# Retailer’s Joint Ordering, Pricing, and Preservation Technology Investment Policies for a Deteriorating Item under Permissible Delay in Payments

Authors: Umakanta Mishra; Jacobo Tijerina-Aguilera; Sunil Tiwari; Leopoldo Eduardo Cárdenas-Barrón
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This article develops an inventory model for deteriorating items with controllable deterioration rate (by using preservation technology) under trade credit policy. As in practical scenarios the demand of an item is directly associated with its selling price, keeping this in mind, it is assumed to be a price dependent demand. The main objective of the inventory model is to determine jointly the optimal ordering, pricing, and preservation technology investment policies for retailer so that the total profit is maximized. The effects of key parameters on optimal solution are studied through a sensitivity analysis with the aim of examining the behavior of the inventory model with controllable deterioration under the permissible delay in payments.


---

## Body

## 1. Introduction

Inventory control is the activity which organizes the availability of items to the customers. On the one hand, storing large quantities is not a right solution for stockout condition and for meeting the demand of consumers. On the other hand, most of products are deteriorating in nature and the deterioration process cannot be stopped but it can be controlled with the aim of delaying the speed of the deterioration.It is well known that the profit can be increased by increasing sales or reducing costs. Longer trade credit period can cause sales making the demand increase, and the consequent rise in the ordering quantity produces a larger deterioration cost. The natural process of deterioration can be able to stop; it may be reduced by specific processes when products are at risk of deterioration. For example, cold storage reduces the deterioration of sea products and seasonal products. Hence, the degree of deterioration of the items depends on the investment in the preservation technology of the inventory at the facility as well as the environmental conditions. Due to advancement in technologies nowadays one can slow down the deterioration rate of items by investing money on preservation technology. It is well known that Ghare and Schrader [1] developed the first inventory model for deteriorating items. Recently, many researchers have studied the effect of preservation technology investment on reducing item’s deterioration. For example, Hsu et al. [2] were the first ones who investigated the impact of preservation technology investment on an exponentially decaying inventory model involving partial backorders. Later, Lee and Dye [3] and Dye and Hsieh [4] established an inventory model for a deteriorating item under the usage of preservation technology considering different scenarios. Afterwards, Dye [5] explored the effect of preservation technology investment on a noninstantaneously deteriorating item. Subsequently, Zhang et al. [6] and Liu et al. [7] studied the characteristics of preservation technology investment for deteriorating items. Yang et al. [8] investigated the tradeoff between preservation technology investment and the optimal dynamic trade credit for a deteriorating inventory model. After, Tsao [9] extended the inventory model of Dye [5] to consider a joint location and preservation technology investment decision-making problem for noninstantaneous deteriorating items taking into consideration trade credit. Recently, Mishra et al. [10] studied the characteristics of preservation technology investment for deteriorating items with price dependent demand.In real life business via share marketing, trade credit financing becomes an influential tool to improve sales and reduce on-hand stock. The permissible delay in payments reduces the cost of holding stock since it reduces the amount of capital invested in stock for the duration of the permissible period. Furthermore, during the delay period, the buyer can accumulate revenue on sales and earn interest on that revenue via share market investment or banking business. It gives economic sense for the retailer to accept the delay period allowed by the supplier. In addition, trade credit offered by the supplier encourages a lot the retailer to buy more products. Hence, the trade credit plays a major role in inventory control for both supplier and retailer.The fact that trade credit allows retailers to increase the size of their orders makes it an important issue in supply chain management. Perhaps, Goyal [11] was the first to develop an economic order quantity (EOQ) inventory model with permissible delay in payment. The inventory model proposed by Goyal [11] has attracted the attention of several researchers and academicians. For example, Jaggi and Aggarwal [12] extended Goyal’s [11] inventory model to perishable items using the discounted cash-flows (DCF) approach to determine the optimum order quantity in situations of trade credit. In a subsequent paper, Aggarwal and Jaggi [13] extended Goyal's [11] inventory model for deteriorating items. After, Jamal et al. [14] developed an optimal ordering policy for perishable items under conditions of permissible delays in payment and allowable shortages. Later, Chung [15] revisited Goyal’s [11] inventory model and he proposed an alternative solution procedure. Afterwards, Jamal et al. [16] further extended Goyal’s [11] inventory model to allowable shortages. Then Teng [17] modified Goyal's [11] inventory model by considering different unit price and unit cost. Through his model, Teng [17] justified that it is more beneficial for a buyer (in economical point) to order less quantity and take the benefits of delay in payments more frequently. Subsequently, Chang [18] built an EOQ inventory model for perishable items under inflation in cases where supplier credit is linked to order quantity. Next, Chang et al. [19] determined that delays in payment depend on the quantity ordered. They also concluded that a fixed trade credit period is permissible when the order quantity exceeds the quantity at which the delay payment is permitted; otherwise, the retailer must pay for the item immediately. Huang [20] developed an inventory model to determine the optimal order quantity considering permissible delays in payment in cases where the order quantity is below a predetermined quantity. After that, Chung and Huang [21] presented a more comprehensive inventory model in which the total variable cost was shown to include various forms of convexity. Next, Chang et al. [19] determined that delays in payment depend on the quantity ordered. They also concluded that a fixed trade credit period is permissible when the order quantity exceeds the quantity at which the delay payment is permitted; otherwise, the retailer must pay for the item immediately. Chung and Lin [22] developed a simple algorithm to overcome the shortcomings of Jaggi and Aggarwal’s [12] inventory model. Geetha and Udayakumar [23] built an inventory model for deteriorating items considering trade credit financing when the demand is a linear function of time. Optimal solution procedures to find the optimal order quantity and cycle time are discussed. The literature on trade credit is vast; the following research works are worth mentioning: Liao [24], Jaggi et al. [25], Teng [26], Teng and Chang [27], Chang et al. [28, 29], Teng et al. [30], Teng et al. [31], Teng et al. [32], Chung et al. [33], Ouyang and Chang [34], Ouyang et al. [35], Khanra et al. [36], Chen et al. [37], Wu and Chan [38], Chen and Teng [39], Geetha and Udayakumar [23], Chang et al. [40], Wu et al. [41], Zhou et al. [42], Tiwari et al. [43], Giri and Sharma [44], Jaggi et al. [45, 46], Mahata and De [47], and Wu et al. [48]. For an extensive and updated review of the trade credit literature, the reader can read the papers of Seifert et al. [49] and Molamohamadi et al. [50].The aim of this study is to provide a solution for the inventory and preservation effort problem under trade credits. The contributions of this work to the literature are as follows. First, this is the first study to consider inventory decision-making problem for deteriorating items with price dependent demand under trade credits. Second, this study also considers the preservation effort decision. This copes with more practical situations. Therefore, this paper studies the retailer’s replenishment and pricing policies for a deteriorating item where retailer invests money on preservation technology in order to reduce the deterioration rate when supplier provides a credit period to retailer for settling the amount. The consequences of price dependent demand, preservation technology investment, and trade credit are incorporated into the traditional deteriorating inventory model. The main objective of this paper is to find the optimal replenishment, pricing, and preservation technology investments strategies which maximize the total profit. Table1 shows a comparison of some recent works related to preservation technology.Table 1
Comparison of some existing literature on preservation technology.
Authors Demand type Trade credit Preservation technology Hsu et al. [2] Constant demand No Yes Dye et al. (2012) Constant demand No Yes Dye [5] Constant demand No Yes He and Huang [51] Price dependent No Yes Zhang et al. [6] Price dependent No Yes Yang et al. [8] Time dependent Yes Yes Tsao [9] Poisson distributed No Yes Mishra et al. [10] Price dependent No Yes This paper Price dependent Yes YesThe rest of the paper is organized as follows. Section2 provides notation, assumptions, and the mathematical formulation for the inventory model with controllable deterioration under the permissible delay in payments. Section 3 proposes a solution procedure for maximizing the total profit. Section 4 illustrates the inventory model with the help of numerical examples. Section 5 presents a sensitivity analysis of the optimal solution with respect to some parameters of the inventory system. Section 6 gives some conclusions and establishes suggestions for future research.


## 2. Mathematical Formulation of the Inventory Model with Controllable Deterioration under the Permissible Delay in Payments

### 2.1. Notation

The notation that is used in the development of the inventor model is shown below.Parameters λ ( α ): deterioration rate when there is an investment on preservation technology (units/time unit) λ 0: deterioration rate without preservation technology investment (units/time unit) δ: sensitive parameter of investment to the deterioration rate D ( p ) = a - b p: demand rate function which is a function of selling price p (units/time unit) a: demand scale b: price sensitive parameter C: purchasing cost ($/unit) h: inventory holding cost ($/unit/time unit) A: ordering cost per order ($/order) M: credit period offered by supplier (time unit) I e: the interest earned by the retailer (%/time unit) I p: the interest payable by the retailer (%/time unit) I ( t ): inventory level (units) TP T , α , p: total profit ($/time unit)Dependent Decision Variables Q: ordering quantity (units)Decision Variables T: inventory cycle length (time unit) α: cost of preservation technology investment per unit time ($/unit/time unit) p: selling price ($/unit).


### 2.2. Assumptions

The inventory model with controllable deterioration under the permissible delay in payments is based on the following assumptions.(i) The demand rate,D(p)=a-bp, is assumed to be price dependent, where a and b are nonnegative constants.(ii) The inventory planning horizon is infinite and the inventory system involves only one item.(iii) It is considered the assumption of He and Huang [51] which is as follows: λ(α)=λ0e-δα, where λ(α) is the deterioration rate after investing on preservation technology, λ0 is the deterioration rate without preservation technology investment, and δ is the sensitive parameter of investment to the deterioration rate. The relationship of deterioration rate and the preservation technology investment parameter satisfies the following conditions: ∂λ(α)/∂α<0,∂2λ(α)/∂α>0.(iv) The cost of preservation technology investment per unit time is constrained toα∈0,α-.(v) There is no shortage of inventory.(vi) Under a trade credit periodM, the retailer would settle the account at time t=M and pays for the interest charge on items in stock with rate Ip over the interval [M,T] as T≥M. Alternatively, when the retailer settles the account at time t=M, it is not required to pay any interest charge for items in the stock during the whole cycle as T≤M.(vii) The retailer accumulates revenue and earns interest from the beginning of the inventory cycle until the end of the trade credit period offered by the supplier. That is, the retailer collects revenue and receives interest during the period fromt=0 to t=M with rate Ie under trade credit conditions.


### 2.3. Inventory Model with Controllable Deterioration under the Permissible Delay in Payments

Figure1 shows the inventory level through time. The inventory model is described briefly as follows: the retailer orders and receives an order size of Q units of a single product from the supplier at the beginning of the cycle, that is, at time t=0. During the time interval [0,T], the inventory level I(t) is depleted gradually due to demand from the customers. At time t=T, the inventory level drops to zero.Figure 1
Graphical representation of the inventory behavior through time.The differential equation representing the change of inventory levelI(t) with respect to time is as follows:(1)dItdt+λαIt=-Dp,0≤t≤T.Considering the initial condition I(0)=Q and the boundary conditionI(T)=0, hence the solution to the differential equation (1) is(2)It=DpλαeλαT-t-1,0≤t≤T.Thus, the retailer’s order size per cycle is given by(3)I0=Q=DpλαeλαT-1.The retailer’s profit function per cycle is comprised of the elements listed below:(a) The total sales revenue =pDp.(b) Ordering cost =A(c) Purchasing cost =CQ=CD(p)/λ(α)eλαT-1.(d) Inventory holding cost =h∫0TItdt=hDp/λ2αeλαT-1-λαT.(e) Preservation technology cost =α.(f) Interest payable: depending on the retailer’s trade credit periodM offered by the supplier, the following two cases occur: Case 1 when M≤T and Case 2 when M≥T which are shown in Figure 1. Thus, to compute the interest payable, we need to consider these two cases. The calculation of interest payable is detailed below.Case 1 (M≤T).
When the credit periodM is less than or equal to the replenishment cycle time T, the retailer begins to pay interest for the items in stock after time M with a rate Ip. Hence, the interest payable is calculated as follows:(4)IpC∫MTDpT-tdt=IpCDp2T-M2.Case 2 (M≥T).
BecauseT is less than or equal to the credit period M, there is no interest paid for financing the inventory in stock. Therefore, the interest payable in this case is zero.(g) Interest earned: to compute the interest earned, it is necessary to consider the two mentioned cases above.Case 1 (M≤T).
In this case, the retailer sells the goods, accumulates sales revenue, and earns interest with rateIe until time M. Therefore, the interest earned from time 0 to M per cycle is computed with(5)Iep∫0TDpT-tdt=IepDpT22.Case 2 (M≥T).
Because the cycle timeT is less than or equal to the credit period M, from time 0 to T, the retailer sells the goods and continues to accumulate sales revenue to earn interest. Therefore, the interest earned from time 0 to M per cycle is determined as(6)DpIepTM-T2.Hence, the total profit per unit time is given by(7)TPT,α,p=sales revenue-1Tordering cost+purchasing cost+inventory holding cost+preservation technology cost+interest payable-interest earned(8)TPT,α,p=TP1T,α,pifM≤TTP2T,α,pifM≥T.In (8), TP1(T,α,p) and TP2(T,α,p) represent the total profit per unit time when M≤T and M≥T, respectively, where(9)TP1T,α,p=pDp-1TA+CDpλαeλαT-1+hDpλ2αeλαT-1-λαT+α+IpCDp2T-M2-IepDpT22,(10)TP2T,α,p=pDp-1TA+CDpλαeλαT-1+hDpλ2αeλαT-1-λαT+α-DpIepTM-T2.It is easy to verify that TP1M,α,p=TP2M,α,p when T=M. In the following section, we determined the optimal solution of the inventory model for the two cases: namely, M≤T and M≥T.


## 2.1. Notation

The notation that is used in the development of the inventor model is shown below.Parameters λ ( α ): deterioration rate when there is an investment on preservation technology (units/time unit) λ 0: deterioration rate without preservation technology investment (units/time unit) δ: sensitive parameter of investment to the deterioration rate D ( p ) = a - b p: demand rate function which is a function of selling price p (units/time unit) a: demand scale b: price sensitive parameter C: purchasing cost ($/unit) h: inventory holding cost ($/unit/time unit) A: ordering cost per order ($/order) M: credit period offered by supplier (time unit) I e: the interest earned by the retailer (%/time unit) I p: the interest payable by the retailer (%/time unit) I ( t ): inventory level (units) TP T , α , p: total profit ($/time unit)Dependent Decision Variables Q: ordering quantity (units)Decision Variables T: inventory cycle length (time unit) α: cost of preservation technology investment per unit time ($/unit/time unit) p: selling price ($/unit).


## 2.2. Assumptions

The inventory model with controllable deterioration under the permissible delay in payments is based on the following assumptions.(i) The demand rate,D(p)=a-bp, is assumed to be price dependent, where a and b are nonnegative constants.(ii) The inventory planning horizon is infinite and the inventory system involves only one item.(iii) It is considered the assumption of He and Huang [51] which is as follows: λ(α)=λ0e-δα, where λ(α) is the deterioration rate after investing on preservation technology, λ0 is the deterioration rate without preservation technology investment, and δ is the sensitive parameter of investment to the deterioration rate. The relationship of deterioration rate and the preservation technology investment parameter satisfies the following conditions: ∂λ(α)/∂α<0,∂2λ(α)/∂α>0.(iv) The cost of preservation technology investment per unit time is constrained toα∈0,α-.(v) There is no shortage of inventory.(vi) Under a trade credit periodM, the retailer would settle the account at time t=M and pays for the interest charge on items in stock with rate Ip over the interval [M,T] as T≥M. Alternatively, when the retailer settles the account at time t=M, it is not required to pay any interest charge for items in the stock during the whole cycle as T≤M.(vii) The retailer accumulates revenue and earns interest from the beginning of the inventory cycle until the end of the trade credit period offered by the supplier. That is, the retailer collects revenue and receives interest during the period fromt=0 to t=M with rate Ie under trade credit conditions.


## 2.3. Inventory Model with Controllable Deterioration under the Permissible Delay in Payments

Figure1 shows the inventory level through time. The inventory model is described briefly as follows: the retailer orders and receives an order size of Q units of a single product from the supplier at the beginning of the cycle, that is, at time t=0. During the time interval [0,T], the inventory level I(t) is depleted gradually due to demand from the customers. At time t=T, the inventory level drops to zero.Figure 1
Graphical representation of the inventory behavior through time.The differential equation representing the change of inventory levelI(t) with respect to time is as follows:(1)dItdt+λαIt=-Dp,0≤t≤T.Considering the initial condition I(0)=Q and the boundary conditionI(T)=0, hence the solution to the differential equation (1) is(2)It=DpλαeλαT-t-1,0≤t≤T.Thus, the retailer’s order size per cycle is given by(3)I0=Q=DpλαeλαT-1.The retailer’s profit function per cycle is comprised of the elements listed below:(a) The total sales revenue =pDp.(b) Ordering cost =A(c) Purchasing cost =CQ=CD(p)/λ(α)eλαT-1.(d) Inventory holding cost =h∫0TItdt=hDp/λ2αeλαT-1-λαT.(e) Preservation technology cost =α.(f) Interest payable: depending on the retailer’s trade credit periodM offered by the supplier, the following two cases occur: Case 1 when M≤T and Case 2 when M≥T which are shown in Figure 1. Thus, to compute the interest payable, we need to consider these two cases. The calculation of interest payable is detailed below.Case 1 (M≤T).
When the credit periodM is less than or equal to the replenishment cycle time T, the retailer begins to pay interest for the items in stock after time M with a rate Ip. Hence, the interest payable is calculated as follows:(4)IpC∫MTDpT-tdt=IpCDp2T-M2.Case 2 (M≥T).
BecauseT is less than or equal to the credit period M, there is no interest paid for financing the inventory in stock. Therefore, the interest payable in this case is zero.(g) Interest earned: to compute the interest earned, it is necessary to consider the two mentioned cases above.Case 1 (M≤T).
In this case, the retailer sells the goods, accumulates sales revenue, and earns interest with rateIe until time M. Therefore, the interest earned from time 0 to M per cycle is computed with(5)Iep∫0TDpT-tdt=IepDpT22.Case 2 (M≥T).
Because the cycle timeT is less than or equal to the credit period M, from time 0 to T, the retailer sells the goods and continues to accumulate sales revenue to earn interest. Therefore, the interest earned from time 0 to M per cycle is determined as(6)DpIepTM-T2.Hence, the total profit per unit time is given by(7)TPT,α,p=sales revenue-1Tordering cost+purchasing cost+inventory holding cost+preservation technology cost+interest payable-interest earned(8)TPT,α,p=TP1T,α,pifM≤TTP2T,α,pifM≥T.In (8), TP1(T,α,p) and TP2(T,α,p) represent the total profit per unit time when M≤T and M≥T, respectively, where(9)TP1T,α,p=pDp-1TA+CDpλαeλαT-1+hDpλ2αeλαT-1-λαT+α+IpCDp2T-M2-IepDpT22,(10)TP2T,α,p=pDp-1TA+CDpλαeλαT-1+hDpλ2αeλαT-1-λαT+α-DpIepTM-T2.It is easy to verify that TP1M,α,p=TP2M,α,p when T=M. In the following section, we determined the optimal solution of the inventory model for the two cases: namely, M≤T and M≥T.


## 3. Optimal Solution Procedure

This section determines the optimum values ofT, α, and p which maximize the total profit TPT,α,p. The necessary conditions to maximize the total profit are proved by following propositions.Proposition 1.
Whenα and p are fixed, the total profit is concave with respect to T.Proof.
The first- and second-order partial derivatives of the total profit with respect toT are as follows:(11)∂TP1T,α,p∂T=A+αT2-CDp2T22TT-M-T-M2Ip+T2Ie-Tee-αδTλ0-eαδ-1+ee-αδTλ0λ0-hDpT2λ0Te2αδ-e-αδλ0+e-αδ+e-αδTλ0λ0-e2αδ-1-Te-αδλ0+e-αδ+e-αδTλ0,∂2TP1T,α,p∂T2=-2A+αT3-12CDp2T-4T-MT2+2T-M2T3Ip-CDp-2ee-αδTλ0T2+2eαδee-αδTλ0-1T3λ0+e-αδ+e-αδTλ0λ0T-hDpee-αδTλ0hT-2e2αδhe-αδ+e-αδTλ0λ0-e-αδλ0T2λ02+2e2αδhee-αδTλ0-e-αδTλ0-1T3λ02<0.Therefore, for an optimum solution, it is considered that ∂TP1(T,α,p)/∂T=0(12)∂TP1T,α,p∂T=A+αT2-CDp2T22TT-M-T-M2Ip+T2Ie-Tee-αδTλ0-eαδ-1+ee-αδTλ0λ0-hDpT2λ0Te2αδ-e-αδλ0+e-αδ+e-αδTλ0λ0-e2αδ-1-Te-αδλ0+e-αδ+e-αδTλ0=0.Proposition 2.
WhenT and p are fixed, the total profit is concave with respect to α.Proof.
The first- and second-order partial derivatives of the total profit with respect toα are as follows:(13)∂TP1T,α,p∂α=-DpTλ02λ02Dp-Cee-αδTλ0Tδλ02+λ0Ceαδ-1+ee-αδTλ0δ+2e2αδhδ-1+ee-αδTλ0-e-αδTλ0+e2αδhe-αδTδλ0-ee-αδ+αδTλ0Tδλ0,∂2TP1T,α,p∂α2=-1Ta-bp-Cee-αδTλ0Tδ2+Ceαδee-αδTλ0-1δ2λ0+Ce-αδ+e-αδTλ0T2δ2λ0+4e2αδhδ2ee-αδTλ0-1-e-αδTλ0λ02+4e2αδhδe-αδTδλ0-e-αδ+e-αδTλ0Tδλ0λ02+e2αδh-e-αδTδ2λ0-e-αδ+e-αδTλ0Tδλ0-δ-e-αδTδλ0λ02<0.Consequently, for an optimum solution, it is necessary that ∂TP1(T,α,p)/∂α=0(14)∂TP1T,α,p∂α=-DpTλ02λ02Dp-Cee-αδTλ0Tδλ02+λ0Ceαδ-1+ee-αδTλ0δ+2e2αδhδ-1+ee-αδTλ0-e-αδTλ0+e2αδhe-αδTδλ0-ee-αδ+αδTλ0Tδλ0=0.Proposition 3.
WhenT and α are fixed, the total profit is concave with respect to p.Proof.
The first- and second-order partial derivatives of the total profit with respect top are as follows: (15)∂TP1T,α,p∂p=a-2bp-bTCλαeλαT-1+hλ2αeλαT-1-λαT-IepT22+IpC2T-M2=0,∂2TP1T,α,p∂p2=-2b+12bTIe<0.Hence, for an optimum solution, it is needed that ∂TP1(T,α,p)/∂p=0(16)∂TP1T,α,p∂p=a-2bp-bTCλαeλαT-1+hλ2αeλαT-1-λαT-IepT22+IpC2T-M2=0.Considering that (12), (14), and (16) are satisfied, thus the sufficient conditions are ∂TP1(T,α,p)/∂T2<0,∂TP1(T,α,p)/∂α2<0, and ∂TP1(T,α,p)/∂p2<0. The Hessian determinant H1 for sufficient conditions is given by(17)H1=∂2TP1T,α,p∂T2∂2TP1T,α,p∂T∂α∂2TP1T,α,p∂T∂p∂2TP1T,α,p∂α∂T∂2TP1T,α,p∂α2∂2TP1T,α,p∂α∂p∂2TP1T,α,p∂p∂T∂2TP1T,α,p∂p∂α∂2TP1T,α,p∂p2<0.Proposition 4.
Whenα and p are fixed, the total profit is concave with respect to T.Proof.
The first- and second-order partial derivatives of the total profit with respect toT are as follows:(18)∂TP2T,α,p∂T=Dpλ02T2λ02A+αDp-pIeM-T2Tλ02+λ0Ceαδ-1+ee-αδTλ0+e2αδh-1+ee-αδTλ0-e-αδTλ0-DpTCee-αδTλ0-IepM-T2+pTIe2+e2αδhλ0ee-αδ+e-αδTλ0-e-αδ,∂2TP2T,α,p∂T2=-Dpee-αδTλ0h+pIe+Ce-αδ+e-αδTλ0λ0T+2DpCee-αδTλ0-pM-T/2Ie+1/2pTIe+e2αδh-e-αδλ0+e-αδ+e-αδTλ0λ0/λ02T2-2DpA/Dp+α/Dp-pM-T/2TIe+Ceαδee-αδTλ0-1/λ0+e2αδhee-αδTλ0-1-e-αδTλ0/λ02T3<0.Thus, for an optimum solution, it is required that ∂TP2(T,α,p)/∂T=0(19)∂TP2T,α,p∂T=Dpλ02T2λ02A+αDp-pIeM-T2Tλ02+λ0Ceαδ-1+ee-αδTλ0+e2αδh-1+ee-αδTλ0-e-αδTλ0-DpTCee-αδTλ0-IepM-T2+pTIe2+e2αδhλ0ee-αδ+e-αδTλ0-e-αδ=0.Proposition 5.
WhenT and p are fixed, the total profit is concave with respect to α.Proof.
The first- and second-order partial derivatives of the total profit with respect toα are as follows:(20)∂TP2T,α,p∂α=-Dpλ02Tλ02Dp-Cee-αδTλ0Tδλ02+Ceαδ-1+ee-αδTλ0δλ0+2e2αδhδ-1+ee-αδTλ0-e-αδTλ0+e2αδhe-αδTδλ0-ee-αδ+αδTλ0Tδλ0,∂2TP2T,α,p∂α2=-1TDp-Cee-αδTλ0Tδ2+Ceαδee-αδTλ0-1δ2λ0+Ce-αδ+e-αδTλ0T2δ2λ0+4e2αδhδ2ee-αδTλ0-1-e-αδTλ0λ02+4e2αδhδe-αδTδλ0-e-αδ+e-αδTλ0Tδλ0λ02+e2αδh-e-αδTδ2λ0-e-αδ+e-αδTλ0Tδλ0-δ-e-αδTδλ0λ02<0.As a result, for an optimum solution, it is obligatory that ∂TP2(T,α,p)/∂α=0(21)∂TP2T,α,p∂α=-Dpλ02Tλ02Dp-Cee-αδTλ0Tδλ02+Ceαδ-1+ee-αδTλ0δλ0+2e2αδhδ-1+ee-αδTλ0-e-αδTλ0+e2αδhe-αδTδλ0-Tδλ0e-αδ+e-αδTλ0=0.Proposition 6.
WhenT and α are fixed, the total profit is concave with respect to p.Proof.
The first- and second-order partial derivatives of the total profit with respect top are as follows:(22)∂TP2T,α,p∂p=a-2bp-bTCλαeλαT-1+hλ2αeλαT-1-λαT-IepTM-T2,∂2TP2T,α,p∂p2=-2b+bM-T2Ie<0.For that reason, for an optimum solution, it is mandatory that ∂TP2(T,α,p)/∂p=0:(23)∂TP2T,α,p∂p=a-2bp-bTCλαeλαT-1+hλ2αeλαT-1-λαT-IepTM-T2=0.Considering that (19), (21), and (23) are satisfied, then the sufficient conditions are ∂TP2(T,α,p)/∂T2<0,  ∂TP2(T,α,p)/∂α2<0, and ∂TP2(T,α,p)/∂p2<0. The Hessian determinant H2 for sufficient conditions is given by(24)H2=∂2TP1T,α,p∂T2∂2TP1T,α,p∂T∂α∂2TP1T,α,p∂T∂p∂2TP1T,α,p∂α∂T∂2TP1T,α,p∂α2∂2TP1T,α,p∂α∂p∂2TP1T,α,p∂p∂T∂2TP1T,α,p∂p∂α∂2TP1T,α,p∂p2<0.Furthermore, the concavity of the total profit functions TP1(T,α,p) and TP2(T,α,p) is proved graphically (see Section 4). Using the mathematical results derived above, the following algorithm is proposed with the aim of determining the optimal values of T,α, and p for the inventory model.Algorithm 7.
Step  1. Input all parameters.
Step  2. Determine the value of T1, α1, and p1 from (12), (14), and (16) and T2, α2, and p2 from (19), (21), and (23).
Step  3. Calculate TP1∗(T,α,p),  TP2∗T,α,p, and TP2∗(M,α,p)for the following cases: (1), (2), (3), or (4).
( 1 ) If M≤T1 and M≤T2, then the optimal solution is T∗=T1, and hence compute the total profit TP1∗(T∗,α∗,p∗)=TP1∗(T1,α,p) with (9) and go to Step 4.
( 2 ) If M≥T1 and M≥T2, then the optimal solution is T∗=T2, and hence calculate the total profit TP∗(T∗,α∗,p∗)=TP2∗(T2,α,p) with (10) and go to Step 4.
( 3 ) If M≤T1 and M≥T2, then the optimal solution is T∗=M, and hence determine the total profit TP∗(T∗,α,p)=TP2∗(M,α,p) with (10) and go to Step 4.
( 4 ) If T1≤M and T2≤M, then calculate TP1(T1,α,p) and TP2(T2,α,p) with (9) and with (10), respectively.
IfTP2(T2,α,p)≥TP1(T1,α,p)⇒TP(T,α,p)=TP2(T2,α,p), then the optimal value of T is T∗=T2, else TP(T,α,p)=TP1(T1,α,p) and the optimal value of T is T∗=T1.
Step  4. Report the optimal solution for T,  α,  p, and total profit TP(T,α,p).


## 4. Numerical Example

This section presents and solves two numerical examples.Example 1.
Consider the following parameters:A=$80/order, a=250 units, b=8 units, C=$4/unit, h=$4/unit/year, Ip=0.2/year, Ie=0.8/year, M=0.6 years, λ0=0.2units/year, and δ=20.
By applying the proposed algorithm exhibited in Section3 and with the help of the software Mathematica-9.0, the following optimal solution is obtained after several iterations: T∗=0.324 year, α∗=2.383 $/unit/year, and p∗=14.143$/unit.
Here, notice thatT=0.324 years are less than M=0.6 years. The maximum total profit is TP1∗(T,α,p)=1276.01, the order size is Q∗=44.946 units, and the conditions ∂2TP1T,α,p/∂T2=-5999.70,  ∂2TP1(T,α,p)/∂α2=-1.25338<0, and ∂2TP1(T,α,p)/∂p2=-14.9632<0. For the values of T,  α, and p, the determinant of Hessian matrix of H1 is as follows: H1=-98695.40<0. Therefore, the values of T∗, α∗, and p∗ are optimal and maximize the total profit. Figures 2, 3, and 4 show the concavity of TP1(T,α,p) with respect to α, p, and α and p, respectively. These figures demonstrate that the solution is a global optimal solution.Figure 2
Concavity ofTP1(T,α,p) with respect to α.Figure 3
Concavity ofTP1(T,α,p) with respect to p.Figure 4
Concavity ofTP1(T,α,p) with respect to α and p.Example 2.
Consider the following parameters:A=$100/order, a=250 units, b=8 units, C=$6/unit, h=$4/unit/year, Ip=0.2/year, Ie=0.8/year, M=0.2 year, λ0=0.2 units/year, and δ=20.
By using the proposed algorithm given in Section3 and with the help of the software Mathematica-9.0, the following optimal solution is found after some iterations: T∗=0.309year,α∗=3.287$/unit/year, and p∗=12.515$/unit. Here, note that T=0.309 years are greater than M=0.2 years. The maximum total profit is TP2∗(T,α,p)=609.841, the lot size is Q∗=46.785units, and the conditions are ∂2TP2(T,α,p)/∂T2=-190001<0,  ∂2TP2(T,α,p)/∂α2=-1.30625<0, and ∂2TP1(T,α,p)/∂p2=-15.7088<0.
For the values ofT, α, and p, the determinant of Hessian matrix of H2 is as follows: H2=-3.7674×106<0. Consequently, the values of T∗,  α∗, and p∗ are optimal and maximize the total profit. Figures 5, 6, and 7 demonstrate the concavity of TP2(T,α,p) with respect to α, p, and α and p, respectively. These figures confirm that the solution is a global optimal solution.Figure 5
Concavity ofTP2(T,α,p) with respect to α.Figure 6
Concavity ofTP2(T,α,p) with respect to p.Figure 7
Concavity ofTP2(T,α,p) with respect to α and p.


## 5. Sensitivity Analysis

This section studies the effects of changes in the inventory system parameters(A,a,b,C,h,Ip,Ie,M,λ0, and δ) by changing of the parameters, taking one parameter at a time and keeping the remaining parameters unchanged. Tables 2 and 3 present the results.Table 2
Effect of changes of parameters on optimal solution for Example1.
Parameters % change T ∗ α ∗ p ∗ Q ∗ TP 1 ∗ ( T , α , p ) A 30% 0.363374 2.96185 14.2406 49.9998 1266.68 15% 0.344390 2.68839 14.1939 47.5481 1271.26 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.303001 2.03812 14.0877 42.1644 1280.93 −30% 0.27607 1.56739 14.0142 38.6333 1286.86  a 30% 0.296699 2.49107 19.0606 51.7499 2578.27 15% 0.309264 2.43699 16.606 48.3907 1880.09 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.342803 2.33074 11.6696 41.3988 766.148 −30% 0.366018 2.28028 9.18029 37.7238 350.614  b 30% 0.318707 2.40211 10.2789 46.1674 714.264 15% 0.321468 2.39273 11.9585 45.5598 961.768 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.327326 2.37393 17.1008 44.3273 1692.24 −30% 0.330438 2.36453 21.3278 43.7018 2275.92  C 30% 0.340667 3.31461 13.5291 48.735 1079.15 15% 0.332418 2.86733 13.8361 46.8019 1179.81 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.319014 2.63483 14.3482 43.7508 1365.74 −30% 0.308379 1.25346 14.7579 41.456 145.81  h 30% 0.236743 0.857263 13.8228 33.567 1083.083 15% 0.273515 1.55704 13.9636 38.3852 1173.36 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.40066 3.40017 14.3936 54.5707 1400.42 −30% 0.52838 4.74473 14.7992 70.0918 1569.93  I p 30% 0.331227 2.49215 14.1469 45.8786 1282.11 15% 0.327832 2.4388 14.145 45.4196 1279.09 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.320743 2.3256 14.1419 44.4585 1272.88 −30% 0.31704 2.26545 14.1407 43.9553 1269.69  I e 30% 0.33946 2.56897 14.4979 46.0505 1379.62 15% 0.331642 2.47447 14.3142 45.4914 1335.50 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.317493 2.29533 13.9837 44.4161 1218.95 −30% 0.311059 2.21026 13.8341 43.9007 1164.14  M 30% 0.346356 2.72841 14.1222 48.0147 1284.36 15% 0.334823 2.55017 14.1343 46.4038 1280.56 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.315005 2.2309 14.1485 43.6575 1270.62 −30% 0.306925 2.09605 14.1488 42.5525 1264.28  λ 0 30% 0.325465 3.05686 14.1462 44.8493 1275.75 15% 0.324939 2.74214 14.1449 44.9509 1275.87 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.323639 1.96606 14.1415 44.9552 1276.17 −30% 0.3228 1.4675 14.1394 44.989 1276.37  δ 30% 0.325952 2.384942 14.1543 45.1516 1280.88 15% 0.325284 2.38413 14.1496 45.0624 1276.50 0% 0.324339 2.38333 14.1433 44.9465 1276.01 −15% 0.322938 2.30928 14.1345 44.7898 1272.30 −30% 0.320723 2.07688 14.1213 44.5685 1267.02Table 3
Effect of changes of parameters on optimal solution for Example2.
Parameters % change T ∗ α ∗ p ∗ Q ∗ TP 2 ∗ ( T , α , p ) A 30% 0.353741 3.99375 12.3624 53.8351 503.202 15% 0.332412 3.66378 12.4358 50.4186 554.753 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.284807 2.84892 12.6013 42.8798 669.313 −30% 0.257541 2.32217 12.6967 38.6175 734.442  a 30% 0.24563 2.60305 17.585 45.6694 1817.05 15% 0.273548 2.92294 15.0486 46.1036 1162.56 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.356499 3.71082 9.98532 47.6621 124.5162 −30% 0.420237 4.21518 9.4614 48.8362 159.119  b 30% 0.34024 3.91769 8.76068 54.446 51.0562 15% 0.325896 3.6252 10.3903 50.708 299.213 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.290785 2.88979 15.3985 42.6373 1016.37 −30% 0.26891 2.40899 19.5319 38.2024 1581.20  C 30% 0.311735 4.05493 11.5925 49.3265 220.946 15% 0.310579 3.68945 12.0539 48.0355 420.428 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.308668 2.83729 12.9758 45.2796 789.217 −30% 0.307852 2.32114 13.4365 44.4094 958.638  h 30% 0.296803 3.12304 12.4686 44.9767 577.478 15% 0.302993 3.20371 12.4916 45.8539 593.516 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.316558 3.37453 12.5389 47.7763 626.478 −30% 0.324019 3.4652 12.5634 48.8331 643.455  I e 30% 0.279986 2.74691 12.7117 41.9136 654.146 15% 0.293719 3.00535 12.6112 44.1856 630.984 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.32813 3.59871 12.4235 49.8067 591.033 −30% 0.350298 3.94664 12.3371 53.386 574.957  M 30% 0.309362 3.24232 12.8288 45.9783 732.704 15% 0.309453 3.26505 12.67 46.3848 671.106 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.309697 3.30954 12.3637 47.180 548.941 −30% 0.309847 3.3313 12.2159 47.569 488.425  λ 0 30% 0.3106 3.96098 12.5114 46.9494 607.348 15% 0.310117 3.64626 12.5131 46.8728 608.514 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.308922 2.87019 12.5172 46.6836 611.386 −30% 0.308151 2.37164 12.5199 46.5615 613.242  δ 30% 0.310117 3.03516 12.5193 46.7683 612.975 15% 0.309907 3.16408 12.5173 46.7806 611.539 0% 0.309565 3.28746 12.515 46.7854 609.841 −15% 0.308994 3.3845 12.5122 46.7748 607.791 −30% 0.307991 3.340288 12.5088 46.7397 605.29Based on the results of Tables2 and 3, the following managerial insights are provided.(a) When the value of scaling factora increases and other parameters’ values are fixed, it can be observed that, for both cases, the optimal total profit per unit time TPi∗(T,α,p); i=1,2, the optimal selling price p∗, and the optimal order quantity Q∗ increase, while the optimal length of replenishment cycle T∗ and the optimal preservation technology investment α∗ fluctuate (increasing for Example 1 and decreasing for Example 2). This indicates that when the scaling factor a is very large, the retailer will establish the selling price higher and will order more products to gain more profit. In addition, the retailer will also invest more funds into the improvement of preservation technology to decrease the item’s deterioration, so the retailer can vend more products.(b) When the price elasticityb increases, the optimal cycle length T∗, the optimal selling price p∗, and the optimal total profit per unit time TPi∗(T,α,p); i=1,2, decrease, whereas the optimal preservation technology investment α∗ and the optimal order quantity Q∗ increase. Apparently, as price elasticity b increases, the retailer will reduce the selling price to prevent the demand rate from decreasing rapidly. And with a reduction in the selling price, the order quantity will decrease and the total profit per unit time will shrink significantly.(c) When the value of the holding cost per unit timeh increases, it can be noted that the optimal selling price p∗, the optimal profit per unit time TPi∗(T,α,p); i=1,2, the optimal length of replenishment cycle T∗, preservation technology investment α∗, and the optimal order quantity Q∗ decrease. This result reveals that while facing a higher holding cost, the enterprise will tend to reduce the optimal preservation technology investment, shorten the length of replenishment cycle, and order a smaller quantity each time for keeping inventory level as low as possible. From this sense, in order to enhance the competitiveness, the enterprise must pay more attention to storage process control to diminish the holding cost.(d) Upon increasing the credit period(M), the optimal total profit per unit time TPi∗(T,α,p); i=1,2, the optimal cycle length T∗, preservation technology investment α∗, the optimal order quantity Q∗, and the optimal selling price p∗ decrease.(e) With the increment in the value of the deterioration rate parameterλ0, then preservation technology investment α∗, the optimal selling price p∗, the optimal order quantity Q∗, and the optimal total profit per unit time TPi∗(T,α,p); i=1,2, decrease. This is an obvious result, because if the number of deteriorated items increases, then we have less profit.(f) As the value of sensitive parameter of investmentδ increases, the optimal total profit per unit time TPi∗(T,α,p); i=1,2, the optimal cycle length T∗, preservation technology investment α∗, and the optimal selling price p∗ increase, whereas the optimal order quantity Q∗ decreases.(g) With increase in the value of interest earned rateIe, the optimal total profit per unit time TPi∗(T,α,p); i=1,2, the optimal cycle length T∗, preservation technology investment α∗, and the optimal selling price p∗ increase, whereas the optimal order quantity Q∗ decreases, whereas an increase in the value of interest payable rate Ip shows exactly opposite effect, which is a traditional result, and this validate the correctness of the proposed inventory model.


## 6. Conclusion

This paper investigates the retailer’s replenishment policy for deteriorating items under trade credit policy when retailer invests money on preservation technology in order to reduce the deterioration rate. The main objective of this paper is to determine the optimal replenishment, selling price, and preservation technology investments strategies which maximize the total profit. There are numerous opportunities for future research work. For example, if the demand cannot be predicted, it is suggested to develop the inventory model with a stochastic demand. The proposed inventory model accepts that the demand only depends on the price under credit period and does not take customers’ memories of the retailer’s trade credit history into consideration. Since marketing and consumer behavior literature has empirically validated that the reference effect plays an important role in customer purchase decisions, hence, another possible extension of this work would need the incorporation of the reference effect into the proposed inventory model. Also, the effects of environmental issues on the suggested inventory model would be worth studying for future research.



---

*Source: 6962417-2018-02-21.xml*