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A multi-warehouse inventory model for items with time-varying demand and shortages
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Computers & Operations Research 30 (2003) 2115 – 2134
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A multi-warehouse inventory model for items with time-varying demand and shortages Yong-Wu Zhou Department of Mathematics, Hefei University of Technology, Hefei, Anhui 230009, China Received 1 November 2001; received in revised form 1 April 2002
Abstract This paper develops a deterministic replenishment model with multiple warehouses (one is an owned warehouse and others are rented warehouses) possessing limited storage capacity. In this model, the replenishment rate is in/nite. The demand rate is a function of time and increases at a decreasing rate. The stocks of rented warehouses are transported to owned warehouse in continuous release pattern. The model allows shortages in owned warehouse and permits part of the backlogged shortages to turn into lost sales—which is assumed to be a function of the currently backlogged amount. The solution procedure for /nding the optimal replenishment policy is shown. As a special case of the model, the corresponding models with completely backlogged shortages and without shortages are also presented. The models are illustrated with the help of numerical examples. Sensitivity analysis of parameters is given in graphical form. Scope and purpose In practical inventory management, there exist many factors like an attracted price discount for bulk purchase, etc. to make retailers buy goods more than the capacity of their owned warehouse. In this case, retailers will need to rent other warehouses or to rebuild a new warehouse. However, from economical point of views, they usually choose to rent other warehouses. If there are multiple warehouses available, an important problem faced by the retailers is which warehouses to be selected to hold items replenished, when to replenish as well as what size to replenish. For such a problem, the existing two-warehouse models, based on an unrealistic assumption that the rented warehouse has unlimited storage capacity, presented some procedures for determining the optimal replenishment policy. This paper extends the existing two-warehouse models in three directions. Firstly, the traditional two-warehouse models assumed the storage capacity of the rented warehouse unlimited. The present paper relaxes this impractical assumption and considers the situation with multiple rented warehouses having a limited capacity. Secondly, the traditional two-warehouse models considered a constant demand rate or a linearly increasing demand rate. In this model, the demand rate varies over time and increases at a decreasing rate, which implies an increasing market going to saturation. Thirdly, we extend the two-warehouse models to the case with partially backlogged shortages. The purpose of this paper is to E-mail address: [email protected] (Y.-W. Zhou). 0305-0548/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 2 ) 0 0 1 2 6 - 0
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build a multi-warehouse replenishment model to help decision-makers solve the problem of which warehouses to be chosen to store items replenished and how to replenish. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Time-varying demand; Inventory; Multi-warehouse; Lost sale
1. Introduction Since Donaldson [1] /rst developed an exact solution procedure to determine both the optimal amount and the optimal time of replenishment, many researchers like Teng [2], Goyal et al. [3], Hill [4], Bhunia and Maiti [5], etc. have paid considerably more attention to the inventory problems with deterministic time-varying demand pattern. By using diEerent methods or from diEerent angles, they either proposed various other heuristic approaches for Donaldson’s problem or extended it to some more practical situations. All these models dealt with the case of a single warehouse and assumed that the available warehouse has unlimited capacity. However, in real life, any warehouse has /nite capacity. On the other hand, due to some reasons such as an attracted price discount for bulk purchase, the order costs higher than one using rented warehouse (RW), and so on, inventory managers usually are attracted to hold more items than can be stored in an owned warehouse (OW). Therefore, multiple warehouses should be required to hold very large inventories. In recent years, various researchers have discussed a two-warehouse inventory system. This system has two warehouses (OW and RW). OW has /nite capacity and RW has in/nite capacity. Since the inventory holding cost is usually higher in RW than in OW, the system /rst supplies items stored in RW to customers and then gives items in OW to customers. Hartely [6] was the /rst to develop the basic two-warehouse inventory model. He proposed a heuristic procedure for determining the optimal order quantity. Sarma [7] further developed the two-warehouse model for deteriorating items with an in/nite replenishment rate and shortages. Pakkala and Achary [8] considered the two-warehouse model for deteriorating items with a /nite replenishment rate and shortages. In all these models, the demand was assumed to be constant and the cost of transporting items from RW to OW was not taken into account. Sarma [9] presented a model with an in/nite replenishment rate by considering the transportation cost from RW to OW to be a /xed constant independent of the quantity being transported, but he did not consider shortages in his model. Goswami and Chaudhuri [10] further developed the model with or without shortages by assuming that the demand varies over time with a linearly increasing trend and that the transportation cost from RW to OW depends on the quantity being transported. In their model, stock was transferred from RW to OW in an intermittent pattern. Through employing a continuous transportation pattern, Bhunia and Maiti [11] developed a two-warehouse model for deteriorating items with a linearly increasing demand and shortages during the in/nite period. Both models developed in [10] and [11] considered the case with a linearly increasing demand and an in/nite time horizon. Nevertheless, a linearly increasing demand with the in/nite horizon looks somewhat less realistic. Employing the /nite planning horizon, Zhou [12] presented a two-warehouse model for deteriorating items with time-varying demand and shortages backordered. In a recent paper, Kar et al. [13] proposed a similar two-warehouse model for non-perishable items with a linear trend in demand and /nite time horizon by considering
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inventory-dependent replenishment cost. However, all these two-warehouse models were based on an impractical assumption that the rented warehouse has unlimited capacity. In this paper, we develop a deterministic inventory model with m warehouses (one is OW and the others are RWs). This model is diEerent from the existing two-warehouse models in three aspects: Firstly, each warehouse possesses limited storage capacity in the present model. Secondly, the existing models with time-varying demand considered the demand rate as a linearly increasing function of time in the in/nite time horizon. It seems to be unreasonable. In this model, the demand rate is a continuous function of time and increases at a decreasing rate. This represents an expanding market approaching saturation. Thirdly, all of the two-warehouse models either did not allow shortages to occur or considered shortages to be backlogged completely. In the practical inventory management, however, not all customers wait for a long time to buy goods from a particular shop although they are likely to be motivated to wait owing to some factors such as a special discount price, goodwill, etc. This means that shortages are not completely backlogged in the real world. This model permits part of the backlogged shortages to turn into lost sales, which makes the model more practical. Like in Bhunia and Maiti’s model [11], where stock is also transported from each RW to OW in continuous pattern. The model with partially backlogged shortages is /rst developed. Then, as a special case of the present model, the corresponding models with completely backlogged shortages and without shortages are also shown. Numerical examples are used to illustrate how the models work. The eEect of all parameters on the optimal replenishment policy is observed by sensitivity analysis.
2. Assumptions and notations To develop the proposed model, the following assumptions and notations are used throughout the whole paper. 1. D(t) is the demand rate, which is a continuous function of time t and increases at a decreasing rate, i.e., D(t) satis/es dD=dt ¿ 0 but d 2 D=dt 2 ¡ 0. 2. T is the length of the replenishment cycle. 3. Q is the replenishment quantity at the beginning of the cycle. 4. m is the number of warehouses that can be used by the system. 5. Due to diEerent storage environment, inventory-holding costs in diEerent warehouses are considered to be diEerent. Inventory-holding cost in OW is usually the lowest. All warehouses are sequenced according to the order of the holding cost from small to big. 6. H1 is the holding cost per unit per unit time in OW and W1 is the storage capacity of OW. 7. Hi is the holding cost per unit per unit time in the ith RW, Wi is the storage capacity of the ith RW and Ci is the transportation cost per unit of items moved from the ith RW to OW (i = 2; 3; : : : ; m). Then we have H1 ¡ H2 ¡ · · · ¡ Hm because of assumption 5. 8. A is the /xed cost per replenishment. 9. Shortages are allowed and assumed to backlog partially. T1 is the time at which shortages occur during a replenishment cycle. Generally speaking, customers are usually impatient. Some of them will not wait for goods in a special shop if many other customers have been waiting. Moreover, the more customers have been waiting, the more customers among them choose to leave the
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shop. So we here assume that the lost sale is proportional to the current shortage level. Let be the scale parameter and s the shortage cost per unit per unit time (0 ¡ ¡ 1). 10. Replenishment rate is in/nite and lead time is zero. 11. Lk denotes an inventory system that uses k warehouses (one is an owned warehouse and others are rented warehouses) to store items (k = 2; 3; : : : ; m). L1 denotes an inventory system that uses only OW. k is the minimal average total cost of Lk -system. 12. L∗k denotes the optimal Lk -system in which its minimal average total cost is the smallest among those of all Lk -systems.
3. Model formulation for Lk -system In order to develop the inventory model of Lk -system, we arbitrarily select (k − 1) warehouses among all rented warehouses. Then the selected (k − 1) warehouses and OW become an Lk -system. For convenience, k warehouses are sequenced again according to the order of the holding cost from small to big. Based on assumption 5, it is obvious that the /rst warehouse is OW here. And let wi be the storage capacity of the ith warehouse, hi the holding cost per unit per unit time in the ith warehouse and ci the transportation cost per unit of items shipped from the ith warehouse to OW (i = 1; 2; : : : ; k). It is very clear that c1 = 0. Next, we study the Lk -system neglecting the capacity constraint of the kth warehouse. In this system, wj units are kept in the jth warehouse (j = 1; 2; : : : ; k − 1) and Q − qk −1 units in the kth warehouse, where qj = ji=1 wi (j = 1; 2; : : : ; k) and q0 = 0. For economic reasons, if more than one warehouse has been used in practice, items of warehouse with lower holding cost are usually consumed only after stock in the warehouse with higher holding cost is used up. Therefore, items in the kth warehouse are /rstly used to meet the demand. Then, items in the (k − 1)th warehouse are used, and so on. If we let tj be the time at which stock in the (k + 1 − j)th warehouse drops to zero (j = 1; 2; : : : ; k), it is obvious that tk = T1 . The graphical representation of the Lk -system is shown in Fig. 1 when k = 3. As stated before, we have t1 tk+1−j D(u) du = Q − qk −1 and D(u) du = wj (j = k − 1; k − 2; : : : ; 1): (1) 0
tk − j
And the sum of the above equations is T1 D(u) du = Q: 0
(2)
During the interval [0; t1 ], the stock in the kth warehouse is depleted only due to demand and drops to zero at time t = t1 . So the holding cost of items in the kth warehouse during [0; t1 ] is given by t hk 0 1 D(u)u du and the holding cost of items in the jth warehouse is hj wj t1 (j = 1; 2; : : : ; k − 1). Thus, the holding cost of items in the whole system during [0; t1 ] is hk
0
t1
D(u)u du +
k −1 j=1
hj w j t1 :
(3)
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Inventory level Q_q2
The 3rd warehouse
The 2nd warehouse
w2
The 1st warehouse w1 t1
t3 = T1
t2
T
time
Fig. 1. Graphical representation of the L3 -system with shortages.
Similarly, in the interval [tj−1 ; tj ], the holding cost of items in the system is given by tj k −j hk+1−j D(u)(u − tj−1 ) du + hi wi (tj − tj−1 ) tj − 1
(4)
i=1
and the holding cost of the system over the interval [tk −1 ; tk ] is tk h1 D(u)(u − tk −1 ) du:
(5)
tk − 1
Therefore, the holding cost of the inventory system in the replenishment cycle can be written as k −j tj k k −1 where t0 = 0: hk+1−j D(u)(u − tj−1 ) du + (tj − tj−1 ) hi w i (6) HC = j=1
tj − 1
j=1
i=1
After some simple operations, the above formula becomes tj k hk+1−j D(u)u du: HC = j=1
(7)
tj − 1
The cost of transporting units from all RWs to OW is given by TC =
k −1 j=1
cj wj + ck (Q − qk −1 ) =
k −1
(cj − ck )wj + ck Q
where c1 = 0:
(8)
j=1
According to assumption 9, let I (t) be the inventory level at time t during the shortage period. Then I (t) satis/es dI (t) = −D(t) − I (t); T1 6 t 6 T: (9) dt
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Using the boundary condition I (T1 ) = 0, we can obtain the solution to Eq. (9): T1 I (t) = D(u) exp(−(t − u)) du T1 6 t 6 T: t
Consequently, the shortage cost for the entire cycle is T T s (−I (t)) dt = D(u)[1 − exp(−(T − u))] du: SC = s T1 T1
(10)
(11)
Thus, the average total cost of the system during the replenishment cycle is given by = {A + HC + TC + SC }=T tj k k −1 = A+ hk+1−j D(u)u du + (cj − ck )wj tj − 1 j=1
+ ck
0
T1
j=1
s D(u) du +
T
T1
D(u)(1 − exp(−(T − u))) du
T:
(12)
From (1) and (2), it can be sen that the average total cost, , is a function of variables T1 and T (denoted by (T1 ; T )). The necessary condition for (T1 ; T ) to be minimum is @(T1 ; T ) @(T1 ; T ) = 0; = 0 and @T1 @T which gives k
s hk −j+1 (tj − tj−1 ) + ck − (1 − exp(−(T − T1 ))) = 0; j=1 T tj k s D(u)[(T + 1) exp(−(T − u)) − 1] du − A + hk −j+1 D(u)u du T1 tj − 1
(13)
j=1
+
k −1 j=1
(cj − ck )wj + ck
0
T1
D(u) du
= 0:
(14)
Theorem 1. The replenishment schedule satisfying the 3rst-order optimality conditions (13) and (14) satis3es the second-order conditions for a minimum. (Proof seen in Appendix A) Let T1∗ and T ∗ be the optimal values of T1 and T . Then, from (2) we can derive the optimal replenishment quantity, Qk , of the Lk -system ignoring the capacity constraints of the kth warehouse. If qk −1 6 Qk 6 qk , it is obvious that T1∗ and T ∗ are the feasible solutions of the Lk -system and k = (T1∗ ; T ∗ ). Otherwise, k is equal to the optimal boundary cost (Tqk ; T # ) when Qk = qk . Now we determine the optimal boundary cost (Tqk ; T # ). First, by /xing Q at qk , from (2) and (1)
Y.-W. Zhou / Computers & Operations Research 30 (2003) 2115 – 2134
we have Tq 0
k
D(u) du = qk
and
tj
tj − 1
D(u) du = wk −j+1 ; j = 1; 2; : : : ; k:
2121
(15)
In this case, the boundary cost function of the system is given by tj k k hk+1−j D(u)u du + cj w j (Tqk ; T ) = A + tj − 1 j=1
+
s
T
Tq k
j=1
D(u)[1 − exp(−(T − u))] du
T;
which depends only on T . The necessary condition for (Tqk ; T ) to be minimum is d(Tqk ; T )=dT = 0; which gives T s D(u)[(T + 1) exp(−(T − u)) − 1] du Tq k tj k hk −j+1 = 0: D(u)u du + cj wj − A+ tj − 1
(16)
(17)
j=1
It can be easily proved that the solution to Eq. (17) also satis/es the second-order suOcient condition. Generally, Eq. (17) can be solved by using any one-dimensional search technique. Substituting the optimal value T # of T into (16), we obtain the optimal boundary cost (Tqk ; T # ). For obtaining the optimal L∗k -system, one should reselect (k − 1) warehouses from all RWs. Thus, another Lk -system is obtained. Using the method presented above, one can derive the replenishment policy and the corresponding minimal average total cost of this Lk -system. After repeating this process till all diEerent Lk -systems have been discussed, one can get a list of the minimal average total costs related to those Lk -systems. Then the L∗k -system can be determined by comparing these costs. The method for /nding the L∗k -system presented above can be summarized as the following algorithm: Algorithm 1. Set c1 = C1 ; w1 = W1 ; h1 = H1 ; k∗ = +∞. If k = 1, go to step 5; Otherwise, go to the next step. Take (k − 1) integers from 2 to m (say 2 6 i2 ¡ i3 ¡ · · · ¡ ik 6 m). Set hj = Hij ; cj = Cij and wj = Wij for j = 2; : : : ; k. Solve Eqs. (13) and (14) for determining T1 and T . Then calculate Qk and k from (1), (2) and (12). 6. If Qk ¿ qk , go to step 7; Otherwise, go to step 8. 7. Set Qk = qk and compute Tqk from (15). Then solve Eq. (17) for getting T # , compute (Tqk ; T # ) from (16) and set k = (Tqk ; T # ); T1 = Tqk T = T # .
1. 2. 3. 4. 5.
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8. If k ¡ k∗ , then k∗ = k , T1∗ = T1 , T ∗ = T Qk∗ = Qk and Ik∗ = {1; i2 ; i3 ; : : : ; ik }, go to step 9; Otherwise, go to step 9. 9. If k = 1, output k∗ , T1∗ , T ∗ , Qk∗ and Ik∗ , stop; Otherwise, go to step 10. 10. If another group of (k − 1) integers, which is diEerent from those taken earlier, can be got from 2 to m, then draw a new group of (k − 1) integers from 2 to m and go to step 4; Otherwise, output k∗ , T1∗ , T ∗ , Qk∗ and Ik∗ , stop. 4. Solution procedure for a multi-warehouse inventory system Now we can determine the optimal replenishment schedule of the proposed system according to the method given below. First, consider an L1 -system which only uses OW to store items and determine its optimal replenishment quantity, Q1∗ , and average total cost, 1∗ . If Q1∗ ¡ w1 (=q1 ), the system does not need to rent any warehouse. Q1∗ is the optimal replenishment quantity of the system. Otherwise, determine the optimal L∗2 -system, the corresponding replenishment schedule and average total cost, 2∗ , by using the method provided in the previous section. If Q2∗ ¡ q2 , decision-makers will not need to rent any warehouse. At this moment, 2∗ has to be compared with 1∗ . When 1∗ 6 2∗ , decision-makers of the system should use the L1 -system. Conversely, they should use the L∗2 -system. Otherwise, decision-makers have to determine the L∗3 -system and its corresponding replenishment schedule and average total cost. If Q3∗ ¡ q3 , decision-makers should use the Li -system while i∗ = min3j=1 {j∗ }; otherwise, the above process is repeated up to the Lm -system. This solution procedure can be written as the following algorithm: Algorithm 2. 1. Input the values of all parameters. 2. Set k = 1 and ∗ = +∞. 3. Determine L∗k -system and the corresponding replenishment schedule and average total cost (Ik∗ ; T1∗ ; T ∗ ; Qk∗ and k∗ ) by using Algorithm 1. 4. If ∗ ¿ k∗ , then set ∗ = k∗ , T1∗∗ = T1∗ , T ∗∗ = T ∗ , Q∗ = Qk∗ and I ∗ = Ik∗ , go to step 5; Otherwise, go to step 5. 5. If Qk∗ ¡ qk , output ∗ ; T1∗∗ , T ∗∗ , Q∗ and I ∗ , stop; Otherwise, go to the next step. 6. If k = m, then output ∗ ; T1∗∗ , T ∗∗ , Q∗ and I ∗ , stop; Otherwise, k = k + 1, go to step 3. 5. Numerical examples To illustrate the proposed model, we show some numerical examples with a special non-linear increasing demand. Let D(t) = D − a exp(−bt) (a ¿ 0; b ¿ 0). From (1) and (2), we have Q = DT1 + a(exp(−bT1 ) − 1)=b and D(tk+1−j − tk −j ) + a(exp(−btk+1−j ) − exp(−btk −j ))=b = wj ;
j = 1; 2; : : : ; k − 1:
(18)
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In this situation, the average total cost of the Lk -system ignoring the capacity constraint of the kth warehouse becomes k −1 1 A+ (cj − ck )wj + ck [DT1 + a(exp(−bT1 ) − 1)=b] (T1 ; T ) = T j=1
k D 2 2 a (t − t ) + 2 ((btj + 1) exp( − btj ) − (btj−1 + 1) exp(−btj−1 )) hk+1−j + 2 j j −1 b j=1 1 s (D + a exp(−bT ))(T − T1 ) − (D + a exp(−bT1 ))(1 − exp(−b(T − T1 ))) + b for b = and
(19a)
k −1 1 A+ (cj − ck )wj + ck [DT1 + a(exp(−bT1 ) − 1)=b] (T1 ; T ) = T j=1 k D 2 a (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) hk+1−j + 2 b j=1 − (btj−1 + 1) exp(−btj−1 )) a s + D(T − T1 ) + (exp(−bT ) − exp(−bT1 )) b( − b) D a exp(−bT1 ) − (1 − exp(−(T − T1 ))) + −b
for b = :
The corresponding optimality conditions for (T1 ; T ) to be minimum is given by k s hk −j+1 (tj − tj−1 ) + ck − (1 − exp(−(T − T1 ))) = 0; h k t1 + T j=2 s (D − a exp(−bu))[(T + 1) exp(−(T − u)) − 1] du T1 k −1 k a (cj − ck )wj + ck (exp(−bT1 ) − 1) + ck DT1 + hk −j+1 − A+ b j=1 j=1 D 2 a (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) − (btj−1 + 1) exp(−btj−1 )) × = 0: 2 b If /xing Q at qk , then the values of t1 ; t2 ; : : : ; tk (=T1 ) can be obtained easily from (18).
(19b)
(20)
(21)
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Table 1 Optimum solution of the model for diEerent values of demand parameters Examples Example Example Example Example
1 2 3 4
a
b
T∗
Q∗
∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.714317 0.512434 0.574863 0.500487
451.528 500 700 700
974.905 1355.09 1423.54 1609.07
L2 -system L2 -system L3 -system L3 -system
At this moment, the boundary cost of the Lk -system becomes k D 2 1 a A+ (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) − (btj−1 + 1) (Tqk ; T ) = hk+1−j T 2 b j=1
T k s cj w j + (D − a exp(−bu))[1 − exp(−(T − u))] du : × exp(−btj−1 )) + Tq j=1 k
(22) The corresponding optimality condition for (Tqk ; T ) to be minimum is T k s D 2 2 hk −j+1 (D − a exp(−bu))[(T + 1) exp(−(T − u)) − 1] du − A + (t − t ) Tq 2 j j −1 j=1
k
+
a ((btj + 1) exp(−bj t) − (btj−1 + 1) exp(−bj t1 )) b2
+ c j wj
= 0:
(23)
The proposed model is illustrated by the following numerical examples. Data 1. Let m = 3; H1 = 0:5; H2 = 0:85; H3 = 1:2; W1 = 100; W2 = 200; W3 = 400; C2 = 0:4; C3 = 0:25; s = 6; D = 2500; = 0:8 and A = 400 in appropriate units. According to the proposed solution procedure, we present the optimal replenishment schedule and the corresponding minimal average cost for diEerent values of demand parameters with the help of the Mathematica software. Results are given in Table 1. Table 2 shows the eEect of change of parameter on the optimal replenishment policy. From Table 2, we can observe that has little impact on the optimal replenishment policy. After the /rst cycle is completed, inventory managers can transfer value of the constant ‘a’ into a exp(−bT ) (where T is the associated value of the previous cycle) and employ this model again. Thus, they can obtain the replenishment schedule for the next cycle, and so on. According to this approaching, inventory managers can obtain the replenishment policies of subsequent replenishment cycles. Here we show the replenishment schedule of the /rst four cycles only for Example 1. The result is given in Table 5.
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Table 2 EEect of parameter on the optimal replenishment policy
T1
T
Q
0.1 0.2 0.4 0.7 0.9
0.56582 0.56543 0.564453 0.563083 0.562109
0.710405 0.710976 0.711991 0.71373 0.714923
454.99 454.57 453.52 452.052 451.004
978.317 977.847 976.89 975.41 974.393
6. Special cases 6.1. Model with completely backlogged shortages If → 0 in the proposed model, we can obtain the corresponding multi-warehouse model with completely backlogged shortages. For this special case, the average cost of the Lk -system ignoring the capacity constraint of the kth warehouse is given by tj k k −1 1 A+ 1 = hk −j+1 D(u)u du + (cj − ck )wj T tj − 1 j=1
+ ck where
t1
0
0
T1
j=1
D(u) du + s
T1
D(u) du = Q − qk −1 =
T
T1
0
D(u)(T − u) du
;
(24)
D(u) du − qk −1 and
tj
tj − 1
D(u) du = wk −j+1 ; j = 2; 3; : : : ; k:
The optimality condition for 1 to be minimum is given by k
hk −j+1 (tj − tj−1 ) + ck − s(T − T1 ) = 0; tj T k D(u)u du − A + hk −j+1 D(u)u du s T1 tj − 1
(25)
j=1
j=1
+
k −1
(cj − ck )wj + ck
j=1
0
T1
D(u) du
= 0:
(26)
Under this case, the corresponding boundary cost function of the system is given by tj T k k 1 (Tqk ; T ) = A + hk+1−j D(u)u du + cj wj + s D(u)(T − u) du T; Tq tj − 1 j=1
j=1
k
(27)
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Table 3 Optimum solution of the model with completely backlogged shortages Examples Example Example Example Example
5 6 7 8
a
b
T∗
Q∗
1∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.67539 0.477257 0.362696 0.487305
415.561 437.017 300 673.112
973.896 1377.38 1684.77 1675.58
L3 -system L3 -system L2 -system L3 -system
where Tqk and tj satisfy Tq k D(u) du = qk 0
and
tj
tj − 1
D(u) du = wk −j+1 (j = 1; 2; : : : ; k):
And the optimality condition for 1 (Tqk ; T ) to be minimum is tj T k hk −j+1 = 0: D(u)u du − A + D(u)u du + cj wj s Tq tj − 1
(28)
j=1
k
Using similar solution procedure to the model proposed in previous section, we can determine the optimal replenishment schedule of the multi-warehouse inventory system with completely backlogged shortages. To illustrate the solution procedure, we also show four numerical examples for the same kind of non-linear increasing demand. The values of all parameters are given in Data 2 except a and b. Computed results are shown in Table 3. Data 2. Let m = 3; H1 = 0:5; H2 = 0:85; H3 = 1:2; W1 = 100; W2 = 200; W3 = 400; C2 = 0:25; C3 = 0:40; s = 6; D = 2500 and A = 400 in appropriate units. After the /rst cycle is completed, we can also get the corresponding replenishment schedule of subsequent replenishment cycles by using the same approach presented in the previous section. Here we give the replenishment schedule of the /rst four cycles only for Example 5 in Table 5. 6.2. Model without shortages If T1 = T in the proposed model, we can also obtain the corresponding multi-warehouse inventory model with no consideration of shortages. In this special case, the average cost of the Lk -system ignoring the capacity constraint of the kth warehouse will become tj T k 1 X A+ hk −j+1 D(u)u du + (cj − ck )wj + ck D(u) du ; (29) 2 (T ) = = T T 0 tj − 1 j=1
where
0
t1
D(u) du = Q − qk −1 =
0
T
D(u) du − qk −1
and
tj
tj − 1
D(u) du = wk −j+1 ; j = 2; 3; : : : ; k:
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Table 4 Optimum solution of the model without shortages Examples Example Example Example Example
9 10 11 12
a
b
T∗
Q∗
2∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.607422 0.429883 0.432319 0.372517
500.476 574.772 700 700
1098.15 1588.73 1724.64 1963.59
L3 -system L3 -system L3 -system L3 -system
Table 5 Replenishment schedule of the /rst four cycles for partially backlogged shortage case, completely backlogged shortage case and no-shortage case Cycle
1 2 3 4
Partially backlogged shortage
Completely backlogged shortage
No shortage
Q
T
Q
T
Q
T
451.528 700 700 700
0.714217 0.651465 0.549402 0.503031
415.561 618.209 700 700
0.67539 0.60588 0.556856 0.506288
500.476 700 700 700
0.607422 0.533803 0.434188 0.387477
The optimality condition for 1 to be minimum is given by k hk −j+1 (tj − tj−1 ) + ck − X = 0: TD(T )
(30)
j=1
The optimal replenishment schedule of the multi-warehouse inventory system without shortages can also be determined by using solution procedure similar to the proposed model. In order to illustrate how the model with no shortages works, we use again the four numerical examples just presented above for the same kind of non-linear increasing demand. Computed results are shown in Table 4. Using the same approach presented in the previous section, we can also get the corresponding replenishment schedule of subsequent replenishment cycles for the no-shortage case. The replenishment schedule of the /rst four cycles for Example 9 is given in Table 5. 7. Sensitivity analysis Using Example 5, we study the eEect of under- or overestimation of various parameters on the optimal replenishment policy and minimum average cost. Here we employ RQ = (Q − Q)=Q × 100%; RT = (T − T )=T × 100% and R = ( − )= × 100% as measures of sensitivity, where Q; T and are the true values and Q ; T and the estimated values. By increasing the values of one parameter or more at a time from −20% to 20% and making the other parameters at their true values, we observe the sensitivity analysis. The results of this analysis are presented in Figs. 2–10.
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Y.-W. Zhou / Computers & Operations Research 30 (2003) 2115 – 2134 6 4 2
-20
-10
10
20
-2 -4
Fig. 2. EEects of H1 , H2 , H3 and s on Q.
1
0.5
-20
-10
10
20
-0.5
Fig. 3. EEects of W1 , W2 , W3 and b on Q. 30 20 10 -20
-10
10
20
-10 -20
Fig. 4. EEects of A, a, all cost P., all demand P. and all P. on Q.
The following inferences can be derived from Figs. 2–10. (1) The optimal replenishment quantity, the optimal length of replenishment cycle and the average total cost of the system are slightly sensitive to the parameters A, a, all demand parameters and all parameters as compared to all other parameters.
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3 2 1
-20
-10
10
20
-1 -2
Fig. 5. EEects of H1 , H2 , H3 and s on T .
0.3 0.2 0.1 -20
-10
10
20
-0.1 -0.2
Fig. 6. EEects of W1 , W2 , W3 on T .
15 10 5 -20
-10
10
20
-5 -10
Fig. 7. EEects of A, a, all cost P., all demand P., b and all P. on T .
(2) The over- or underestimations of W1 and W2 have little eEect on the optimal replenishment quantity and the length of replenishment cycle. Moreover, it can be observed that the average total cost of the system decreases as W1 and W2 increase. It means that enlarging the capacity of warehouses with lower holding cost is pro/table to the system.
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1
-20
-10
10
20
-1
-2
Fig. 8. EEects of H1 , H2 , H3 and s on . 4
2
-20
-10
10
20
-2
-4
Fig. 9. EEects of W1 , W2 , W3 and b on . 30 20 10 -20
-10
10
20
-10 -20 -30
Fig. 10. EEects of A, a, all cost P., all demand P. and all P. on .
(3) For the change of all cost parameters, the optimal replenishment quantity and the length of replenishment cycle remain the same, but the average cost is considerably sensitive. It varies from −20% to 20%. (4) The variation of W3 has no eEect on the optimal replenishment policy and the average cost. It implies that increase of capacity of the warehouse with the highest holding cost will not aEect inventory policy of the system presented here.
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(5) The impact of parameter a and all demand parameters on the optimal replenishment policy and average cost of system is more signi/cant than that of parameter b. (6) The optimal replenishment quantity and the optimal length of replenishment cycle are slightly sensitive to parameters s, H3 and H2 as compared to H1 . The average total cost of the system is hardly sensitive to parameters s, H1 and H3 as compared to H2 . 8. Conclusions In the present paper, we have extended the existing two-warehouse models to the case with multiple warehouses by removing the unrealistic assumption in the existing two-warehouse systems. In the proposed model, we have also incorporated a type of partial lost sale into an inventory system by assuming it to be a function of shortages already backlogged. We assume that each warehouse has limited capacity, and that holding costs in diEerent warehouses are diEerent due to diEerent environments. As a result of these realistic assumptions, multi-warehouse models for determining the optimal replenishment schedule are presented for three cases (partially backlogged shortages, completely backlogged shortages and no shortages). Another feature of this paper is that we have considered a new type of increasing demand, which is not considered by others. The proposed demand has a broad area of applicability. In fact, the real life period of any product contains three phases: the growth phase, the mature phase and the declining phase. During the growth phase, the demand of the product should increase over time because customers knowing the product become more and more as this product is put on market. However, this expanding market is not unlimited but approaches maturity gradually. This implies that the demand of the product in the growth phase should increase at a decreasing rate. Therefore, the proposed demand pattern is applicable for all new types of products like a new type of TV-set, washing machine, refrigerator, etc. This multi-warehouse model can be applied to many practical situations. Now-a-days, due to globalization of the business market, the competition in business becomes more and more /erce. In order to attract more customers, most department stores in China usually provide customers with better buying environment such as the beautifully decorated showroom, enough space to let customers choose items they want, etc. As a result, it will lead to a crisis of storage space in department stores. That is to say, they will have to obtain enough storage space by renting some separated warehouses or rebuilding some new warehouses. However, from the economical point of view, the former maybe more pro/table to them. Therefore, the proposed model will give more bene/ts to these department stores, especially to big supermarket. The detailed explanation can be seen in some literature such as [11] and [13]. Further extensions of this model can be done for a bulk release pattern, deteriorating items, multi-items, incremental discount price, etc. We will consider these extensive problems in our future research. Acknowledgements The author wishes to thank the anonymous referees for their thorough reading and constructive comments and Prof. Hon-Shiang Lau for his constructive suggestions on the earlier version of the
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paper. This work is supported by the National Natural Science Fund of China under grant 79970058 and NSF of Anhui Province under grant 01046104.
Appendix A. Proof of Theorem 1. The result of the theorem will be proved by showing that the associated Hessian matrix of the solution to Eqs. (13) and (14) has positive principal minors. For doing this, we /rst /nd the second-order partial derivatives of (T1 ; T ) with respect to T1 and T . k 2 @ (T1 ; T ) 1 = D(T1 ) s exp(−(T − T1 )) + hk t1 + hk −j+1 (tj − tj−1 ) ; (A.1) 2 T @T1 j=2 where tj denotes the derivative of tj with respect to T1 (j = 1; 2; : : : ; k).
and
1 @2 (T1 ; T ) = − sD(T1 ) exp(−(T − T1 )) @T1 @T T
(A.2)
T 1 @2 (T1 ; T ) = s D(T ) − D(u) exp(−(T − u)) du : @T 2 T T1
(A.3)
Using the monotonicity of the demand function D(t), we have T 1 @2 (T1 ; T ) ¿ s D(T ) − D(T ) exp(−(T − u)) du @T 2 T T1 =
s D(T ) exp(−(T − T1 )) ¿ 0: T
(A.4)
Let "1 and "2 be the order 1 and order 2 principal minors of the associated Hessian matrix of the solution to Eqs. (13) and (14), respectively. Then it is clear that 2 2 @2 @ 2 @2 @ "1 = and " = − : (A.5) 2 @T1 @T @T12 @T12 @T 2 Next, we will prove that "1 and "2 are both greater than zero. Since kj=1 (tj − tj−1 ) = tk = T1 , diEerentiating this equation with respect to T1 , we have
t1 +
k
(tj − tj−1 ) = 1:
(A.6)
j=2
On the other hand, after diEerentiating (1) and (2) with respect to T1 , we obtain t1 = D(T1 )=D(t1 ) ¿ 0
and
D(tj )tj − D(tj−1 )tj−1 = 0 for j = 2; 3; : : : ; k:
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Thus, when D(t) is an increasing function of time t, for j = 2; 3; : : : ; k tj − tj−1 = D(tj )(tj − tj−1 )=D(tj ) ¡ (D(tj )tj − D(tj−1 )tj−1 )=D(tj ) = 0:
(A.7)
Hence, it can be obtained from (A.1)–(A.3) and (A.5) that k 1 (tj − tj−1 ) "1 ¿ D(T1 ) s exp(−(T − T1 )) + hk t1 + T j=2 1 D(T1 )[s exp(−(T − T1 )) + hk ] ¿ 0: T Furthermore, we have from (A.4), (A.5) and (A.8) s "2 ¿ 2 D(T1 ){D(T )[s exp(−(T − T1 )) + hk ] − s2 D(T1 ) exp(−(T − T1 ))} T =
(A.8)
exp(−(T − T1 )) s2 D(T1 )[D(T ) − 2 D(T1 )] exp(−2(T − T1 )) ¿ 0: T2 Therefore, the associated Hessian matrix of the solution to Eqs. (13) and (14) has positive principal minors. The proof of theorem is completed. ¿
References [1] Donaldson WA. Inventory replenishment policy for a linear trend in demand-an analytical solution. Operational Research Quarternary 1977;28:663–70. [2] Teng JT. A note on inventory replenishment policy for increasing demand. Journal of the Operational Research Society 1994;45:619–43. [3] Goyal SK, Hariga M, Alyan A. The trended inventory lot sizing problem with shortages under a new replenishment policy. Journal of the Operational Research Society 1996;47:1286–95. [4] Hill RM. Inventory models for increasing demand followed by level demand. Journal of the Operational Research Society 1995;46:1250–9. [5] Bhunia AK, Maiti M. An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand. Applied Mathematical Modelling 1999;23:301–8. [6] Hartely VR. Operations research-A managerial emphasis, Santa Monica, CA: Good Year, 1976. p. 315 –7 [chapter 12]. [7] Sarma KVS. A deterministic order-level inventory model for deteriorating items with two storage facilities. European Journal of Operational Research 1987;29:70–2. [8] Pakkala TPM, Achary KK. A deterministic inventory model for deteriorating items with two warehouses and /nite replenishment rate. European Journal of Operational Research 1992;57:71–6. [9] Sarma KVS. A deterministic inventory model with two level of storage and an optimum release rate. Opsearch 1983;20:175–80. [10] Goswami A, Chaudhuri KS. An economic order quantity model for items with two levels of storage for a linear trend in demand. Journal of the Operational Research Society 1992;43:157–67. [11] Bhunia AK, Maiti M. A two-warehouse inventory model for deteriorating items with a linear trend in demand and shortages. Journal of the Operational Research Society 1998;49:287–92. [12] Zhou YW. An optimal EOQ model for deteriorating items with two warehouses and time-varying demand. Mathematica Applicata 1998;10:19–23.
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[13] Kar S, Bhunia AK, Maiti M. Deterministic inventory model with two levels of storage, a linear trend in demand and a /xed time horizon. Computers & Operations Research 2001;28:1315–31. Yong-Wu Zhou is a professor of Operations Research at Hefei University of Technology, Peoples’ Republic of China. He received his M.S. degree in applied mathematics from Northwest University of Technology at Xi’an and his Ph.D. in management science from Hefei University of Technology. His major research /elds include production-inventory control, operation research, risk decision, etc. More than 40 papers have appeared, respectively, in Journal of the Operational Research Society, Applied Mathematical Modelling, Computers & Operations Research, Journal of System Engineering and Electronics, Journal of System Science and System Engineering, Systems Engineering-Theory & Practice, etc.
Computers & Operations Research 30 (2003) 2115 – 2134
www.elsevier.com/locate/dsw
A multi-warehouse inventory model for items with time-varying demand and shortages Yong-Wu Zhou Department of Mathematics, Hefei University of Technology, Hefei, Anhui 230009, China Received 1 November 2001; received in revised form 1 April 2002
Abstract This paper develops a deterministic replenishment model with multiple warehouses (one is an owned warehouse and others are rented warehouses) possessing limited storage capacity. In this model, the replenishment rate is in/nite. The demand rate is a function of time and increases at a decreasing rate. The stocks of rented warehouses are transported to owned warehouse in continuous release pattern. The model allows shortages in owned warehouse and permits part of the backlogged shortages to turn into lost sales—which is assumed to be a function of the currently backlogged amount. The solution procedure for /nding the optimal replenishment policy is shown. As a special case of the model, the corresponding models with completely backlogged shortages and without shortages are also presented. The models are illustrated with the help of numerical examples. Sensitivity analysis of parameters is given in graphical form. Scope and purpose In practical inventory management, there exist many factors like an attracted price discount for bulk purchase, etc. to make retailers buy goods more than the capacity of their owned warehouse. In this case, retailers will need to rent other warehouses or to rebuild a new warehouse. However, from economical point of views, they usually choose to rent other warehouses. If there are multiple warehouses available, an important problem faced by the retailers is which warehouses to be selected to hold items replenished, when to replenish as well as what size to replenish. For such a problem, the existing two-warehouse models, based on an unrealistic assumption that the rented warehouse has unlimited storage capacity, presented some procedures for determining the optimal replenishment policy. This paper extends the existing two-warehouse models in three directions. Firstly, the traditional two-warehouse models assumed the storage capacity of the rented warehouse unlimited. The present paper relaxes this impractical assumption and considers the situation with multiple rented warehouses having a limited capacity. Secondly, the traditional two-warehouse models considered a constant demand rate or a linearly increasing demand rate. In this model, the demand rate varies over time and increases at a decreasing rate, which implies an increasing market going to saturation. Thirdly, we extend the two-warehouse models to the case with partially backlogged shortages. The purpose of this paper is to E-mail address: [email protected] (Y.-W. Zhou). 0305-0548/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 2 ) 0 0 1 2 6 - 0
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build a multi-warehouse replenishment model to help decision-makers solve the problem of which warehouses to be chosen to store items replenished and how to replenish. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Time-varying demand; Inventory; Multi-warehouse; Lost sale
1. Introduction Since Donaldson [1] /rst developed an exact solution procedure to determine both the optimal amount and the optimal time of replenishment, many researchers like Teng [2], Goyal et al. [3], Hill [4], Bhunia and Maiti [5], etc. have paid considerably more attention to the inventory problems with deterministic time-varying demand pattern. By using diEerent methods or from diEerent angles, they either proposed various other heuristic approaches for Donaldson’s problem or extended it to some more practical situations. All these models dealt with the case of a single warehouse and assumed that the available warehouse has unlimited capacity. However, in real life, any warehouse has /nite capacity. On the other hand, due to some reasons such as an attracted price discount for bulk purchase, the order costs higher than one using rented warehouse (RW), and so on, inventory managers usually are attracted to hold more items than can be stored in an owned warehouse (OW). Therefore, multiple warehouses should be required to hold very large inventories. In recent years, various researchers have discussed a two-warehouse inventory system. This system has two warehouses (OW and RW). OW has /nite capacity and RW has in/nite capacity. Since the inventory holding cost is usually higher in RW than in OW, the system /rst supplies items stored in RW to customers and then gives items in OW to customers. Hartely [6] was the /rst to develop the basic two-warehouse inventory model. He proposed a heuristic procedure for determining the optimal order quantity. Sarma [7] further developed the two-warehouse model for deteriorating items with an in/nite replenishment rate and shortages. Pakkala and Achary [8] considered the two-warehouse model for deteriorating items with a /nite replenishment rate and shortages. In all these models, the demand was assumed to be constant and the cost of transporting items from RW to OW was not taken into account. Sarma [9] presented a model with an in/nite replenishment rate by considering the transportation cost from RW to OW to be a /xed constant independent of the quantity being transported, but he did not consider shortages in his model. Goswami and Chaudhuri [10] further developed the model with or without shortages by assuming that the demand varies over time with a linearly increasing trend and that the transportation cost from RW to OW depends on the quantity being transported. In their model, stock was transferred from RW to OW in an intermittent pattern. Through employing a continuous transportation pattern, Bhunia and Maiti [11] developed a two-warehouse model for deteriorating items with a linearly increasing demand and shortages during the in/nite period. Both models developed in [10] and [11] considered the case with a linearly increasing demand and an in/nite time horizon. Nevertheless, a linearly increasing demand with the in/nite horizon looks somewhat less realistic. Employing the /nite planning horizon, Zhou [12] presented a two-warehouse model for deteriorating items with time-varying demand and shortages backordered. In a recent paper, Kar et al. [13] proposed a similar two-warehouse model for non-perishable items with a linear trend in demand and /nite time horizon by considering
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inventory-dependent replenishment cost. However, all these two-warehouse models were based on an impractical assumption that the rented warehouse has unlimited capacity. In this paper, we develop a deterministic inventory model with m warehouses (one is OW and the others are RWs). This model is diEerent from the existing two-warehouse models in three aspects: Firstly, each warehouse possesses limited storage capacity in the present model. Secondly, the existing models with time-varying demand considered the demand rate as a linearly increasing function of time in the in/nite time horizon. It seems to be unreasonable. In this model, the demand rate is a continuous function of time and increases at a decreasing rate. This represents an expanding market approaching saturation. Thirdly, all of the two-warehouse models either did not allow shortages to occur or considered shortages to be backlogged completely. In the practical inventory management, however, not all customers wait for a long time to buy goods from a particular shop although they are likely to be motivated to wait owing to some factors such as a special discount price, goodwill, etc. This means that shortages are not completely backlogged in the real world. This model permits part of the backlogged shortages to turn into lost sales, which makes the model more practical. Like in Bhunia and Maiti’s model [11], where stock is also transported from each RW to OW in continuous pattern. The model with partially backlogged shortages is /rst developed. Then, as a special case of the present model, the corresponding models with completely backlogged shortages and without shortages are also shown. Numerical examples are used to illustrate how the models work. The eEect of all parameters on the optimal replenishment policy is observed by sensitivity analysis.
2. Assumptions and notations To develop the proposed model, the following assumptions and notations are used throughout the whole paper. 1. D(t) is the demand rate, which is a continuous function of time t and increases at a decreasing rate, i.e., D(t) satis/es dD=dt ¿ 0 but d 2 D=dt 2 ¡ 0. 2. T is the length of the replenishment cycle. 3. Q is the replenishment quantity at the beginning of the cycle. 4. m is the number of warehouses that can be used by the system. 5. Due to diEerent storage environment, inventory-holding costs in diEerent warehouses are considered to be diEerent. Inventory-holding cost in OW is usually the lowest. All warehouses are sequenced according to the order of the holding cost from small to big. 6. H1 is the holding cost per unit per unit time in OW and W1 is the storage capacity of OW. 7. Hi is the holding cost per unit per unit time in the ith RW, Wi is the storage capacity of the ith RW and Ci is the transportation cost per unit of items moved from the ith RW to OW (i = 2; 3; : : : ; m). Then we have H1 ¡ H2 ¡ · · · ¡ Hm because of assumption 5. 8. A is the /xed cost per replenishment. 9. Shortages are allowed and assumed to backlog partially. T1 is the time at which shortages occur during a replenishment cycle. Generally speaking, customers are usually impatient. Some of them will not wait for goods in a special shop if many other customers have been waiting. Moreover, the more customers have been waiting, the more customers among them choose to leave the
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shop. So we here assume that the lost sale is proportional to the current shortage level. Let be the scale parameter and s the shortage cost per unit per unit time (0 ¡ ¡ 1). 10. Replenishment rate is in/nite and lead time is zero. 11. Lk denotes an inventory system that uses k warehouses (one is an owned warehouse and others are rented warehouses) to store items (k = 2; 3; : : : ; m). L1 denotes an inventory system that uses only OW. k is the minimal average total cost of Lk -system. 12. L∗k denotes the optimal Lk -system in which its minimal average total cost is the smallest among those of all Lk -systems.
3. Model formulation for Lk -system In order to develop the inventory model of Lk -system, we arbitrarily select (k − 1) warehouses among all rented warehouses. Then the selected (k − 1) warehouses and OW become an Lk -system. For convenience, k warehouses are sequenced again according to the order of the holding cost from small to big. Based on assumption 5, it is obvious that the /rst warehouse is OW here. And let wi be the storage capacity of the ith warehouse, hi the holding cost per unit per unit time in the ith warehouse and ci the transportation cost per unit of items shipped from the ith warehouse to OW (i = 1; 2; : : : ; k). It is very clear that c1 = 0. Next, we study the Lk -system neglecting the capacity constraint of the kth warehouse. In this system, wj units are kept in the jth warehouse (j = 1; 2; : : : ; k − 1) and Q − qk −1 units in the kth warehouse, where qj = ji=1 wi (j = 1; 2; : : : ; k) and q0 = 0. For economic reasons, if more than one warehouse has been used in practice, items of warehouse with lower holding cost are usually consumed only after stock in the warehouse with higher holding cost is used up. Therefore, items in the kth warehouse are /rstly used to meet the demand. Then, items in the (k − 1)th warehouse are used, and so on. If we let tj be the time at which stock in the (k + 1 − j)th warehouse drops to zero (j = 1; 2; : : : ; k), it is obvious that tk = T1 . The graphical representation of the Lk -system is shown in Fig. 1 when k = 3. As stated before, we have t1 tk+1−j D(u) du = Q − qk −1 and D(u) du = wj (j = k − 1; k − 2; : : : ; 1): (1) 0
tk − j
And the sum of the above equations is T1 D(u) du = Q: 0
(2)
During the interval [0; t1 ], the stock in the kth warehouse is depleted only due to demand and drops to zero at time t = t1 . So the holding cost of items in the kth warehouse during [0; t1 ] is given by t hk 0 1 D(u)u du and the holding cost of items in the jth warehouse is hj wj t1 (j = 1; 2; : : : ; k − 1). Thus, the holding cost of items in the whole system during [0; t1 ] is hk
0
t1
D(u)u du +
k −1 j=1
hj w j t1 :
(3)
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Inventory level Q_q2
The 3rd warehouse
The 2nd warehouse
w2
The 1st warehouse w1 t1
t3 = T1
t2
T
time
Fig. 1. Graphical representation of the L3 -system with shortages.
Similarly, in the interval [tj−1 ; tj ], the holding cost of items in the system is given by tj k −j hk+1−j D(u)(u − tj−1 ) du + hi wi (tj − tj−1 ) tj − 1
(4)
i=1
and the holding cost of the system over the interval [tk −1 ; tk ] is tk h1 D(u)(u − tk −1 ) du:
(5)
tk − 1
Therefore, the holding cost of the inventory system in the replenishment cycle can be written as k −j tj k k −1 where t0 = 0: hk+1−j D(u)(u − tj−1 ) du + (tj − tj−1 ) hi w i (6) HC = j=1
tj − 1
j=1
i=1
After some simple operations, the above formula becomes tj k hk+1−j D(u)u du: HC = j=1
(7)
tj − 1
The cost of transporting units from all RWs to OW is given by TC =
k −1 j=1
cj wj + ck (Q − qk −1 ) =
k −1
(cj − ck )wj + ck Q
where c1 = 0:
(8)
j=1
According to assumption 9, let I (t) be the inventory level at time t during the shortage period. Then I (t) satis/es dI (t) = −D(t) − I (t); T1 6 t 6 T: (9) dt
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Using the boundary condition I (T1 ) = 0, we can obtain the solution to Eq. (9): T1 I (t) = D(u) exp(−(t − u)) du T1 6 t 6 T: t
Consequently, the shortage cost for the entire cycle is T T s (−I (t)) dt = D(u)[1 − exp(−(T − u))] du: SC = s T1 T1
(10)
(11)
Thus, the average total cost of the system during the replenishment cycle is given by = {A + HC + TC + SC }=T tj k k −1 = A+ hk+1−j D(u)u du + (cj − ck )wj tj − 1 j=1
+ ck
0
T1
j=1
s D(u) du +
T
T1
D(u)(1 − exp(−(T − u))) du
T:
(12)
From (1) and (2), it can be sen that the average total cost, , is a function of variables T1 and T (denoted by (T1 ; T )). The necessary condition for (T1 ; T ) to be minimum is @(T1 ; T ) @(T1 ; T ) = 0; = 0 and @T1 @T which gives k
s hk −j+1 (tj − tj−1 ) + ck − (1 − exp(−(T − T1 ))) = 0; j=1 T tj k s D(u)[(T + 1) exp(−(T − u)) − 1] du − A + hk −j+1 D(u)u du T1 tj − 1
(13)
j=1
+
k −1 j=1
(cj − ck )wj + ck
0
T1
D(u) du
= 0:
(14)
Theorem 1. The replenishment schedule satisfying the 3rst-order optimality conditions (13) and (14) satis3es the second-order conditions for a minimum. (Proof seen in Appendix A) Let T1∗ and T ∗ be the optimal values of T1 and T . Then, from (2) we can derive the optimal replenishment quantity, Qk , of the Lk -system ignoring the capacity constraints of the kth warehouse. If qk −1 6 Qk 6 qk , it is obvious that T1∗ and T ∗ are the feasible solutions of the Lk -system and k = (T1∗ ; T ∗ ). Otherwise, k is equal to the optimal boundary cost (Tqk ; T # ) when Qk = qk . Now we determine the optimal boundary cost (Tqk ; T # ). First, by /xing Q at qk , from (2) and (1)
Y.-W. Zhou / Computers & Operations Research 30 (2003) 2115 – 2134
we have Tq 0
k
D(u) du = qk
and
tj
tj − 1
D(u) du = wk −j+1 ; j = 1; 2; : : : ; k:
2121
(15)
In this case, the boundary cost function of the system is given by tj k k hk+1−j D(u)u du + cj w j (Tqk ; T ) = A + tj − 1 j=1
+
s
T
Tq k
j=1
D(u)[1 − exp(−(T − u))] du
T;
which depends only on T . The necessary condition for (Tqk ; T ) to be minimum is d(Tqk ; T )=dT = 0; which gives T s D(u)[(T + 1) exp(−(T − u)) − 1] du Tq k tj k hk −j+1 = 0: D(u)u du + cj wj − A+ tj − 1
(16)
(17)
j=1
It can be easily proved that the solution to Eq. (17) also satis/es the second-order suOcient condition. Generally, Eq. (17) can be solved by using any one-dimensional search technique. Substituting the optimal value T # of T into (16), we obtain the optimal boundary cost (Tqk ; T # ). For obtaining the optimal L∗k -system, one should reselect (k − 1) warehouses from all RWs. Thus, another Lk -system is obtained. Using the method presented above, one can derive the replenishment policy and the corresponding minimal average total cost of this Lk -system. After repeating this process till all diEerent Lk -systems have been discussed, one can get a list of the minimal average total costs related to those Lk -systems. Then the L∗k -system can be determined by comparing these costs. The method for /nding the L∗k -system presented above can be summarized as the following algorithm: Algorithm 1. Set c1 = C1 ; w1 = W1 ; h1 = H1 ; k∗ = +∞. If k = 1, go to step 5; Otherwise, go to the next step. Take (k − 1) integers from 2 to m (say 2 6 i2 ¡ i3 ¡ · · · ¡ ik 6 m). Set hj = Hij ; cj = Cij and wj = Wij for j = 2; : : : ; k. Solve Eqs. (13) and (14) for determining T1 and T . Then calculate Qk and k from (1), (2) and (12). 6. If Qk ¿ qk , go to step 7; Otherwise, go to step 8. 7. Set Qk = qk and compute Tqk from (15). Then solve Eq. (17) for getting T # , compute (Tqk ; T # ) from (16) and set k = (Tqk ; T # ); T1 = Tqk T = T # .
1. 2. 3. 4. 5.
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8. If k ¡ k∗ , then k∗ = k , T1∗ = T1 , T ∗ = T Qk∗ = Qk and Ik∗ = {1; i2 ; i3 ; : : : ; ik }, go to step 9; Otherwise, go to step 9. 9. If k = 1, output k∗ , T1∗ , T ∗ , Qk∗ and Ik∗ , stop; Otherwise, go to step 10. 10. If another group of (k − 1) integers, which is diEerent from those taken earlier, can be got from 2 to m, then draw a new group of (k − 1) integers from 2 to m and go to step 4; Otherwise, output k∗ , T1∗ , T ∗ , Qk∗ and Ik∗ , stop. 4. Solution procedure for a multi-warehouse inventory system Now we can determine the optimal replenishment schedule of the proposed system according to the method given below. First, consider an L1 -system which only uses OW to store items and determine its optimal replenishment quantity, Q1∗ , and average total cost, 1∗ . If Q1∗ ¡ w1 (=q1 ), the system does not need to rent any warehouse. Q1∗ is the optimal replenishment quantity of the system. Otherwise, determine the optimal L∗2 -system, the corresponding replenishment schedule and average total cost, 2∗ , by using the method provided in the previous section. If Q2∗ ¡ q2 , decision-makers will not need to rent any warehouse. At this moment, 2∗ has to be compared with 1∗ . When 1∗ 6 2∗ , decision-makers of the system should use the L1 -system. Conversely, they should use the L∗2 -system. Otherwise, decision-makers have to determine the L∗3 -system and its corresponding replenishment schedule and average total cost. If Q3∗ ¡ q3 , decision-makers should use the Li -system while i∗ = min3j=1 {j∗ }; otherwise, the above process is repeated up to the Lm -system. This solution procedure can be written as the following algorithm: Algorithm 2. 1. Input the values of all parameters. 2. Set k = 1 and ∗ = +∞. 3. Determine L∗k -system and the corresponding replenishment schedule and average total cost (Ik∗ ; T1∗ ; T ∗ ; Qk∗ and k∗ ) by using Algorithm 1. 4. If ∗ ¿ k∗ , then set ∗ = k∗ , T1∗∗ = T1∗ , T ∗∗ = T ∗ , Q∗ = Qk∗ and I ∗ = Ik∗ , go to step 5; Otherwise, go to step 5. 5. If Qk∗ ¡ qk , output ∗ ; T1∗∗ , T ∗∗ , Q∗ and I ∗ , stop; Otherwise, go to the next step. 6. If k = m, then output ∗ ; T1∗∗ , T ∗∗ , Q∗ and I ∗ , stop; Otherwise, k = k + 1, go to step 3. 5. Numerical examples To illustrate the proposed model, we show some numerical examples with a special non-linear increasing demand. Let D(t) = D − a exp(−bt) (a ¿ 0; b ¿ 0). From (1) and (2), we have Q = DT1 + a(exp(−bT1 ) − 1)=b and D(tk+1−j − tk −j ) + a(exp(−btk+1−j ) − exp(−btk −j ))=b = wj ;
j = 1; 2; : : : ; k − 1:
(18)
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In this situation, the average total cost of the Lk -system ignoring the capacity constraint of the kth warehouse becomes k −1 1 A+ (cj − ck )wj + ck [DT1 + a(exp(−bT1 ) − 1)=b] (T1 ; T ) = T j=1
k D 2 2 a (t − t ) + 2 ((btj + 1) exp( − btj ) − (btj−1 + 1) exp(−btj−1 )) hk+1−j + 2 j j −1 b j=1 1 s (D + a exp(−bT ))(T − T1 ) − (D + a exp(−bT1 ))(1 − exp(−b(T − T1 ))) + b for b = and
(19a)
k −1 1 A+ (cj − ck )wj + ck [DT1 + a(exp(−bT1 ) − 1)=b] (T1 ; T ) = T j=1 k D 2 a (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) hk+1−j + 2 b j=1 − (btj−1 + 1) exp(−btj−1 )) a s + D(T − T1 ) + (exp(−bT ) − exp(−bT1 )) b( − b) D a exp(−bT1 ) − (1 − exp(−(T − T1 ))) + −b
for b = :
The corresponding optimality conditions for (T1 ; T ) to be minimum is given by k s hk −j+1 (tj − tj−1 ) + ck − (1 − exp(−(T − T1 ))) = 0; h k t1 + T j=2 s (D − a exp(−bu))[(T + 1) exp(−(T − u)) − 1] du T1 k −1 k a (cj − ck )wj + ck (exp(−bT1 ) − 1) + ck DT1 + hk −j+1 − A+ b j=1 j=1 D 2 a (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) − (btj−1 + 1) exp(−btj−1 )) × = 0: 2 b If /xing Q at qk , then the values of t1 ; t2 ; : : : ; tk (=T1 ) can be obtained easily from (18).
(19b)
(20)
(21)
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Table 1 Optimum solution of the model for diEerent values of demand parameters Examples Example Example Example Example
1 2 3 4
a
b
T∗
Q∗
∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.714317 0.512434 0.574863 0.500487
451.528 500 700 700
974.905 1355.09 1423.54 1609.07
L2 -system L2 -system L3 -system L3 -system
At this moment, the boundary cost of the Lk -system becomes k D 2 1 a A+ (tj − tj2−1 ) + 2 ((btj + 1) exp(−btj ) − (btj−1 + 1) (Tqk ; T ) = hk+1−j T 2 b j=1
T k s cj w j + (D − a exp(−bu))[1 − exp(−(T − u))] du : × exp(−btj−1 )) + Tq j=1 k
(22) The corresponding optimality condition for (Tqk ; T ) to be minimum is T k s D 2 2 hk −j+1 (D − a exp(−bu))[(T + 1) exp(−(T − u)) − 1] du − A + (t − t ) Tq 2 j j −1 j=1
k
+
a ((btj + 1) exp(−bj t) − (btj−1 + 1) exp(−bj t1 )) b2
+ c j wj
= 0:
(23)
The proposed model is illustrated by the following numerical examples. Data 1. Let m = 3; H1 = 0:5; H2 = 0:85; H3 = 1:2; W1 = 100; W2 = 200; W3 = 400; C2 = 0:4; C3 = 0:25; s = 6; D = 2500; = 0:8 and A = 400 in appropriate units. According to the proposed solution procedure, we present the optimal replenishment schedule and the corresponding minimal average cost for diEerent values of demand parameters with the help of the Mathematica software. Results are given in Table 1. Table 2 shows the eEect of change of parameter on the optimal replenishment policy. From Table 2, we can observe that has little impact on the optimal replenishment policy. After the /rst cycle is completed, inventory managers can transfer value of the constant ‘a’ into a exp(−bT ) (where T is the associated value of the previous cycle) and employ this model again. Thus, they can obtain the replenishment schedule for the next cycle, and so on. According to this approaching, inventory managers can obtain the replenishment policies of subsequent replenishment cycles. Here we show the replenishment schedule of the /rst four cycles only for Example 1. The result is given in Table 5.
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Table 2 EEect of parameter on the optimal replenishment policy
T1
T
Q
0.1 0.2 0.4 0.7 0.9
0.56582 0.56543 0.564453 0.563083 0.562109
0.710405 0.710976 0.711991 0.71373 0.714923
454.99 454.57 453.52 452.052 451.004
978.317 977.847 976.89 975.41 974.393
6. Special cases 6.1. Model with completely backlogged shortages If → 0 in the proposed model, we can obtain the corresponding multi-warehouse model with completely backlogged shortages. For this special case, the average cost of the Lk -system ignoring the capacity constraint of the kth warehouse is given by tj k k −1 1 A+ 1 = hk −j+1 D(u)u du + (cj − ck )wj T tj − 1 j=1
+ ck where
t1
0
0
T1
j=1
D(u) du + s
T1
D(u) du = Q − qk −1 =
T
T1
0
D(u)(T − u) du
;
(24)
D(u) du − qk −1 and
tj
tj − 1
D(u) du = wk −j+1 ; j = 2; 3; : : : ; k:
The optimality condition for 1 to be minimum is given by k
hk −j+1 (tj − tj−1 ) + ck − s(T − T1 ) = 0; tj T k D(u)u du − A + hk −j+1 D(u)u du s T1 tj − 1
(25)
j=1
j=1
+
k −1
(cj − ck )wj + ck
j=1
0
T1
D(u) du
= 0:
(26)
Under this case, the corresponding boundary cost function of the system is given by tj T k k 1 (Tqk ; T ) = A + hk+1−j D(u)u du + cj wj + s D(u)(T − u) du T; Tq tj − 1 j=1
j=1
k
(27)
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Table 3 Optimum solution of the model with completely backlogged shortages Examples Example Example Example Example
5 6 7 8
a
b
T∗
Q∗
1∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.67539 0.477257 0.362696 0.487305
415.561 437.017 300 673.112
973.896 1377.38 1684.77 1675.58
L3 -system L3 -system L2 -system L3 -system
where Tqk and tj satisfy Tq k D(u) du = qk 0
and
tj
tj − 1
D(u) du = wk −j+1 (j = 1; 2; : : : ; k):
And the optimality condition for 1 (Tqk ; T ) to be minimum is tj T k hk −j+1 = 0: D(u)u du − A + D(u)u du + cj wj s Tq tj − 1
(28)
j=1
k
Using similar solution procedure to the model proposed in previous section, we can determine the optimal replenishment schedule of the multi-warehouse inventory system with completely backlogged shortages. To illustrate the solution procedure, we also show four numerical examples for the same kind of non-linear increasing demand. The values of all parameters are given in Data 2 except a and b. Computed results are shown in Table 3. Data 2. Let m = 3; H1 = 0:5; H2 = 0:85; H3 = 1:2; W1 = 100; W2 = 200; W3 = 400; C2 = 0:25; C3 = 0:40; s = 6; D = 2500 and A = 400 in appropriate units. After the /rst cycle is completed, we can also get the corresponding replenishment schedule of subsequent replenishment cycles by using the same approach presented in the previous section. Here we give the replenishment schedule of the /rst four cycles only for Example 5 in Table 5. 6.2. Model without shortages If T1 = T in the proposed model, we can also obtain the corresponding multi-warehouse inventory model with no consideration of shortages. In this special case, the average cost of the Lk -system ignoring the capacity constraint of the kth warehouse will become tj T k 1 X A+ hk −j+1 D(u)u du + (cj − ck )wj + ck D(u) du ; (29) 2 (T ) = = T T 0 tj − 1 j=1
where
0
t1
D(u) du = Q − qk −1 =
0
T
D(u) du − qk −1
and
tj
tj − 1
D(u) du = wk −j+1 ; j = 2; 3; : : : ; k:
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Table 4 Optimum solution of the model without shortages Examples Example Example Example Example
9 10 11 12
a
b
T∗
Q∗
2∗
Selection
2000 2000 1000 1000
0.6 2.8 0.6 2.8
0.607422 0.429883 0.432319 0.372517
500.476 574.772 700 700
1098.15 1588.73 1724.64 1963.59
L3 -system L3 -system L3 -system L3 -system
Table 5 Replenishment schedule of the /rst four cycles for partially backlogged shortage case, completely backlogged shortage case and no-shortage case Cycle
1 2 3 4
Partially backlogged shortage
Completely backlogged shortage
No shortage
Q
T
Q
T
Q
T
451.528 700 700 700
0.714217 0.651465 0.549402 0.503031
415.561 618.209 700 700
0.67539 0.60588 0.556856 0.506288
500.476 700 700 700
0.607422 0.533803 0.434188 0.387477
The optimality condition for 1 to be minimum is given by k hk −j+1 (tj − tj−1 ) + ck − X = 0: TD(T )
(30)
j=1
The optimal replenishment schedule of the multi-warehouse inventory system without shortages can also be determined by using solution procedure similar to the proposed model. In order to illustrate how the model with no shortages works, we use again the four numerical examples just presented above for the same kind of non-linear increasing demand. Computed results are shown in Table 4. Using the same approach presented in the previous section, we can also get the corresponding replenishment schedule of subsequent replenishment cycles for the no-shortage case. The replenishment schedule of the /rst four cycles for Example 9 is given in Table 5. 7. Sensitivity analysis Using Example 5, we study the eEect of under- or overestimation of various parameters on the optimal replenishment policy and minimum average cost. Here we employ RQ = (Q − Q)=Q × 100%; RT = (T − T )=T × 100% and R = ( − )= × 100% as measures of sensitivity, where Q; T and are the true values and Q ; T and the estimated values. By increasing the values of one parameter or more at a time from −20% to 20% and making the other parameters at their true values, we observe the sensitivity analysis. The results of this analysis are presented in Figs. 2–10.
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-20
-10
10
20
-2 -4
Fig. 2. EEects of H1 , H2 , H3 and s on Q.
1
0.5
-20
-10
10
20
-0.5
Fig. 3. EEects of W1 , W2 , W3 and b on Q. 30 20 10 -20
-10
10
20
-10 -20
Fig. 4. EEects of A, a, all cost P., all demand P. and all P. on Q.
The following inferences can be derived from Figs. 2–10. (1) The optimal replenishment quantity, the optimal length of replenishment cycle and the average total cost of the system are slightly sensitive to the parameters A, a, all demand parameters and all parameters as compared to all other parameters.
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3 2 1
-20
-10
10
20
-1 -2
Fig. 5. EEects of H1 , H2 , H3 and s on T .
0.3 0.2 0.1 -20
-10
10
20
-0.1 -0.2
Fig. 6. EEects of W1 , W2 , W3 on T .
15 10 5 -20
-10
10
20
-5 -10
Fig. 7. EEects of A, a, all cost P., all demand P., b and all P. on T .
(2) The over- or underestimations of W1 and W2 have little eEect on the optimal replenishment quantity and the length of replenishment cycle. Moreover, it can be observed that the average total cost of the system decreases as W1 and W2 increase. It means that enlarging the capacity of warehouses with lower holding cost is pro/table to the system.
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1
-20
-10
10
20
-1
-2
Fig. 8. EEects of H1 , H2 , H3 and s on . 4
2
-20
-10
10
20
-2
-4
Fig. 9. EEects of W1 , W2 , W3 and b on . 30 20 10 -20
-10
10
20
-10 -20 -30
Fig. 10. EEects of A, a, all cost P., all demand P. and all P. on .
(3) For the change of all cost parameters, the optimal replenishment quantity and the length of replenishment cycle remain the same, but the average cost is considerably sensitive. It varies from −20% to 20%. (4) The variation of W3 has no eEect on the optimal replenishment policy and the average cost. It implies that increase of capacity of the warehouse with the highest holding cost will not aEect inventory policy of the system presented here.
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(5) The impact of parameter a and all demand parameters on the optimal replenishment policy and average cost of system is more signi/cant than that of parameter b. (6) The optimal replenishment quantity and the optimal length of replenishment cycle are slightly sensitive to parameters s, H3 and H2 as compared to H1 . The average total cost of the system is hardly sensitive to parameters s, H1 and H3 as compared to H2 . 8. Conclusions In the present paper, we have extended the existing two-warehouse models to the case with multiple warehouses by removing the unrealistic assumption in the existing two-warehouse systems. In the proposed model, we have also incorporated a type of partial lost sale into an inventory system by assuming it to be a function of shortages already backlogged. We assume that each warehouse has limited capacity, and that holding costs in diEerent warehouses are diEerent due to diEerent environments. As a result of these realistic assumptions, multi-warehouse models for determining the optimal replenishment schedule are presented for three cases (partially backlogged shortages, completely backlogged shortages and no shortages). Another feature of this paper is that we have considered a new type of increasing demand, which is not considered by others. The proposed demand has a broad area of applicability. In fact, the real life period of any product contains three phases: the growth phase, the mature phase and the declining phase. During the growth phase, the demand of the product should increase over time because customers knowing the product become more and more as this product is put on market. However, this expanding market is not unlimited but approaches maturity gradually. This implies that the demand of the product in the growth phase should increase at a decreasing rate. Therefore, the proposed demand pattern is applicable for all new types of products like a new type of TV-set, washing machine, refrigerator, etc. This multi-warehouse model can be applied to many practical situations. Now-a-days, due to globalization of the business market, the competition in business becomes more and more /erce. In order to attract more customers, most department stores in China usually provide customers with better buying environment such as the beautifully decorated showroom, enough space to let customers choose items they want, etc. As a result, it will lead to a crisis of storage space in department stores. That is to say, they will have to obtain enough storage space by renting some separated warehouses or rebuilding some new warehouses. However, from the economical point of view, the former maybe more pro/table to them. Therefore, the proposed model will give more bene/ts to these department stores, especially to big supermarket. The detailed explanation can be seen in some literature such as [11] and [13]. Further extensions of this model can be done for a bulk release pattern, deteriorating items, multi-items, incremental discount price, etc. We will consider these extensive problems in our future research. Acknowledgements The author wishes to thank the anonymous referees for their thorough reading and constructive comments and Prof. Hon-Shiang Lau for his constructive suggestions on the earlier version of the
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paper. This work is supported by the National Natural Science Fund of China under grant 79970058 and NSF of Anhui Province under grant 01046104.
Appendix A. Proof of Theorem 1. The result of the theorem will be proved by showing that the associated Hessian matrix of the solution to Eqs. (13) and (14) has positive principal minors. For doing this, we /rst /nd the second-order partial derivatives of (T1 ; T ) with respect to T1 and T . k 2 @ (T1 ; T ) 1 = D(T1 ) s exp(−(T − T1 )) + hk t1 + hk −j+1 (tj − tj−1 ) ; (A.1) 2 T @T1 j=2 where tj denotes the derivative of tj with respect to T1 (j = 1; 2; : : : ; k).
and
1 @2 (T1 ; T ) = − sD(T1 ) exp(−(T − T1 )) @T1 @T T
(A.2)
T 1 @2 (T1 ; T ) = s D(T ) − D(u) exp(−(T − u)) du : @T 2 T T1
(A.3)
Using the monotonicity of the demand function D(t), we have T 1 @2 (T1 ; T ) ¿ s D(T ) − D(T ) exp(−(T − u)) du @T 2 T T1 =
s D(T ) exp(−(T − T1 )) ¿ 0: T
(A.4)
Let "1 and "2 be the order 1 and order 2 principal minors of the associated Hessian matrix of the solution to Eqs. (13) and (14), respectively. Then it is clear that 2 2 @2 @ 2 @2 @ "1 = and " = − : (A.5) 2 @T1 @T @T12 @T12 @T 2 Next, we will prove that "1 and "2 are both greater than zero. Since kj=1 (tj − tj−1 ) = tk = T1 , diEerentiating this equation with respect to T1 , we have
t1 +
k
(tj − tj−1 ) = 1:
(A.6)
j=2
On the other hand, after diEerentiating (1) and (2) with respect to T1 , we obtain t1 = D(T1 )=D(t1 ) ¿ 0
and
D(tj )tj − D(tj−1 )tj−1 = 0 for j = 2; 3; : : : ; k:
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Thus, when D(t) is an increasing function of time t, for j = 2; 3; : : : ; k tj − tj−1 = D(tj )(tj − tj−1 )=D(tj ) ¡ (D(tj )tj − D(tj−1 )tj−1 )=D(tj ) = 0:
(A.7)
Hence, it can be obtained from (A.1)–(A.3) and (A.5) that k 1 (tj − tj−1 ) "1 ¿ D(T1 ) s exp(−(T − T1 )) + hk t1 + T j=2 1 D(T1 )[s exp(−(T − T1 )) + hk ] ¿ 0: T Furthermore, we have from (A.4), (A.5) and (A.8) s "2 ¿ 2 D(T1 ){D(T )[s exp(−(T − T1 )) + hk ] − s2 D(T1 ) exp(−(T − T1 ))} T =
(A.8)
exp(−(T − T1 )) s2 D(T1 )[D(T ) − 2 D(T1 )] exp(−2(T − T1 )) ¿ 0: T2 Therefore, the associated Hessian matrix of the solution to Eqs. (13) and (14) has positive principal minors. The proof of theorem is completed. ¿
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[13] Kar S, Bhunia AK, Maiti M. Deterministic inventory model with two levels of storage, a linear trend in demand and a /xed time horizon. Computers & Operations Research 2001;28:1315–31. Yong-Wu Zhou is a professor of Operations Research at Hefei University of Technology, Peoples’ Republic of China. He received his M.S. degree in applied mathematics from Northwest University of Technology at Xi’an and his Ph.D. in management science from Hefei University of Technology. His major research /elds include production-inventory control, operation research, risk decision, etc. More than 40 papers have appeared, respectively, in Journal of the Operational Research Society, Applied Mathematical Modelling, Computers & Operations Research, Journal of System Engineering and Electronics, Journal of System Science and System Engineering, Systems Engineering-Theory & Practice, etc.