A Partial Backlogging Inventory Model for Deteriorating Items with Fluctuating Selling Price and Purchasing Cost

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.


Introduction
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. 1 .Furthermore, the purchasing cost as a percentage of sales is often substantial, which had been mentioned in Heizer and Render 2 .Therefore, from an integrated logistics management perspective, taking the varying selling price and purchasing cost into account is essential.

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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 3 proposed an optimal pricing and lot-sizing policy under the conditions of perishability and partial backordering.Teng et al. 4 considered the partial backlogging inventory model with time varying demand and purchasing cost.Chang et al. 5 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 6 proposed the inventory lot-size models with time varying demand and purchasing cost under generalized holding cost.Abad 7 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 8 provided the optimal pricing policy for partial backlogging inventory model with price-dependent demand.Das Roy et al. 9 proposed an optimal shipment strategy for imperfect items in a stock-out situation.Das Roy et al. 10 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 11 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.

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 H time units.The initial and the final inventory levels are both zero during the time horizon H.
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.

Mathematical Model
For simplicity, we use the same inventory model as in Teng and Yang 11 , which is shown in Figure 1.
As a result, we obtain the time-weighted inventory during the ith cycle as Similarly, the time-weighted backorders due to shortages during the ith cycle is Ordering quantity Inventory level and the total number of lost sales due to shortages during the ith cycle is The order quantity and unit sold at t i in the ith replenishment cycle is Therefore, the purchasing cost during the ith replenishment cycle is and the revenue realized in the ith replenishment cycle is

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Hence, if n replenishment orders are placed in 0, H , then the total realized profit of the inventory system during the planning horizon H is with 0 s 0 < t 1 and s n H.The problem is to determine n, {s i }, and {t i } such that TP n, {s i }, {t i } in 3.8 is maximized.

Theoretical Results and Solution
For a fixed value of n, the necessary conditions for TP n, {s i }, {t i } to be maximized are: ∂T P n, {s i }, {t i } /∂t i 0 for i 1, 2, . . ., n, and ∂T P n, {s i }, {t i } /∂s i 0, for.i 1, 2, . . ., n − 1.Consequently, we obtain Proof.See Appendix A.
The results in Lemma 4.1 can be interpreted as follows.The condition R t i t i , t ≤ 0 implies that p t c h θc v t e θ u−t ≤ c v t e θ u−t .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.Theorem 4.2.For any given n, if R t i t i , t > 0, with t ≥ t i , and P t i t, t i < 0, with t ≤ t i , i 1, 2, . . ., n, then the solution that satisfies the system of 4.1 and 4.2 exists uniquely and 0 ≤ s i−1 < t i < s i , for i 1, 2, . . ., n.

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Next, we show that the total profit TP n, {s * i }, {t * i } is a concave function of the number of replenishments.As a result, the search for the optimal replenishment number, n * , is reduced to find a local maximum.For simplicity, let

4.9
By applying Bellman's principle of optimality 12 , we have the following theorem: Proof.The proof is similar to that of Teng and Yang 11 , the reader can easily prove it.
By a similar discussion as in Teng and Yang 11 , I here use the average backlogging rate β, unit purchasing cost c v and average unit selling price p to replace β t i 1 − s i , c v t i and p t i , respectively.the estimate of the number of replenishments is obtained as where Q H H 0 f t dt.It is obvious that searching for n * by starting with n in 4.10 will speed the computational efficiency significantly, comparing to starting with n 1.The algorithm for determining the optimal number of replenishments n * and schedule is summarized as follows.

Algorithm for Finding Optimal Number and Schedule
Step 1. Choose two initial trial values of n * , say n as in 4.10 and n − 1. Use a standard search method to obtain {t * i } and {s * i }, and compute the corresponding TP n and TP n − 1 , respectively.
Step 3. If TP n < TP n − 1 , then compute TP n − 2 , TP n − 3 , . .., until we find TP k > TP k − 1 .Set n * k and stop.49030.61.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 c f , c h , c b , c l , θ, β t 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.

Numerical Examples
From Table 3, the following phenomena can be obtained.
1 The optimal maximum profit decreases as c f , c h , c b , c l , θ increases, however, it increases as the backlogging rate β t increases. 2 The optimal maximum profit is more sensitive on parameters c f than others.
3 The optimal replenishment number is very slightly sensitive to the change of these parameters except c f and β t .
4 The estimated number n 1 is very close to the optimal replenishment number n * , 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 p t and purchasing cost c v t 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.

Conclusions
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.
This completes the proof.

B. Proof of Theorem 4.2
For any given s i−1 and t i , from 4.1 , we set

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We then have and lim x → ∞ F x > 0. Taking the first derivatives of F x with respect to x, we obtain As a result, we know that there exists a unique s i > t i such that F s i 0. Thus, the solution to 4.1 uniquely exists.Similarly, from 4.2 , we set We then have since R t i t i , t > 0, and lim x → ∞ G x −c l < 0. By taking the first derivatives of G x with respect to x, we obtain Consequently, there exists a unique t i 1 > s i such that G t i 1 0, which implies that solution to 4.2 uniquely exists.Therefore, we complete the proof.

C. Proof of Theorem 4.3
Taking the second derivatives with respect to t i and s i on TP n, {s i }, {t i } , we have Proceeding inductively, we have Therefore, Δ 2i−1 < 0 and Δ 2i > 0, for i 2, 3, . ... This completes the proof.

Figure 1 :
Figure 1: Graphical representation of inventory model.

Table 1 :
5ajor characteristic of inventory models on selected articles.Shortages are allowed.Unsatisfied demand is backlogged, and the fraction of shortages backordered is a decreasing function of time t, denoted by β t , where t is the waiting time up to the next replenishment, and 0 ≤ β t ≤ 1 with β 0 1.Note that if β t 1 or 0 for all t, then shortages are completely backlogged or lost .5In 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. 4 .6Intoday'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: H: the time horizon under consideration, f t : the demand rate at time t, without loss of generality, we here assume that f t is increase, positive, differentiable in 0, H ,cv t : the purchasing cost per unit at time t, which is positive, differentiable in 0, H , p t : the selling price per unit at time t, which is positive, differentiable in 0, H , θ: the deterioration rate, c f : the fixed ordering cost per order, c h : the inventory holding cost per unit per unit time, c b : the backlogging cost per unit per unit time, if the shortage is backlogged, c l : the unit opportunity cost of lost sales, if the shortage is lost.We assume without loss of generality that c l > c v t , n: the number of replenishments over 0, H a decision variable , t i : the ith replenishment time a decision variable , i 1, 2, . . ., n, s i : the time at which the inventory level reaches zero in the ith replenishment cycle a decision variable , i 1, 2, . . ., n.
≤ t, respectively.Taking the partial derivative of R t, u and P s, t with respect to t respectively, we obtained the following results: Note that the longer the waiting time, the lower the marginal resultant profit.Consequently, P s, t is a decreasing function of t.Thus, we may assume without loss of generality that P t s, t < 0, for all t > s.Then, we obtain the following result.For any given n, if R t i t i , t ≤ 0, with t ≥ t i , and P t i t, t i < 0, with t ≤ t i , i 1, 2, . . ., n, then the optimal solution is n * 1 and t * with t ≤ u, andP s, t p t − c v t − c b t − s β t − s − c l 1 − β t − s , . See Appendix B.
Proof.See Appendix C.

Table 2 :
The optimal replenishment schedule for Example 5.1.

Table 3 :
Sensitivity analysis on parameters changed for Example 5.2.

Table 4 :
Sensitivity analysis on rate of change for Example 5.3.
− c v t i − θ c h θc v t i − c v t i e θ t−t i f t dt, − c v t i 1 − c b t i 1 − s i β t i 1 − s i − c b β t i 1 − s i − c h θc v t i e θ s i −t i f s i .Let Δ k be the principal minor of order k, then, under condition 4.7 -4.8 , it is clear that TP ∂s1 ∂t 1 −R t 1 t 1 , s 1 f s 1 < 0, C.2 which implies that Δ 1 ∂ 2 TP/∂s 1 ∂t 1 < 0: ∂ 2 TP/∂t 2 ∂s 1 Δ 1 > 0.For principal minor of higher order, i 2, 3, . .., it is not difficult to show that they satisfy the following recursive relation: 2 p t i − c v t i − c b β t i − t p t i c l − c v t i − c b t i − t β t i − t f t dt