Information and Management Sciences Volume 18, Number 2, pp. 113-125, 2007
Deterministic Inventory Lot-Size Models with Time-Varying Demand and Cost under Generalized Holding Costs Jinn-Tsair Teng
Hui-Ling Yang
The William Paterson University
Hung Kuang University
U.S.A.
R.O.C. Abstract
In this paper, we generalize the EOQ model by Khouja and Park [6] to allow for not only time-varying demand but also unequal cycle time. We prove that there exists a unique optimal replenishment schedule and show that the total relevant cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Therefore, a simple iterative algorithm to obtain the optimal replenishment number and time scheduling is provided. In addition, an easy and simple heuristic algorithm to obtain the optimal solution for the lot-sizing model by Khouja and Park [6] is also proposed. Finally, numerical examples for illustrating the model are provided.
Keywords: Inventory, Lot-Size, Fluctuating Demand, Fluctuating Cost, Shortages.
1. Introduction The classical economic order quantity, EOQ, model assumes not only a constant demand rate but also a fixed unit purchasing cost. However, as we know, the demand rate remains stable only in the maturity stage of a product life cycle. Moreover, in today’s time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle. For example, the cost of a personal computer drops almost linearly with time as shown in Lee et al. [9]. Therefore, using the EOQ formulation in stages other than the maturity stage of a product life cycle or for a high-tech product with constantly declining cost will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial (52% for all industry) as Received May 2006; Revised and Accepted November 2006. Supported by the Assigned Released Time for research and a summer research grant from the William Paterson University of New Jersey, USA.
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shown in Heizer and Render [7]. Consequently, adding the purchasing strategy into EOQ model is vital. One method of dealing with EOQ models with time-varying demand and cost over a finite-planning horizon is the use of discrete dynamic programming (e.g., Wagner and Whitin, [14]). Based on our decades of teaching experiences, students do not have any difficulty learning the continuous version of EOQ. However, there are many students who have difficulty handling tedious and cumbersome dynamic programming. As stated in Friedman [4], “In particular, Wagner and Whitin use this approach (i.e., dynamic programming) to formulate a dynamic version of the economic lot size model. Although this may be a satisfactory approach, it is generally preferable to solve analytically for the optimal replenishment policy, whenever possible.” As a result, for easy understanding and applying, we will solve the EOQ problem here by a continuous version with a simple analytical solution, instead of using a discrete version of dynamic programming. In the growth stage of a product life cycle, the demand rate can be well approximated by a linear form. Consequently, Resh et al. [10], and Donaldson [3] developed an algorithm to determine the optimal replenishment number and timing for a linearly increasing demand pattern. Henery [8] then extended the demand to any log-concave demand function. Following the approach of Donaldson, Dave [2] developed an exact replenishment policy for an inventory model with shortages. In contrast to the traditional replenishment policy that does not start with shortages, Goyal et al. [5] proposed an alternative that starts with shortages in every cycle. By using two numerical examples, they suggested that their policy outperforms the traditional approach. Lately, Teng et al. [11] investigated various inventory replenishment models with shortages, and mathematically proved that the alternative by Goyal et al. [5] is indeed less expensive to operate than the traditional policy. Recently, Teng and Yang [12] dealt with deterministic EOQ models allowed for shortages when both demand and cost are fluctuating with time. Notice that all models mentioned above assume that the inventory holding cost is related to its size, not its value. In contrast to the above models, by assuming that the inventory cost is related to its value, Khouja and Park [6] concurrently developed the optimal lot-sizing model under an exponential cost decrease but a constant demand rate. Teunter [13] then used the net present value concept to study Khouja and Park’s EOQ model, and derived a simple modified EOQ formula. Several years ago, the authors visited a high-tech e-warehouse in Taiwan, and learned that the e-warehouse charged
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customers based on either the total sales per month (i.e., the inventory charge is related to its value) or the average rented boxes per month (i.e., the charge is related to its size). Therefore, for generality, in this paper, we assume that the inventory holding cost is the sum of the size-related holding cost (such as the warehouse cost) and the value-related holding cost (such as the insurance cost). Consequently, the inventory holding cost here becomes a generalized case of all the above models mentioned. In addition, we assume that not only the demand function but also the unit cost is positive and fluctuating with time (which is more general than increasing, decreasing, and log-concave functions). As a result, the proposed model is suitable for any given time horizon in a product life cycle including high-tech products. We then prove that there exists a unique optimal replenishment schedule. Moreover, we show that the total relevant cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Consequently, we simplify the search process by providing an intuitively good starting search point. We also propose a heuristic algorithm which is simpler to understand and easier to use than the approximation as in (9) by Khouja and Park [6]. Finally, numerical examples for illustrating the model are provided.
2. Assumptions and Notation The mathematical model of the inventory replenishment problem here is based on the following assumptions: a. Lead time is zero. b. Shortages are allowed and backlogged. c. The initial inventory level is zero. d. We assume here that the ordering, holding and shortage costs are constant. The reader can easily extend the model to allow for time-varying ordering, holding and shortage costs. In addition, the following notation is used throughout this paper. H = the time horizon under consideration. f (t) = the demand rate at time t. We assume without loss of generality that f (t) is greater than zero in (0, H], and continuous differentiable in the planning horizon [0, H].
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c(t) = the purchase cost per unit at time t, we assume that c(t) is greater than zero, and continuous differentiable. c0 = the ordering cost per order. ch = the size-related inventory holding cost per unit per unit time. r = the value-related inventory holding cost per dollar per unit time = the cost of having one dollar tied up in inventory for a unit time. cs = the shortage cost per unit per unit time. n = the total number of replenishments over [0, H] (a decision variable). ti = the ith replenishment time (a decision variable), i = 1, 2, . . . , n. si = the time at which the inventory level reaches zero in the ith replenishment cycle (a decision variable), i = 1, 2, . . . , n, with t i ≤ si ≤ ti+1 . For generality, we assume that the inventory holding cost is the sum of the size-related holding cost (such as the warehouse cost) and the value-related holding cost (such as the insurance cost). Consequently, the inventory holding cost at replenishment time t i per unit per unit time is ch + rc(ti ). 3. Mathematical Model and Solution The ith replenishment is made at time t i . The quantity received at ti is used partly to meet the accumulated shortages in the previous cycle from time s i−1 to ti (si−1 ≤ ti ). The inventory at ti gradually reduces to zero at si (si ≥ ti ). Then the accumulated shortages increase until the i + 1th replenishment at t i+1 (si ≤ ti+1 ). The decision maker wishes to know how many orders to place, when to place the orders, and how much to order each time so that the total relevant cost (which is the sum of the ordering, holding, shortage and purchase costs) over a finite time horizon [0, H] is minimized. Based on whether the inventory is permitted to start and/or end with shortages, we have four possible models, which were introduced in Teng et al. [11]. The inventory model proposed here is depicted graphically in Figure 1. The total relevant cost associated with this inventory model is established as follows: W (n, {si }, {ti }) = nc0 +
n X
[ch + rc(ti )]
i=1
+
n X i=1
c(ti )
Z
Z
si ti
(t − ti )f (t)dt + cs
n Z X
ti
i=1 si−1
(ti − t)f (t)dt
si
f (t)dt, si−1
(1)
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where the first term is the ordering costs, the second term is the inventory holding costs, the third term is the shortage costs, and the last term is the purchase costs. Thus, the problem here is to find an integer n and a vector of 2n components ht 1 , s1 , t2 , . . . , tn , sn i such that the total relevant cost W (n, {s i }, {ti }) in (1) is minimized.
Figure 1. Graphical representation of inventory model. For any fixed n, the necessary conditions for W (n, {s i }, {ti }) to be minimized are as follows: ∂W (n, {si }, {ti })/∂si = 0, i = 1, 2, . . . , n − 1 and ∂W (n, {si }, {ti })/∂ti = 0, i = 1, 2, . . . , n. Consequently, we have c(ti+1 ) + cs (ti+1 − si ) = c(ti ) + [ch + rc(ti )](si − ti ),
i = 1, 2, . . . , n − 1,
(2)
and [ch + rc(ti ) − c0 (ti )] 0
= [cs + c (ti )]
Z
Z
si ti
f (t)dt − rc0 (ti )
Z
si ti
(t − ti )f (t)dt
ti
f (t)dt,
i = 1, 2, . . . , n.
(3)
si−1
Note that 0 denotes the first derivative with respect to time throughout the paper. Equations (1)-(3) now imply the following theorems: Theorem 1. (a) If ch + rc(ti ) ≤ c0 (ti ), for all ti , then n = 1 and t1 = 0. (b) If c0 (ti ) ≤ −cs , for all ti , then n = 1 and t1 = H. (c) If ch + rc(ti ) > c0 (ti )[1 + r(si − ti )] and c0 (ti ) > −cs , for all ti , then the solution to (2) and (3) exists uniquely (i.e., the optimal values of {s ∗i } and {t∗i } are uniquely determined).
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Proof. See Appendix 1. The results in (a) and (b) of Theorem 1 can be interpreted as follows. For (a), the condition c0 (ti ) ≥ ch +rc(ti ) > 0 means that the rate of increase in purchase cost is higher than or equal to the inventory holding cost. Therefore, buying and storing a unit now is less expensive than buying it later. Using a similar argument for (b), if c 0 (ti ) ≤ −cs , then −c0 (ti ) ≥ cs > 0 which implies that the total relevant cost due to a decrease in the unit purchase cost is larger than or equal to the shortage cost. Consequently, delaying the purchase until the end of the planning horizon H is cheaper than buying it earlier. In reality, we know that the rate of increase in purchase cost cannot be higher than or equal to the inventory holding cost forever, especially in today’s low inflation economy. Consequently, we know that Case (a) rarely happens in the real world. Likewise, Case (b) hardly happens either. As a result, we assume without loss of generality that the conditions in Case (c) hold throughout the rest of the paper. Next, we show that the total relevant cost W (n, {s ∗i }, {t∗i }) is a convex function of the number of replenishments. As a result, the search for the optimal replenishment number, n∗ , is reduced to finding a local minimum. For simplicity, let W (n) = W (n, {s∗i }, {t∗i }).
(4)
By applying Bellman’s principle of optimality [1], we have the following theorem: Theorem 2. W (n) is convex in n. Proof. Using the same argument as in Friedman [4], the proof can be constructed. 4. An Algorithm The result in (c) of Theorem 1 reduces the 2n-dimensional problem of finding {s ∗i } and {t∗i } to a one-dimensional problem. For any fixed n, since s 0 = 0, we only need to find t∗1 to generate s∗1 by (3), t∗2 by (2), and then the rest of {s∗i } and {t∗i } uniquely by repeatedly using (3) and (2). For any chosen t ∗1 , if s∗n = H, then t∗1 is chosen correctly. Otherwise, we can easily find the optimal t ∗1 by standard search techniques. Next, we establish an estimate of the optimal number of replenishments, n ∗ . Using the similar estimate of the optimal number of replenishments as in Teng et al. [11], we propose an estimate as n1 = the rounded integer of
h (c + ra + b)(c − b)Q(H)H i1/2 s h
2c0 (ch + ra + cs )
,
(5)
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where Q(H) is the cumulative demand over the time interval [0, H], a = [c(0) + c(H)]/2 (as an approximate average unit purchase cost), b = [c(0) − c(H)]/H (as an approximate average declining rate of the unit purchase cost), c h + ra + b is an approximate total inventory cost per unit per unit time, and c s −b is an approximate total shortage cost per unit per unit time. It is obvious that searching for the optimal number of replenishments by starting with n1 in (5) instead of n = 1 will increase the computational efficiency significantly (e.g., see Example 1 below). The algorithm for determining the optimal number of replenishments n∗ and the optimal timing {t∗i } and {s∗i } is similar to that in Yang et al. [15] with n1 in (5), L = 0 and U = H/n1 as follows. Algorithm for Finding Optimal Number and Schedule Step 0. Choose two initial trial values of n ∗ , say n as in (5) and n − 1. Use a standard search method to obtain {t∗i } and {s∗i }, and compute the corresponding W (n) and W (n − 1), respectively. Step 1. If W (n) ≥ W (n − 1), then compute W (n − 2), W (n − 3), . . ., until we find W (k) < W (k − 1). Set n∗ = k and stop. Step 2. If W (n) < W (n − 1), then compute W (n + 1), W (n + 2), . . ., until we find W (k) < W (k + 1). Set n∗ = k and stop.
5. A Special Case of No Shortages As discussed as above, if shortages are not allowed, then the total relevant cost associated with this inventory model can be established as follows: W (n, {ti }) = nc0 +
n X
[ch + rc(ti )]
i=1
Z
ti+1 ti
(t − ti )f (t)dt +
n X
c(ti )
i=1
Z
ti+1
f (t)dt,
(6)
ti
where t1 = 0 and tn+1 = H. For any fixed n, the necessary condition for W (n, {t i }) to be minimized is as follows: ∂W (n, {t i })/∂ti = 0, i = 2, 3, . . . , n. Consequently, we have [ch + rc(ti ) − c0 (ti )]
Z
ti+1 ti
f (t)dt − rc0 (ti )
Z
ti+1 ti
(t − ti )f (t)dt
= [(ch + rc(ti−1 ))(ti − ti−1 ) + c(ti−1 ) − c(ti )]f (ti ),
i = 2, 3, . . . , n.
(7)
Since shortages are not permitted, it implies that the shortage cost c s = ∞. From (5), we can easily obtain the estimate of the optimal number of replenishments as n1 = the rounded integer of
h (c + ra + b)Q(H)H i1/2 h
2c0
.
(8)
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Next, we propose a simple to understand and easy to use heuristic algorithm to obtain the optimal solution for the lot-sizing model, in which both the demand and the purchasing cost are fluctuating with time with equal-length cycle. Heuristic Algorithm for Time-Varying Demand and Cost with Equal-length Cycle Step 1. Set the heuristic replenishment number n to be n 1 as in (8). Step 2. Let the heuristic replenishment time be ti = (i − 1)H/n1 , for i = 1, 2, 3, . . . , n1 .
(9)
Substituting n and ti into (6), we obtain the total relevant cost W (n, {t i }), and stop. Note that in Khouja and Park [6], they assumed that the demand rate is a constant. As a result, if f (t) = D, then (8) can be simplified as follows. (ch + ra + b)D 1/2 H. (10) 2c0 The proposed heuristic solution n1 in (10) is simpler to understand and easier to use than n1 = the rounded integer of
the approximation (9) in Khouja and Park [6]. In addition, we show from Examples 2 and 3 below that our proposed heuristic algorithm obtains the optimal solution n ∗ to Examples 1 and 2 as in Khouja and Park [6], respectively. 6. Numerical Examples In this section, we first investigate the optimal ordering strategy for high-tech products in which the demand is increasing while the unit cost is strictly declining. We then use the same two examples as in Khouja and Park [6] to obtain a less expensive total relevant cost than theirs by relaxing the assumption of constant replenishment cycles. Furthermore, our proposed heuristic solution n 1 in (10) obtains the optimal solution n ∗ to both Examples 1 and 2 as given in Khouja and Park [6]. Example 1. Suppose that the demand function for a high-tech product at time t is f (t) = 200 exp(0.8t), the unit cost is c(t) = 250 − 100t, H = 1, c 0 = 80, ch = 50, r = 0.08 and cs = 120 in appropriate units. Applying (5), we have n 1 = 6, and W (5) = 60184.11, W (6) = 60177.61 and W (7) = 60195.52. The corresponding n ∗ = 6, and W (n∗ , {s∗i }, {t∗i }) = 60, 177.61. The computational results are shown in Table 1. Note that Qi in Tables 1, 2, and 3 stands for the order quantity in the ith cycle.
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Table 1. Optimal solution of Example 1. i 1 2 3 4 5 6
t∗i 0.1779 0.3618 0.5330 0.6931 0.8435 0.9853
s∗i 0.1974 0.3801 0.5503 0.7094 0.8590 1.0000
c(t∗i ) 232.211 213.816 196.697 180.687 165.649 151.473
Q1 42.767 46.087 49.406 52.724 56.042 59.359
Example 2. As Example 1 in Khouja and Park [6], we assume that f (t) = 100, 000 units/year, c0 = $300/order, ch = 0, the unit cost is c(t) = 8e−0.52262t , r = 0.08 and H = 1, and cs = ∞. By (10), we have n1 = 25, and W (24) = 638082.17, W (25) = 638068.23 and W (26) = 638078.54. Consequently, we know that n ∗ = 25, and W (n∗ , {t∗i }) = 638, 068.23. The numerical solution is shown in Table 2. By comparison, our total relevant cost is slightly less than the total relevant cost of 638,112 obtained by Khouja and Park [6], in which the replenishment interval is constant. Table 2. Optimal solution of Example 2. i 1 2 3 4 5 6 7 8 9 10 11 12 13
t∗i 0.0000 0.0354 0.0711 0.1071 0.1435 0.1803 0.2173 0.2548 0.2926 0.3308 0.3694 0.4084 0.4477
c(t∗i ) 8.00 7.85 7.71 7.56 7.42 7.28 7.14 7.00 6.87 6.73 6.60 6.46 6.33
Qi 3538.17 3571.06 3604.57 3638.71 3673.49 3708.95 3745.10 3781.96 3819.55 3857.89 3897.01 3936.93 3977.67
i 14 15 16 17 18 19 20 21 22 23 24 25
t∗i 0.4875 0.5277 0.5683 0.6094 0.6509 0.6928 0.7352 0.7781 0.8215 0.8653 0.9097 0.9546 1.0000
c(t∗i ) 6.20 6.07 5.94 5.82 5.69 5.57 5.45 5.33 5.21 5.09 4.97 4.86
Qi 4019.27 4061.74 4105.11 4149.42 4194.70 4240.97 4288.27 4336.64 4386.10 4436.71 4488.49 4541.49
Example 3. As in Example 2, we use H = 0.5, instead of 1. By (10), we have n 1 = 13, and W (12) = 360041.07, W (13) = 359998.59 and W (14) = 360005.23. Consequently, we know that n∗ = 13, and W (n∗ , {t∗i }) = 359, 998.59. The numerical solution is shown in Table 3.
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Table 3. Optimal solution of Example 3. i 1 2 3 4 5 6 7
t∗i 0.0000 0.0362 0.0728 0.1097 0.1470 0.1847 0.2227
c(t∗i ) 8.00 7.85 7.70 7.55 7.41 7.26 7.12
Qi 3623.47 3657.97 3693.13 3728.97 3765.51 3802.77 3840.78
i 8 9 10 11 12 13
t∗i 0.2611 0.2999 0.3391 0.3787 0.4187 0.4591 0.5000
c(t∗i ) 6.98 6.84 6.70 6.56 6.43 6.29
Qi 3879.55 3919.11 3959.48 4000.70 4042.78 4085.74
From Examples 1, 2, and 3, we know that our proposed formula for the estimate of the optimal number of replenishments n ∗ is extremely accurate. 7. Conclusions It is a common phenomenon for high-tech products that the demand is increasing while the unit purchase cost is declining. In this paper, we modify the traditional EOQ model with constant demand and unit cost to allow for time-varying demand and unit purchase cost. We also assume that the inventory holding cost is the sum of the size-related holding cost (such as the warehouse cost) and the value-related holding cost (such as the insurance cost). We provide an easy-to-use algorithm for finding the optimal ordering policy. In addition, the numerical results reveals that our proposed algorithm obtains the optimal solution and the total relevant cost is less than that of Khouja and Park [6] by relaxing the assumption of equal length cycle. The proposed model can be extended in several ways. Firstly, we can easily extend the backlogging rate of unsatisfied demand to any decreasing function β(x), where x is the waiting time up to the next replenishment, and 0 ≤ β(x) ≤ 1 with β(0) = 1. Secondly, we can generalize the model to allow for deteriorating items. Thirdly, we can consider the demand is a function of price. Finally, we can also incorporate the quantity discount, and the learning curve phenomenon into the model. Appendix 1. Proof of Theorem 1 Let Z Wi (si−1 , ti , si ) =
si n
ti
o
Z
[ch +rc(ti )](t−ti )+c(ti ) f (t)dt+
L(ti ) = [ch + rc(ti )](t − ti ) + c(ti ),
for t ≥ ti ,
ti
[cs (ti −t)+c(ti )]f (t)dt,
(A1)
si−1
(A2)
Deterministic Inventory Lot-Size Models with Time-Varying Demand and Cost
and K(ti ) = cs (ti − t) + c(ti ),
for t ≤ ti .
123
(A3)
We can easily obtain ∂Wi = ∂ti
Z
si ti
+
Z
n
o
c0 (ti )[1 + r(t − ti )] − [ch + rc(ti )] f (t)dt
ti si−1
[cs + c0 (ti )]f (t)dt,
for t ≥ ti ,
L0 (ti ) = −[ch + rc(ti )] + c0 (ti )[1 + r(t − ti )],
(A4) for t ≥ ti ,
(A5)
and K 0 (ti ) = cs + c0 (ti ).
(A6)
If c0 (ti ) ≥ ch + rc(ti ) > 0 then we know from (A4) and (A5) that ∂W i /∂ti ≥ 0 and L0 (ti ) ≥ 0. Therefore, both L(ti ) and Wi are increasing with ti . Setting ti to be si−1 , we get Wi (si−1 , ti , si ) ≥ Wi (si−1 , si−1 , si ) for all i. Similarly, L(ti ) ≥ L(t1 ) = L(0) for all i. Consequently, we obtain W (n, {si }, {ti }) ≥ nc0 + ≥ c0 +
n Z X
si
i=1 si−1 Z Hn
L(ti )f (t)dt ≥ nc0 +
n Z X
si
L(0)f (t)dt
i=1 si−1
o
[ch + rc(0)]t + c(0) f (t)dt.
0
(A7)
This completes the proof of (a). Similarly, if c0 (t) ≤ −cs then we have ∂Wi /∂ti ≤ 0 and K 0 (ti ) ≤ 0. Setting ti to be si , we obtain Wi (si−1 , ti , si ) ≥ Wi (si−1 , si , si ). Similarly, K(ti ) ≥ K(tn ) = K(H) for all i. Hence, we get W (n, {si }, {ti }) ≥ nc0 + ≥ c0 +
n Z X
si
i=1 si−1 Z H 0
K(ti )f (t)dt ≥ nc0 +
[cs (H − t) + c(H)]f (t)dt,
n Z X
si
K(H)f (t)dt
i=1 si−1
(A8)
which proves (b). To prove (c), we first prove that there exists a unique optimal t i to the problem. For any fixed n, differentiating W (n, {si }, {ti }) with respect to ti and simplifying terms, we obtain ∂W (n, {si }, {ti })/∂t1 =
n n X i=1
o
[ch + rc(ti )](si − ti ) + c(ti ) f (si )
X dsi n−1 dsi − [cs (ti+1 − si ) + c(ti+1 )]f (si ) dt1 i=1 dt1
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− +
n n X
i=1 n n X
0
[ch + rc(ti ) − c (ti )] [cs + c0 (ti )]
i=1
Z
Z
si ti
ti
f (t)dt si−1
0
f (t)dt − rc (ti ) o dt
i
dt1
Z
si t1
(t − ti )f (t)dt
o dt
i
dt1
.
(A9)
If we relax sn to be any number, then we know from (2) and (3) that n
o
∂W (n, {si }, {ti })/∂t1 = [ch + rc(tn )](sn − tn ) + c(tn ) f (sn )
dsn > 0, dt1
(A10)
which implies that W (n, {si }, {ti }) without the boundary condition of s n = H is an increasing function of t1 . From (2) and (3), we know that if t1 = 0 (or H), then sn (t1 ) = 0 (or > H). As a result, for any given n, there exists a unique t 1 such that W (n, {si }, {ti }) in (1) is minimized with sn = H. Since s0 = 0 and t∗1 is unique, if we can prove that both s∗i generated by (3) and t∗i+1 by (2) are uniquely determined, then we prove (c). Let F (x) = c(x) + cs (x − si ) − c(ti ) − [ch + rc(ti )](si − ti ).
(A11)
Since c(t) is differentiable, there exists a k (with t i ≤ k ≤ si ) such that n
o
F (si ) = c(si ) − c(ti ) − [ch + rc(ti )](si − ti ) = c0 (k) − [ch + rc(k)] (si − ti ) < 0. (A12) From F 0 (x) = c0 (x) + cs > 0, we know that there exists a unique x > s i such that F (x) = 0. This implies that ti+1 is uniquely determined by (2). Similarly, we can easily prove that si is uniquely determined by (3). This completes the proof of (c). Acknowledgements The authors would like to thank the anonymous referees for their constructive comments and valuable suggestions. References [1] Bellman, R. E., Dynamic Programming, Princeton University Press, Princeton, N.J., 1957. [2] Dave, U., A deterministic lot-size inventory model with shortages and a linear trend in demand, Naval Research Logistics, Vol.36, No.6, pp.507-514, 1989. [3] Donaldson, W. A., Inventory replenishment policy for a linear trend in demand: an analytical solution, Operational Research Quarterly, Vol.28, No.3, pp.663-670, 1977. [4] Friedman, M. F., Inventory lot-size models with general time-dependent demand and carrying cost function, INFOR, Vol.20, No.2, pp.157-167, 1982. [5] Goyal, S. K., Morin, D. and Nebebe, F., The finite horizon trended inventory replenishment problem with shortages, Journal of the Operational Research Society, Vol.43, No.12, pp.1173-1178, 1992.
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Authors’ Information Jinn-Tsair Teng received a B.S. degree in Mathematical Statistics from Tamkang University, an M.S. degree in Applied Mathematics from National Tsing Hua University in Taiwan, and a Ph.D. in Industrial Administration from Carnegie Mellon University in USA. He joined the Department of Marketing and Management Sciences at William Peterson University of New Jersey in 1992. His research interests include supply chain management and marketing research. He has published research articles in Management Sciences, Marketing Science, Journal of the Operational Research Society, Operations Research Letters, Naval Research Logistics, European Journal of Operational Research, Applied Mathematical Modelling, Journal of Global Optimization, International Journal of Production Economics, and others. Department of Marketing and Management Sciences, Cotsakos College of Business, The William Paterson University of New Jersey, Wayne, New Jersey 07470, U.S.A. E-mail:
[email protected]
TEL : +1-973-720-2651
Hui-Ling Yang received B.S. and M.S. degrees in Mathematics from National Taiwan Normal University and a Ph.D. degree in Industrial Engineering from National Tsing Hua University, Taiwan, ROC. She conducts research in operations research, mathematical programming and statistical analysis. Her research area is focus on inventory management. She has published research articles in various journals such as European Journal of Operational Research, International Journal of Production Economics, Journal of the Chinese Institute of Engineers, Journal of the Chinese Institute of Industrial Engineers, Journal of the Operational Research Society, Naval Research Logistics, Operations Research Letters, etc. Department of Computer Science and Information Engineering, Hung Kuang University, 34, Chung-Chie Rd., Shalu, Taichung, Taiwan 43302, R.O.C. E-mail:
[email protected]
TEL : +886-4-26318652
