are assumed to be exponentially distributed. Finite backlog of the demands is allowed in the system. A cost of cB is assigned to each unit of backlog and a cost of cI is assigned to each unit of inventory. Using the inventory level as an indicator, our interest is to determine the production strategy such that the average pro®t is maximized. We employ the (s, S) policy as the production (inventory) control. An (s, S) policy is characterized by two integers s and S. The machine keeps on producing until an inventory level of S is reached. Once this inventory level is reached, the production is stopped by shutting down the machine. We allow the inventory level fall to s, (s , S), and re-start the production. The machine requires a set-up time that is assumed to be exponentially distributed. In many manufacturing systems, set-up time is a vital procedure. Moreover, our model can be further extended to multi-item case. In this case, set-up procedure becomes necessary and important. The hedging point policy discussed in Akella and Kumar (1986) is a particular case of the (s, S) policy with S s 1 1: When the optimal hedging policy is a zero-inventory policy (i.e. the hedging point is zero), the policy matches with the just-in-time (JIT) policy. The JIT policies have strongly been favored in real-life production systems for process discipline reasons even when they are not optimal. By using the JIT policy, the TOYOTA company can manage to reduce work-in-process and cycle time, see Monden (1983), pp. 13±34. In this paper, we obtain the analytical form of the steady state probability of the inventory levels under (s, S) production policies. The average pro®t of the system can then be written in terms of this probability distribution, and the optimal values of s and S are obtained for any given unit price P of the product and the guaranteed delivery time T. The remainder of this paper is organized as follows. In Section 2, we present the model of manufacturing system under (s, S) policy. In Section 2.1, we derive a desirable maximum backlog when the guaranteed delivery time T is given. In Section 2.2, we give the balanced equations for the steady state
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
251
distribution of the machine-inventory system. Analytic form of this probability distribution is obtained in Section 2.3. In Section 3, the average pro®t for the systems is written in terms of the steady state probability distribution and numerical examples are also given. In Section 4, concluding remarks are given to discuss possible extensions of our model.
2. The manufacturing system In this section, we present a one-machine manufacturing system with product delivery time guarantee. Let us de®ne the following notations: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
m 21, the mean processing time for a demand for one unit of the product, l 21, the mean inter-arrival time for one unit of the product, t 21, the mean set-up time of the machine, O, operating cost for one unit of the product, P (P . O), unit price of the product, T, the guaranteed time of delivery, s , S, the parameters of the (s, S) policy, cI, inventory carrying cost per unit of the product (dollars), cP, penalty cost per unit of the product (dollars), cS, machine set-up cost (dollars), cD, penalty cost per unit of delay product (dollars), m, the maximum allowable backlog, Imax, the maximum allowable inventory.
We employ (s, S) policy as the production control. The processing time for one unit of product by the machine and the inter-arrival time for one unit of demand are assumed to be exponentially distributed. The demand rate of the product depends on the unit price P and the guaranteed delivery time T, and is given by Eq. (1). Backlog of products is allowed in the system. We reject any arrival demand when the inventory level is 2m, i.e. our maximum backlog capacity is m. A penalty cost of cP is assigned to each unit of rejected demand. To determine the optimal values of s and S, we have to set an appropriate guaranteed delivery time. 2.1. Determination of maximum allowable backlog In manufacturing system of limited production capacity, to guarantee customers a delivery time of the product, there should be an upper limit for the maximum allowable backlog m. Here we propose a simple estimation of the maximum allowable backlog m. Suppose we have k backlogged orders, the probability that they can be offered within the guarantee time T should be greater than a pre-determined value g (0 , g , 1). The reliability value (or the service level) g can be 90 or 95% for example. The following proposition gives an inequality relating the quantities g , T and the maximum allowable backlog m. Proposition 1. Given the delivery guarantee time T and the reliability value g , the maximum allowable
252
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
Fig. 1. The Markov Chain of the System
backlog m is the largest positive integer k which satis®es the inequalities: 8 kX 21 j j > T m 2 mk m 2 t j2k mk e2 t2mT > mT > g e $ for m ± t; 1 1 2 > > j! m 2 tk < j0
> > > > > :
mT
1 2 g e
$
kX 21 j0
mTj mk T k e2mT 1 j! k!
2
for m t:
2.2. The inventory levels When the desirable guaranteed delivery time T and the unit price P of the product are decided, the maximum allowable backlog m (Proposition 1) and the demand rate l (cf. Eq. (1)) can also be determined. We then employ the (s, S) policy in the production planning of the manufacturing system. We note that under the (s, S) policy, the inventory levels can take integer values in [2m, S]. When the inventory level reaches S, the machine is shut off. When the machine is shut off, the inventory levels can take integer values in [s, S] and once it falls to s, the machine is turned on and a set-up time is required. During the set-up time, the inventory level can take integer values in [2m, s]. There are (2(S 1 m) 1 1) possible states in the inventory system, see Fig. 1. We order the states as follows. The ®rst (S 2 s) states correspond to the situation that the machine is turned off, and the inventory levels are ordered from S down to (s 1 1). The next (m 1 s 1 1) states correspond to the situation that the machine is in the set-up time, and we order the inventory levels from (S 2 1) down to 2m. The next (m 1 S) states correspond to the situation that the machine is in the operating state, the inventory level takes integer values in [2m, S 2 1]. We order the inventory levels from (S 2 1) down to 2m. We construct the generator matrix for the machine-inventory system. A generator matrix A of a Markov process gives the transition rate [A]ij from state j to state i. Under our
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
253
ordering, we obtain the following generator matrix A for the machine-inventory system: 2
l
6 6 2l l 6 6 6 ] ] 6 6 6 6 6 6 6 6 6 6 6 6 6 A6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0
2l
l1t ]
] 2l
t
0
0
m1l 2l
2m
m 1 l 2m ]
2t ] ]
0
2t
]
]
2l
m1l
2m
]
]
]
2l
l1m 2l
3
S .. . .. .
7 7 7 7 7 7 7 7 s 7 7 7 .. 7 7 . 7 7 7 2m 1 1 7 7 7 2m 7 7 7 7 S21 7 7 7 S22 7 7 7 .. 7 . 7 7 7 .. 7 . 7 7 2m 5 2m 1 1 m 2m
The generator matrix A is irreducible, has zero column sum, positive diagonal entries and nonpositive off-diagonal entries. Therefore A has one-dimensional null space; see Varga (1963), p. 30. The steady state probability distribution is the normalized positive null vector of A. 2.3. The steady state distribution of the inventory levels We derive the analytic form of the steady state probability distribution of the inventory levels. We let probability distribution vector p be written as follows: p pS ; pS21 ; ¼p2m ; qS21 ; qS22 ; ¼q2m t : Here qi is steady state probability that the inventory level is i and the machine is operating and pi is steady state probability that the inventory level is i and the machine is shut off or in set-up stage. By letting qS21 1 and direct veri®cation, we have the following proposition. Proposition 2. The steady state probability distribution p of A can be written as follows: p a pS ; pS21 ; ¼p2m ; qS21 ; qS22 ; ¼q2m t
254
W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
where
8 > > > > > > > > > > > > > > > > <
pS2i ps2i
m ; l
i 0; ¼; S 2 s 2 1;
mli ; l 1 ti11
i 0; ¼; m 2 s;
qS21 1; > > mi 2 li > > > qS2i i21 ; i 2; ¼; S 2 s; > > m m 2 l > > > > 22 j > > mS2s1i 2 lS2s1i tli21 t l 1 m i21 iX lm > > 2 ; q 2 > : s2i l1t m l 1 t l 1 m l 1 ti mS2s1i21 m 2 l j0
Here a ps 1
PS 2 1
i 1; ¼; m 1 s:
3
i2 m
pi 1 qi is the normalization constant.
3. The average pro®t and numerical examples The average pro®t of the system, C(s, S) can be written as follows: C s; S P 2 O£ S s; S 2 I s; S 2 U s; S 2 P s; S 2 D s; S; where S(s, S), I(s, S), U(s, S), P(s, S) and D(s, S) are the average sales, the average inventory holding cost, the average set-up cost, the average demand rejection penalty cost and average product delay cost, respectively. Here P and O are the unit price and unit operation cost of the product, respectively. In terms of the inventory distribution, the average sales S(s, S), the average inventory cost I(s, S), the average set-up cost U(s, S) and the average demand rejection penalty cost P(s, S) can be written as ! ! SX 21 SX 21 pi 1 qi ; I s; S cI SpS 1 i pi 1 qi ; S s; S l pS 1 i1
u s; S cS
s X i2 m
pi ;
i1
and
P s; S cP l p2m 1 q2m ;
respectively. For the average product delay cost D(s, S), we note that an arrival demand will not be delayed if the inventory level is positive. When a demand arrived, with probability p2k that the backlog is k and the machine is in the set-up state and with probability q2k that the backlog is k and the machine is the production state. In the ®rst case, the probability that the demand will delay is given by hk 1 2
ZT m mtk e2mt 1 2 e2t T2t dt: k! 0
In the second case, the probability that the demand will delay is given by gk 1 2
k ZT m mtk X mTj e2mt dt e2mT : k! j! 0 j0
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Table 1 The optimal (m p, s p, S p, pro®t) for different d and r
r
d
0.1 0.3 0.5 0.7 0.9
0.1
0.3
0.5
0.7
0.9
(1,16,17,151.6) (202,16,17,125.0) (236,16,17,109.0) (266,15,16,99.9) (294,15,16,92.6)
(63,19,20,142.1) (1,19,20,120.3) (67,19,20,104.4) (266,19,20,97.6) (294,19,20,91.9)
(60,19,20,134.2) (15,19,20,116.2) (1,19,20,104.6) (17,19,20,97.3) (9,19,20,91.0)
(51,19,20,128.1) (16,19,20,112.8) (7,19,20,102.7) (1,19,20,96.2) (9,19,20,90.6)
(40,19,20,123.2) (14,20,21,111.2) (7,19,20,100.8) (4,19,20,95.1) (1,19,20,89.8)
Thus the average product delay cost is given D s; S cD
m X k0
p2k hk 1 q2k gk :
In what follows, we give some numerical examples. Recall that the demand rate
l sP2t1 T 2t2 : We let the elasticity constants t 1, t 2, s to be 1, 2, 1000 respectively. The unit price P is $120 and the unit operating cost O is $20. The guaranteed delivery time T is one day. The parameters T0 and b of the delivery distribution are to be 0.1 and 10, respectively. The maximum inventory capacity Imax is 100. The inventory cost cI per unit of product is $10. Furthermore, we let the reliability factor to be 90%, the penalty cost cP to be $100. For the set-up cost cU, we set it to be $50 and the unit product delay cost to be $120. In Table 1 we give the optimal pair (s p, S p) and their corresponding average pro®t C (s p, S p) for different values of r l=m and d l=t:
4. Further researches In this paper, we discuss optimal (s, S) policies for production planning with delivery time guarantees. We establish a relationship (Proposition 2) between the desirable maximum allowable backlog m and the guaranteed delivery time T. Analytic form of the average pro®t is also derived. By varying different possible values of s and S, the optimal (s, S) policy can be obtained. Many interesting problems remain open. 1. 2. 3. 4.
We may consider the case when the machine is unreliable. Extension to the case of multi-item is also possible. Furthermore, other distributions (e.g. the Weibull distribution) can be used to model the delivery time. Another possible extension of our model is the following non-linear programming problem. We do not pre-determine the unit price P and the guaranteed delivery time T (Recall that the demand rate is a function of P and T cf. Eq. (1)). Rather, they are part of the decisions. The generalized formulation is
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W.K. Ching / Computers & Industrial Engineering 40 (2001) 249±257
given as follows: ( max l P 2 O 2 cI SpS 1
P;T;s;S
SX 21 i1
l P 2 O 2 cI iqi 2 cP l q2m 1 p2m 2 cS
s X i2 m
) pi
subject to Eq. (2), s , S # Imax, and 1 # m.
Acknowledgements I would like to thank the two anonymous referees for their helpful comments. Their constructive suggestions improved the readability of this paper. Appendix A A.1. Proof of Proposition 1 Proof. The processing time for one product is exponentially distributed with mean 1/m , therefore the probability density function for the total processing time of k products is given by the Erlangian function fk t m mtk21 e2mt = k 2 1!: The delivery time is modeled by the shifted exponential distribution t e 2t t. Hence we have ZT m mtk21 e2mt 1 2 e2t T2t dt $ g: 0 k 2 1! By making use of the formula kX 21 ZT m mtk21 mTj e2mt dt 1 2 e2mT ; j! 0 k 2 1! j0
we get the inequalities (2). Therefore the maximum allowable backlog m is the largest positive integer k satisfying the above inequality. A A.2. Proof of Proposition 2 Proof. We let qS21 1; we can solve pS2i
m ; l
i 0; ¼; S 2 s 2 1
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by backward substitution. Then by forward substitution, we have qS2i and qs2i
mi 2 li ; mi21 m 2 l
i 2; ¼; S 2 s
j 22 mS2s1i 2 lS2s1i tli21 t l 1 m i21 iX lm S2s1i21 2 ; 2 l1t m l 1 t l 1 m l 1 ti m m 2 l j0
i 1; ¼; m 1 s: The steady state probability is then obtained by the normalization. A References Akella, R., & Kumar, P. (1986). Optimal control of production rate in a failure prone manufacturing systems. IEEE Transaction on Automatic Control, 31, 116±126. Buss, A., Lawrence, S., & Kropp, D. (1994). Volume and capacity interaction in facility design. IIE Transaction, 26, 36±49. Ching, W. (1998). An inventory model for manufacturing systems with delivery time guarantees. Computers and Operational Research, 25, 367±377. Dewan, S., & Mendelson, H. (1990). User delay costs and internal pricing for a service facility. Management Science, 36, 1502±1517. Donohue, K. (1994). The economics of capacity and marketing measure in a simple manufacturing environment. Production and Operations Management, 3, 78±99. Freeland, J. (1980). Coordination strategies for production and marketing in a functionally decentralized ®rm. IIE Transaction, 12, 126±132. Hill, A., & Khosla, I. (1992). Models for optimal leadtime reduction. Production and Operations Management, 1, 185±197. Karmarkar, U. (1994). A robust forecasting techniques for inventory and leadtime management. Journal of Operations Management, 12, 45±54. Li, L. (1992). The role of inventory in delivery-time completion. Management Science, 38, 182±197. Monden, Y. (1983). Toyota production system, Atlanta, GA: Industrial Engineering and Management Press. Varga, R. (1963). Matrix iterative analysis, New Jersey: Prentice-Hall.