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Advances in Operations Research
Volume 2012 (2012), Article ID 385371, 15 pages
A Partial Backlogging Inventory Model for Deteriorating Items with Fluctuating Selling Price and Purchasing Cost
Department of Computer Science and Information Engineering, Hung Kuang University, 34 Chung-Chie Road, Shalu, Taichung 43302, Taiwan
Received 2 October 2011; Revised 12 January 2012; Accepted 14 February 2012
Academic Editor: I. L. Averbakh
Copyright © 2012 Hui-Ling Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In today’s competitive markets, selling price and purchasing cost are usually fluctuating with economic conditions. Both selling price and purchasing cost are vital to the profitability of a firm. Therefore, in this paper, I extend the inventory model introduced by Teng and Yang (2004) to allow for not only the selling price but also the purchasing cost to change from one replenishment cycle to another during a finite time horizon. The objective is to find the optimal replenishment schedule and pricing policy to obtain the profit as maximum as possible. The conditions that lead to a maximizing solution guarantee that the existence, uniqueness, and global optimality are proposed. An efficient solution procedure and some theoretical results are presented. Finally, numerical examples for illustration and sensitivity analysis for managerial decision making are also performed.
In today’s time-based competitive market, the unit selling price of product may increase significantly while its demand increases such as fashionable or valuable goods. On the other hand, the selling prices of items may drop dramatically throughout their life cycles due to advances in technology, competition, and so forth. Thus, the selling price is fluctuating. From the other aspect, some products of the purchasing cost decreases as the demand increases such as 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 constantly as shown in Lee et al. . Furthermore, the purchasing cost as a percentage of sales is often substantial, which had been mentioned in Heizer and Render . Therefore, from an integrated logistics management perspective, taking the varying selling price and purchasing cost into account is essential.
Moreover, in reality, for fashionable commodities and high-tech products with short life cycles, the backorder rate is diminishing with the length of waiting time. Customers who experience stock-out will be less likely to buy again from the suppliers, they may turn to another store to purchase the goods. The sales for the product may decline due to the introduction of more competitive product or the change in consumers’ preferences. The longer the waiting time, the lower the backlogging rate is. This leads to a larger fraction of lost sales and a less profit. As a result, take the factor of partial backlogging into account is necessary. Abad  proposed an optimal pricing and lot-sizing policy under the conditions of perishability and partial backordering. Teng et al.  considered the partial backlogging inventory model with time varying demand and purchasing cost. Chang et al.  considered the inventory model with selling price and purchasing cost and then provided an optimal replenishment policy for a retailer to achieve its maximum profit. To study the effect of the factors (selling pricing and/or purchasing cost), several authors had developed various models with the related issue. For example, Teng and Yang  proposed the inventory lot-size models with time varying demand and purchasing cost under generalized holding cost. Abad  added the pricing strategy into consideration and provided the optimal price and lot-size for a retailer when the demand was a function of the selling price. Recently, Sana  provided the optimal pricing policy for partial backlogging inventory model with price-dependent demand. Das Roy et al.  proposed an optimal shipment strategy for imperfect items in a stock-out situation. Das Roy et al.  again provided an economic-order quantity model of imperfect quality items with partial backlogging. The major assumptions and objective used in the above research articles are summarized in Table 1.
Thus, in contrast to the above articles mentioned, the inventory model here is developed as introduced by Teng and Yang  to allow for not only selling price but also purchasing cost to be fluctuating, which are changed from one replenishment cycle to another during a finite time horizon. The objective is to find the optimal replenishment schedule and pricing policy to obtain the maximum profit rather than minimum cost. The total profit associated with the inventory system is a concave function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local maximum. Moreover, an intuitively estimate for finding the optimal replenishment number is provided. Some numerical examples for illustration and sensitivity analysis for managerial decision making are performed. Finally, summary and suggestions for future research are provided.
2. Assumptions and Notation
The mathematical model of the inventory replenishment problem is based on the following assumptions:(1)The planning horizon of the inventory problem here is finite and is taken as time units. The initial and the final inventory levels are both zero during the time horizon .(2)Replenishment is instantaneous and lead time is zero.(3)In reality, the item may deteriorate with time varying. For simplicity, we assume that the deterioration rate here is constant and there is no repair or replacement of the deteriorated items.(4)Shortages are allowed. Unsatisfied demand is backlogged, and the fraction of shortages backordered is a decreasing function of time , denoted by , where is the waiting time up to the next replenishment, and with . Note that if = 1 (or 0) for all , then shortages are completely backlogged (or lost).(5)In the lost-sales case, the opportunity cost due to lost sale is the sum of the revenue loss and the cost of lost goodwill. Hence, the opportunity cost due to lost sale here is greater than the unit purchasing cost. For details, see Teng et al. .(6)In today’s global high competition environment, we here assume that the selling price and purchasing cost are fluctuating with time and changed from one replenishment cycle to another during a finite time horizon.
For convenience, the following notation is used throughout this paper:: the time horizon under consideration,: the demand rate at time , without loss of generality, we here assume that is increase, positive, differentiable in ,: the purchasing cost per unit at time , which is positive, differentiable in ,: the selling price per unit at time , which is positive, differentiable in [0, ],: the deterioration rate,: the fixed ordering cost per order,: the inventory holding cost per unit per unit time,: the backlogging cost per unit per unit time, if the shortage is backlogged,: the unit opportunity cost of lost sales, if the shortage is lost. We assume without loss of generality that ,: the number of replenishments over [0, ] (a decision variable),: the th replenishment time (a decision variable), ,: the time at which the inventory level reaches zero in the th replenishment cycle (a decision variable), .
3. Mathematical Model
As a result, we obtain the time-weighted inventory during the th cycle as Similarly, the time-weighted backorders due to shortages during the th cycle is and the total number of lost sales due to shortages during the th cycle is The order quantity and unit sold at in the th replenishment cycle is and unit sold at in the th replenishment cycle is Therefore, the purchasing cost during the th replenishment cycle is and the revenue realized in the th replenishment cycle is Hence, if replenishment orders are placed in [0, ], then the total realized profit of the inventory system during the planning horizon is with and . The problem is to determine , , and such that in (3.8) is maximized.
4. Theoretical Results and Solution
For a fixed value of , the necessary conditions for to be maximized are: for , and , for. . Consequently, we obtain respectively. Note that (4.1) and (4.2) are coincident with the following articles’(1)Equations (12) and (11) in Teng and Yang , if ,(2)Equations (15) and (14) in Teng et al. , if and ,(3)Equations (11) and (10) in Chang et al. , if and .
Thus, the model here proposed is a generalization of the above three mentioned models. For simplicity, from (4.2), let the marginal resultant profit per unit during no-shortage and shortage period be with , and with , respectively. Taking the partial derivative of and with respect to respectively, we obtained the following results: Note that the longer the waiting time, the lower the marginal resultant profit. Consequently, is a decreasing function of . Thus, we may assume without loss of generality that , for all . Then, we obtain the following result.
Lemma 4.1. For any given n, if , with , and , with , then the optimal solution is and (i.e., purchase at the beginning).
Proof. See Appendix A.
The results in Lemma 4.1 can be interpreted as follows. The condition implies that . This means that the increasing rate of the unit purchasing cost is higher than or equal to the sum of the marginal selling price and marginal inventory carrying cost per unit (which includes inventory and deterioration costs). Therefore, buying and storing a unit and then selling now are more profitable than buying and selling it later.
Proof. See Appendix B.
The result in Theorem 4.2 reduces the 2-dimensional problem of finding and to a one-dimensional problem. Since , we only need to find to generate by (4.1), by (4.2), and then the rest of and uniquely by repeatedly using (4.1) and (4.2). For any chosen , if , then is chosen correctly. Otherwise, we can easily find the optimal by standard search techniques.
Having calculated the second partial derivatives of the function shows that the Hessian matrix is negative definite if
Proof. See Appendix C.
Next, we show that the total profit is a concave function of the number of replenishments. As a result, the search for the optimal replenishment number, , is reduced to find a local maximum. For simplicity, let By applying Bellman’s principle of optimality , we have the following theorem:
Theorem 4.4. is concave in .
Proof. The proof is similar to that of Teng and Yang , the reader can easily prove it.
By a similar discussion as in Teng and Yang , I here use the average backlogging rate , unit purchasing cost and average unit selling price to replace , and , respectively. the estimate of the number of replenishments is obtained as where . It is obvious that searching for by starting with in (4.10) will speed the computational efficiency significantly, comparing to starting with . The algorithm for determining the optimal number of replenishments and schedule is summarized as follows.
Algorithm for Finding Optimal Number and Schedule
Step 1. Choose two initial trial values of , say as in (4.10) and . Use a standard search method to obtain and , and compute the corresponding and , respectively.Step 2. If , then compute , , …, until we find . Set and stop.Step 3. If , then compute , , …, until we find . Set and stop.
5. Numerical Examples
Example 5.1. Let , , , , , , , , in appropriate units. After calculation, we have , , and . By (4.10), we obtain the estimate number of replenishments . From computational results, we have , and . Therefore, the optimal number of replenishments is 14, and the optimal profit is 49044.31. The optimal replenishment schedule is shown in Table 2.
Example 5.2. To understand the effect of changes in parameters , , , , , on the optimal solution, the sensitivity analysis is performed by changing one parameter at a time and keeps the others unchanged. The parameter values are the same as in Example 5.1. The results obtained are shown as in Table 3.
From Table 3, the following phenomena can be obtained.
(1)The optimal maximum profit decreases as , , , , increases, however, it increases as the backlogging rate increases.(2)The optimal maximum profit is more sensitive on parameters than others.(3)The optimal replenishment number is very slightly sensitive to the change of these parameters except and .(4)The estimated number is very close to the optimal replenishment number , no matter what magnitude of the parameters changed.
Example 5.3. Using the same numerical values as in Example 5.1, we consider the influence of changes of the rate of change of selling price and purchasing cost on the total profit. The results are obtained as shown in Table 4.
From Table 4, it is obviously that the phenomena are obtained.(1)The percentage change in total maximum profit is significantly sensitive on the variation of rate of change.(2)The total profit increases as the rate of change of selling price increases, while decreases as the rate of change of purchasing cost increases.(3)The estimated number is also close to the optimal replenishment number .(4)The optimal replenishment number is slightly sensitive to the change of rate of change.
In this paper, a partial-backlogging inventory lot-size model for deteriorating items with time dependent demand, fluctuating selling price and purchasing cost is considered. We show that the optimal replenishment schedule exists uniquely and the total profit associated with the inventory system is a concave function of the number of replenishments. An intuitively estimate for finding the optimal replenishment number is provided. From the sensitivity analysis, the results indicate that the effect of variation of rate of change of the selling price and purchasing cost on the system behavior is significant. Thus, to incorporate the selling price and purchasing cost into the inventory model is vital, especially in the present changeable market.
With more practical features, the model developed here can be further extended by incorporating with other kind of functions or parameters, such as considering the demand as a function of selling price or stock dependent, or time varying deterioration rate, or to develop the model by employing other factors, such as inflation and price discount.
A. Proof of Lemma 4.1
Let We then have If , then we know from (A.2) that . Therefore, for any given , is decreasing with . This implies that for any fixed . Consequently, we obtain This completes the proof.
B. Proof of Theorem 4.2
For any given and , from (4.1), we set We then have and . Taking the first derivatives of with respect to , we obtain As a result, we know that there exists a unique such that . Thus, the solution to (4.1) uniquely exists. Similarly, from (4.2), we set We then have since , and . By taking the first derivatives of with respect to , we obtain Consequently, there exists a unique (>) such that , which implies that solution to (4.2) uniquely exists. Therefore, we complete the proof.
C. Proof of Theorem 4.3
which implies that : which implies that . For principal minor of higher order, , it is not difficult to show that they satisfy the following recursive relation: with the initial . From (4.7)-(4.8) and the relation between second-order partial derivatives, we have For in (C.5), we obtain Thus, Proceeding inductively, we have Therefore, and , for . This completes the proof.
The author would like to thank the editor and anonymous reviewers for their valuable and constructive comments which have led to a significant improvement in the manuscript. This research was supported by the National Science Council of the Republic of China under Grant NSC 96–2221-E-241-001.
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