Retailer's Optimal Pricing and Ordering Policies for Non-Instantaneous Deteriorating Items with Price-Dependent Demand and Partial Backlogging

Abstract

An inventory system for non-instantaneous deteriorating items with price-dependent demand is formulated and solved. A model is developed in which shortages are allowed and partially backlogged, where the backlogging rate is variable and dependent on the waiting time for the next replenishment. The major objective is to determine the optimal selling price, the length of time in which there is no inventory shortage, and the replenishment cycle time simultaneously such that the total profit per unit time has a maximum value. An algorithm is developed to find the optimal solution, and numerical examples are provided to illustrate the theoretical results. A sensitivity analysis of the optimal solution with respect to major parameters is also carried out.

1. Introduction

The Economic Order Quantity (EOQ) model proposed by Harris [1] has been widely used by enterprises in order to reduce the cost of stock. Due to the variability in economic circumstances, many scholars constantly modify the basic assumptions of the EOQ model and consider more realistic factors in order to make the model correspond with reality. One such modification is the inclusion of the deterioration of items. In general, deterioration is defined as damage, spoilage, dryness, vaporization, and so forth, which results in a decrease of the usefulness of the original item. Ghare and Schrader [2] were the first to consider deterioration when they presented an EOQ model for an exponentially constant deteriorating inventory. Later, Covert and Philip [3] formulated the model with a variable deterioration rate with a two-parameter Weibull distribution. Philip [4] then developed the inventory model with a three-parameter Weibull distribution rate and no shortages. Tadikamalla [5] further developed the inventory model with deterioration using Gamma distribution. Shah [6] extended Philip’s [4] model by allowing shortages and complete backlogging. However, when shortages occur, one cannot be certain that customers are willing to wait for a backorder. Some customers are willing to wait, while others will opt to buy from other sellers. Park [7] and Wee [8] considered constant partial backlogging rates during a shortage period in their inventory models. In some instances the backlogging rate was variable. Abad [9, 10] investigated EOQ models for deteriorating items allowing shortage and partial backlogging. He assumed that the backlogging rate was variable and dependent on the length of waiting time for the next replenishment. Two backlogging rates which are dependent on the length of waiting time for the next replenishment arose: 𝑘₀𝑒^−𝛿𝑥 and 𝑘₀/(1+𝛿𝑥), where 𝑥 is the length of waiting time for the next replenishment, 0<𝑘₀≤1 and 𝛿>0. Dye [11] revised Abad’s [10] model by adding both the backorder cost and the cost of lost sales to the total profit function. There is a vast amount of literature on inventory models for deteriorating items. Review articles by Goyal [12], Sarma [13], Raafat et al. [14], Pakkala and Achary [15], Goyal and Giri [16], Ouyang et al. [17], Dye et al. [18], and others provide a summary of this material. In this literature, all the inventory models for deteriorating items assume that the deterioration of inventory items starts as soon as they arrive in stock. However, in real life most goods would have an initial period in which the quality or original condition is maintained, namely, no deterioration occurs. This type of phenomenon is common, for example, firsthand fruit and vegetables have a short span during which fresh quality is maintained and there is almost no spoilage. Wu et al. [19] defined a phenomenon of “non-instantaneous deterioration” and developed a replenishment policy for non-instantaneous deteriorating items with stock-dependent demand such that the total relevant inventory cost per unit time had a minimum value.

In addition to deterioration, price has a great impact on demand. In general, a decrease in selling price leads to increased customer demand and results in a high sales volume. Therefore, pricing strategy is a primary tool that sellers or retailers use to maximize profit and consequently models with price-dependent demand occupy a prominent place in the inventory literature. Eilon and Mallaya [20] were the first to investigate a deteriorating inventory model with price-dependent demand. Cohen [21] determined both the optimal replenishment cycle and price for inventory that was subject to continuous decay over time at a constant rate. Wee [22] studied pricing and replenishment policy for a deteriorating inventory with a price elastic demand rate that declined over time. Abad [9] considered the dynamic pricing and lot sizing problem of a perishable good under partial backlogging of demand. He assumed that the fraction of shortages backordered was variable and a decreasing function of the waiting time. Wee [23, 24] extended Cohen’s [21] model to develop a replenishment policy for deteriorating items with price-dependent demand, with Weibull distribution deterioration and separately considered with/without a quantity discount. Wee and Law [25] developed an inventory model for deteriorating items with price-dependent demand in which the time value of money was also taken into account. Abad [26] presented a model of pricing and lot sizing under conditions of perishability, finite production, and partial backlogging. Mukhopadhyay et al. [27, 28] re-established Cohen’s [21] model by taking a price elastic demand rate and considering a time-proportional and two-parameter Weibull distribution deterioration rate separately. Chang et al. [29] introduced a deteriorating inventory model with price-time dependent demand and partial backlogging.

In order to match realistic circumstances, a non-instantaneous deteriorating inventory model for determining the optimal pricing and ordering policies with price-dependent demand is considered in this study. In the model, shortages are allowed and partially backlogged where the backlogging rate is variable and dependent on the waiting time for the next replenishment. The purpose is to simultaneously determine the optimal selling price, the length of time in which there is no inventory shortage, and the replenishment cycle time, such that the total profit per unit time has a maximum value for the retailer. There are two possible scenarios in this study. The length of time in which there is no shortage is (i) larger than or equal to, or (ii) shorter than or equal to the length of time in which the product has no deterioration. The optimal pricing and ordering policies are obtained through theoretical analysis. It is first proven that for any given selling price, the optimal values of the length of time in which there is no inventory shortage and the replenishment cycle time not only exist, but are unique. Next, this paper proves that there exists a unique selling price to maximize the total profit per unit time when the time in which there is no inventory shortage and the replenishment cycle time are given. Furthermore, an algorithm is developed to find the optimal solution. Numerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the optimal solution with respect to major parameters is also carried out.

2. Notation and Assumptions

The following notation and assumptions are used throughout the paper.

Notation. 𝐴: The ordering cost per order
𝑐: The purchasing cost per unit𝑐₁: The holding cost per unit per unit time𝑐₂: The shortage cost per unit per unit time𝑐₃: The unit cost of lost sales𝑝: The selling price per unit, where 𝑝>𝑐𝜃: The parameter of the deterioration rate function𝑡_𝑑: The length of time in which the product exhibits no deterioration𝑡₁: The length of time in which there is no inventory shortage𝑇: The length of the replenishment cycle time𝑄: The order quantity𝑝^∗: The optimal selling price per unit𝑡^∗₁: The optimal length of time in which there is no inventory shortage𝑇^∗: The optimal length of the replenishment cycle time𝑄^∗: The optimal order quantity𝐼₁(𝑡): The inventory level at time 𝑡∈[0,𝑡_𝑑]𝐼₂(𝑡): The inventory level at time 𝑡∈[𝑡_𝑑,𝑡₁], where 𝑡₁>𝑡_𝑑𝐼₃(𝑡): The inventory level at time 𝑡∈[𝑡₁,𝑇]𝐼₀: The maximum inventory level𝑆: The maximum amount of demand backlogged𝑇𝑃(𝑝,𝑡₁𝑇): The total profit per unit time of the inventory system𝑇𝑃^∗∶ The optimal total profit per unit time of the inventory system, that is, 𝑇𝑃^∗=𝑇𝑃(𝑝^∗,𝑡^∗₁,𝑇^∗).

Assumptions. (1) A single non-instantaneous deteriorating item is modeled.
(2)The replenishment rate is infinite and the lead time is zero.(3)The demand rate 𝐷(𝑝)is a non-negative, continuous, decreasing, and concave function of the selling price𝑝, that is, 𝐷^′(𝑝)<0 and 𝐷^′′(𝑝)<0.(4)During the fixed period, 𝑡_𝑑, there is no deterioration and at the end of this period, the inventory item deteriorates at the constant rate 𝜃.(5)There is no replacement or repair for deteriorated items during the period under consideration.(6)Shortages are allowed to occur. It is assumed that only a fraction of demand is backlogged. Furthermore the longer the waiting time, the smaller the backlogging rate. Let 𝐵(𝑥) denote the backlogging rate given by 𝐵(𝑥)=1/(1+𝛿𝑥), where 𝑥 is the waiting time until the next replenishment and 𝛿 is a positive backlogging parameter.

3. Model Formulation

First a short problem description is provided. The replenishment problem of a single non-instantaneous deteriorating item with partial backlogging is considered in this study. The inventory system is as follows. 𝐼₀ units of item arrive at the inventory system at the beginning of each cycle and drop to zero due to demand and deterioration. Then shortage occurs until the end of the current order cycle. Based on the values of 𝑡₁ and 𝑡_𝑑, there are two possible cases: (1)𝑡₁≥𝑡_𝑑 and (2)𝑡₁≤𝑡_𝑑 (see Figure 1). These cases are discussed as follows.

(a)

(b)

Figure 1 

Graphical representation of inventory system.

Case 1 (𝑡₁≥𝑡_𝑑). In this case, the length of time in which there is no shortage is larger than or equal to the length of time in which the product has no deterioration. During the time interval [0,𝑡_𝑑], the inventory level decreases due to demand only. Subsequently the inventory level drops to zero due to both demand and deterioration during the time interval [𝑡_𝑑,𝑡₁]. Finally, a shortage occurs due to demand and partial backlogging during the time interval [𝑡₁,𝑇]. The whole process is repeated.
As described before, the inventory level decreases according to demand only during the time interval [0,𝑡_𝑑]. Hence the differential equation representing the inventory status is given by
𝑑𝐼₁(𝑡)𝑑𝑡=−𝐷(𝑝),0<𝑡<𝑡_𝑑, (3.1) with the boundary condition 𝐼₁(0)=𝐼₀. By solving (3.1) over time t, it yields. 𝐼₁(𝑡)=𝐼₀−𝐷(𝑝)𝑡,0≤𝑡≤𝑡_𝑑. (3.2)
During the time interval [𝑡_𝑑,𝑡₁], the inventory level decreases due to demand as well as deterioration. Thus, the differential equation representing the inventory status is given by
𝑑𝐼₂(𝑡)𝑑𝑡+𝜃𝐼₂(𝑡)=−𝐷(𝑝),𝑡_𝑑<𝑡<𝑡₁, (3.3) with the boundary condition 𝐼₂(𝑡₁)=0. The solution of (3.3) is 𝐼₂(𝑡)=𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡)}−1],𝑡_𝑑≤𝑡≤𝑡₁. (3.4)
Considering continuity of 𝐼₁(𝑡) and 𝐼₂(𝑡) at point 𝑡=𝑡_𝑑, that is, 𝐼₁(𝑡_𝑑)=𝐼₂(𝑡_𝑑), the maximum inventory level for each cycle can be obtained and is given by
𝐼₀=𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝐷(𝑝)𝑡_𝑑. (3.5) Substituting (3.5) into (3.2) gives 𝐼₁(𝑡)=𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝐷(𝑝)(𝑡_𝑑−𝑡),0≤𝑡≤𝑡_𝑑. (3.6)
During the shortage time interval [𝑡₁,𝑇], the demand at time 𝑡 is partially backlogged according to the fraction𝐵(𝑇−𝑡). Thus, the inventory level at time 𝑡 is governed by the following differential equation:
𝑑𝐼₃(𝑡)𝑑𝑡=−𝐷(𝑝)𝐵(𝑇−𝑡)=−𝐷(𝑝)1+𝛿(𝑇−𝑡),𝑡₁<𝑡<𝑇, (3.7) with the boundary condition 𝐼₃(𝑡₁)=0. The solution of (3.7) is 𝐼₃(𝑡)=−𝐷(𝑝)𝛿{ln[1+𝛿(𝑇−𝑡₁)]−ln[1+𝛿(𝑇−𝑡)]},𝑡₁≤𝑡≤𝑇. (3.8) Putting 𝑡=𝑇 into (3.8), the maximum amount of demand backlogged per cycle is obtained as follows: 𝑆≡−𝐼₃(𝑇)=𝐷(𝑝)𝛿ln[1+𝛿(𝑇−𝑡₁)]. (3.9) From (3.5) and (3.9), one can obtain the order quantity per cycle as 𝑄=𝐼₀=+𝑆𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝐷(𝑝)𝑡_𝑑+𝐷(𝑝)𝛿ln[1+𝛿(𝑇−𝑡₁)]. (3.10)
Next, the total relevant inventory cost and sales revenue per cycle consist of the following six elements.
(a)The ordering cost is𝐴.(b)The inventory holding cost (denoted by 𝐻𝐶) is 𝐻𝐶=𝑐₁[^𝑡𝑑₀𝐼₁(𝑡)𝑑𝑡+^𝑡1_𝑡𝑑𝐼₂(𝑡)𝑑𝑡]=𝑐₁{^𝑡𝑑₀{𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝐷(𝑝)(𝑡_𝑑−𝑡)}𝑑𝑡+^𝑡1_𝑡𝑑𝐷(𝑝)𝜃[𝑒^{𝜃(𝑡1−𝑡)}−1]𝑑𝑡}=𝑐₁𝑡𝐷(𝑝){^𝑑𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}𝑡−1]+^2𝑑2+1𝜃²[𝑒^{𝜃(𝑡1−𝑡𝑑)}−𝜃(𝑡₁−𝑡_𝑑)−1]}. (3.11)(c)The shortage cost due to backlog (denoted by𝑆𝐶) is𝑆𝐶=𝑐₂^𝑇_𝑡1[−𝐼₃𝑐(𝑡)]𝑑𝑡=²𝐷(𝑝)𝛿^𝑇_𝑡1{ln[1+𝛿(𝑇−𝑡₁=𝑐)]−ln[1+𝛿(𝑇−𝑡)]}𝑑𝑡₂𝐷(𝑝)𝛿{𝑇−𝑡₁−ln[1+𝛿(𝑇−𝑡₁)]𝛿}. (3.12)(d)The opportunity cost due to lost sales (denoted by𝑂𝐶) is𝑂𝐶=𝑐₃^𝑇_𝑡1𝐷(𝑝)[1−𝐵(𝑇−𝑡)]𝑑𝑡=𝑐₃^𝑇_𝑡11𝐷(𝑝)[1−1+𝛿(𝑇−𝑡)]𝑑𝑡=𝑐₃𝐷(𝑝){𝑇−𝑡₁−ln[1+𝛿(𝑇−𝑡₁)]𝛿}. (3.13)(e)The purchase cost (denoted by𝑃𝐶) is1𝑃𝐶=𝑐𝑄=𝑐𝐷(𝑝){𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝑡_𝑑+1𝛿ln[1+𝛿(𝑇−𝑡₁)]}. (3.14)(f)The sales revenue (denoted by 𝑆𝑅) is𝑆𝑅=𝑝[^𝑡1₀𝐷(𝑝)𝑑𝑡−𝐼₃(𝑇)]=𝑝𝐷(𝑝){𝑡₁+1𝛿ln[1+𝛿(𝑇−𝑡₁)]}. (3.15) Therefore, the total profit per unit time of Case 1 (denoted by𝑇𝑃₁(𝑝,𝑡₁,𝑇)) is given by 𝑇𝑃₁(𝑝,𝑡₁,𝑇)=(𝑆𝑅−𝐴−𝐻𝐶−𝑆𝐶−𝑂𝐶−𝑃𝐶)𝑇=𝐷(𝑝)𝑇𝑐{(𝑝−𝑐+²+𝛿𝑐₃𝛿)[𝑡₁+ln[1+𝛿(𝑇−𝑡₁)]𝛿]−𝜃(𝑐+𝑐₁𝑡_𝑑)+𝑐₁𝜃²×[𝑒^{𝜃(𝑡1−𝑡𝑑)}−𝜃(𝑡₁−𝑡_𝑑)−1]−𝑐₁𝑡_𝑑𝑡₁+𝑐₁𝑡²_𝑑2−𝑐₂+𝛿𝑐₃𝛿𝐴𝑇−𝐷(𝑝)}. (3.16)

Case 2 (𝑡₁≤𝑡_𝑑). In this case, the length of time in which there is no shortage is shorter than or equal to the length of time in which the product exhibits no deterioration. This implies that the optimal replenishment policy for the retailer is to sell out all stock before the deadline at which the items start to decay. Under these circumstances, the model becomes the traditional inventory model with a shortage. By using similar arguments as in Case 1, the order quantity per order, 𝑄, and the total profit per unit time (denoted by𝑇𝑃₂(𝑝,𝑡₁,𝑇)) can be obtained and are given by 𝑄=𝐷(𝑝)𝑡₁+𝐷(𝑝)𝛿ln[1+𝛿(𝑇−𝑡₁)],𝑇𝑃₂(𝑝,𝑡₁,𝑇)=𝐷(𝑝)𝑇{(𝑝−𝑐)[𝑡₁+ln[1+𝛿(𝑇−𝑡₁)]𝛿𝑐]−¹𝑡²₁2−𝑐₂+𝛿𝑐₃𝛿[𝑇−𝑡₁−ln[1+𝛿(𝑇−𝑡₁)]𝛿𝐴]−𝐷(𝑝)}. (3.17)(3.18) respectively.
From (4.13), the following is obtained:
𝑇=𝑡₁+𝑐₁𝑡₁𝛿(𝑝−𝑐+𝑀−𝑐₁𝑡₁). (4.15) Substituting (4.15) into (4.14) gives 𝑐₁𝑡₁𝛿−𝑝−𝑐+𝑀𝛿ln[𝑝−𝑐+𝑀𝑝−𝑐+𝑀−𝑐₁𝑡₁𝑐]−¹𝑡²₁2+𝐴𝐷(𝑝)=0. (4.16) By using a similar approach as used in Case 1, the following results are found.

Lemma 4.3. For any given p, (a)if Δ(𝑝)≤0, then the solution of (𝑡₁,𝑇)which satisfies (4.13) and (4.14) not only exists but also is unique,(b)if Δ(𝑝)>0, then the solution of (𝑡₁,𝑇)which satisfies (4.13) and (4.14) does not exist.

Proof. The proof is similar to Appendix A, and hence is omitted here.

Lemma 4.4. For any given p, (a)if Δ(𝑝)≤0, then the total profit per unit time 𝑇𝑃₂(𝑝,𝑡₁,𝑇) is concave and reaches its global maximum at the point (𝑡₁,𝑇)=(𝑡₁₂,𝑇₂), where (𝑡₁₂,𝑇₂) is the point which satisfies (4.13) and (4.14),(b)if Δ(p)>0, then the total profit per unit time 𝑇𝑃₂(𝑝,𝑡₁,𝑇)has a maximum value at the point (𝑡₁,𝑇)=(𝑡₁₂,𝑇₂), where 𝑡₁₂=𝑡_𝑑 and 𝑇₂=𝑡_𝑑+𝑐₁𝑡_𝑑/(𝛿(𝑝−𝑐+𝑀−𝑐₁𝑡_𝑑)).

Proof. The proof is similar to Appendix B, and hence is omitted here.

Likewise, for a given (𝑡₁₂,𝑇₂), taking the first-and second-order derivatives of 𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂) in (3.18) with respect to 𝑝, it is found that

𝑑𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂)=𝐷𝑑𝑝^′(𝑝)𝑇₂{(𝑝−𝑐+𝑀)[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿𝑐]−¹𝑡²₁₂2−𝑀𝑇₂}+𝐷(𝑝)𝑇₂[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿𝑑],₂𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂)𝑑𝑝²=𝐷^′′(𝑝)𝑇₂{(𝑝−𝑐+𝑀)[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿𝑐]−¹𝑡²₁₂2−𝑀𝑇₂}+2𝐷^′(𝑝)𝑇₂[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿]. (4.17)(4.18)

It can be shown that 𝑑²𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂)/𝑑𝑝²<0. Consequently, 𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂) is a concave function of 𝑝 for fixed (𝑡₁₂,𝑇₂), and hence there exists a unique value of 𝑝 (say 𝑝₂) which maximizes 𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂). 𝑝₂ can be obtained by solving 𝑑𝑇𝑃₂(𝑝,𝑡₁₂,𝑇₂)/𝑑𝑝=0; that is, 𝑝₂ can be determined by solving the following equation: 𝐷^′(𝑝)𝑇₂{(𝑝−𝑐+𝑀)[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿𝑐]−¹𝑡²₁₂2−𝑀𝑇₂}+𝐷(𝑝)𝑇₂[𝑡₁₂+ln[1+𝛿(𝑇₂−𝑡₁₂)]𝛿]=0. (4.19)

Combining the previous Lemmas 4.2 and 4.4, the following result is obtained.

Theorem 4.5. For any given p, (a)if Δ(𝑝)>0, the optimal length of time in which there is no inventory shortage is 𝑡₁₁ and the optimal replenishment cycle length is 𝑇₁,(b)if Δ(𝑝)<0, the optimal length of time in which there is no inventory shortage is 𝑡₁₂ and the optimal replenishment cycle length is 𝑇₂,(c)if Δ(𝑝)=0, the optimal length of time in which there is no inventory shortage is 𝑡_𝑑 and the optimal replenishment cycle length is 𝑡_𝑑+(𝑐₁𝑡_𝑑/(𝛿(𝑝−𝑐+𝑀−𝑐₁𝑡_𝑑))).

Proof. It immediately follows from Lemmas 4.2, 4.4 and the fact that 𝑇𝑃₁(𝑝,𝑡_𝑑,𝑇)=𝑇𝑃₂(𝑝,𝑡_𝑑,𝑇) for given p.

Now, the following algorithm is established to obtain the optimal solution (𝑝^∗,𝑡^∗₁,𝑇^∗)of the problem. The convergence of the procedure can be proven by adopting a similar graphical technique as used in Hadley and Whitin [30].

Algorithm 4.6. Step 1. Start with 𝑗=0 and the initial value of 𝑝_𝑗=𝑐.Step 2. Calculate Δ(𝑝_𝑗)=(𝑐₁𝑡_𝑑/𝛿)−((𝑝_𝑗−𝑐+𝑀)/𝛿)ln[(𝑝_𝑗−𝑐+𝑀)/(𝑝_𝑗−𝑐+𝑀−𝑐₁𝑡_𝑑)]−(𝑐₁𝑡²_𝑑/2)+(𝐴/𝐷(𝑝_𝑗)) for a given 𝑝_𝑗, (i)if Δ(𝑝_𝑗)>0, determine the values 𝑡_11,𝑗 and 𝑇_1,𝑗 by solving (4.1). Then, put (𝑡_11,𝑗,𝑇_1,𝑗) into (4.12) and solve this equation to obtain the corresponding value 𝑝_1,𝑗+1. Let 𝑝_𝑗+1=𝑝_1,𝑗+1 and (𝑡_1𝑗,𝑇_𝑗)=(𝑡_11,𝑗,𝑇_1,𝑗), go to Step 3.(ii)If Δ(𝑝_𝑗)<0, determine the values 𝑡_12,𝑗 and 𝑇_2,𝑗 by solving (4.13) and (4.14). Then, put (𝑡_12,𝑗,𝑇_2,𝑗) into (4.19) and solve this equation to obtain the corresponding value 𝑝_2,𝑗+1. Let 𝑝_𝑗+1=𝑝_2,𝑗+1 and (𝑡_1𝑗,𝑇_𝑗)=(𝑡_12,𝑗,𝑇_2,𝑗), go to Step 3.(iii)If Δ(𝑝_𝑗)=0, set 𝑡_1,𝑗=𝑡_𝑑 and 𝑇_𝑗=𝑡_𝑑+(𝑐₁𝑡_𝑑/𝛿(𝑝_𝑗−𝑐+𝑀−𝑐₁𝑡_𝑑)), and then put (𝑡_1,𝑗,𝑇_𝑗) into (4.12) or (4.19) to obtain the corresponding value 𝑝_1,𝑗+1(=𝑝_2,𝑗+1). Let 𝑝_𝑗+1=𝑝_1,𝑗+1or 𝑝_2,𝑗+1 and(𝑡_1𝑗,𝑇_𝑗)=(𝑡_𝑑,𝑡_𝑑+(𝑐₁𝑡_𝑑/(𝑝_𝑗−𝑐+𝑀−𝑐₁𝑡_𝑑))), go to Step 3.Step 3. If the difference between𝑝_𝑗and 𝑝_𝑗+1 is enough small (i.e.,|𝑝_𝑗−𝑝_𝑗+1|≤10⁻⁵), then set 𝑝^∗=𝑝_𝑗 and (𝑡^∗₁,𝑇^∗)=(𝑡_1𝑗,𝑇_𝑗). Thus (𝑝^∗,𝑡^∗₁,𝑇^∗) is the optimal solution. Otherwise, set 𝑗=𝑗+1 and go back to Step 2.

The previous algorithm can be implemented with the help of a computer-oriented numerical technique for a given set of parameter values. Once (𝑝^∗,𝑡^∗₁,𝑇^∗) is obtained, 𝑄^∗ can be found from (3.10) or (3.17) and 𝑇𝑃^∗=𝑇𝑃(𝑝^∗,𝑡^∗₁,𝑇^∗) from (3.16) or (3.18).

5. Numerical Examples

In order to illustrate the solution procedure for this inventory system, the following examples are presented.

Example 5.1. This example is based on the following cost parameter values: 𝐴=$250/per order, 𝑐=$20/per unit, 𝑐₁=$1/per unit/per unit time, 𝑐₂=$5/per unit/per unit time, 𝑐₃=$25/per unit, 𝜃=0.08, 𝑡_𝑑=1/12, and 𝐵(𝑥)=1/(1+0.1𝑥). In addition, it is assumed that the demand rate is a linearly decreasing function of the selling price and is given by 𝐷(𝑝)=200−4𝑝, where 0<𝑝<50. Under the given values of the parameters and according to the algorithm in the previous section, the computational results can be found as shown in Table 1. From Table 1, it can be seen that after five iterations, the optimal selling price 𝑝^∗=$35.9722, the optimal length of time in which there is no inventory shortage 𝑡^∗₁=1.56831, and the optimal length of replenishment cycle 𝑇^∗=2.05155. Hence the optimal order quantity 𝑄^∗=119.632 units, and the optimal total profit per unit time of the inventory system 𝑇𝑃(𝑝^∗,𝑡^∗₁,𝑇^∗)=$660.918.

Table 1 

The solution procedure of Example 5.1.

Moreover, if 𝑡_𝑑=0, this model becomes the instantaneous deterioration case, and the optimal solutions can be found as follows: 𝑝^∗=36.0234, 𝑡₁^∗=1.5556, 𝑇^∗=2.05227,𝑄^∗=119.711, and 𝑇𝑃^∗=655.022. The results with instantaneous and non instantaneous deterioration models for 𝑡_𝑑∈{1/12,2/12,3/12} are shown in Table 2. From Table 2, it can be seen that there is an improvement in total profit from the non-instantaneously deteriorating demand model. Moreover, the longer the length of time where no deterioration occurs, the greater the improvement in total profit will be. This implies that if the retailer can extend the length of time in which no deterioration occurs by improving stock equipment, then the total profit per unit time will increase.

Table 2 

The results with instantaneous and non-instantaneous deterioration models.

Example 5.2. This study now investigates the effects of changes in the values of the cost parameters 𝐴, 𝑐, 𝑐₁, 𝑐₂, 𝑐₃, and 𝜃 on the optimal selling price 𝑝^∗, the optimal length of time in which there is no inventory shortage 𝑡^∗₁, the optimal length of replenishment cycle 𝑇^∗, the optimal order quantity 𝑄^∗, and the optimal total profit per unit time of inventory system 𝑇𝑃(𝑝^∗,𝑡^∗₁,𝑇^∗) according to Example 5.1. For convenience, only the case with a linear demand function 𝐷(𝑝)=400−4𝑝, where 0<𝑝<100, is considered. The sensitivity analysis is performed by changing each value of the parameters by +50%, +25%, −25%, and −50%, taking one parameter at a time and keeping the remaining parameter values unchanged. The computational results are shown in Table 3.

Table 3 

Sensitivity analysis with respect to the cost items.

On the basis of the results of Table 3, the following observations can be made.

(a)The optimal selling price 𝑝^∗ increases with an increase in the values of parameters 𝐴, 𝑐, 𝑐₁, 𝑐₂, 𝑐₃,and 𝜃. Moreover, 𝑝^∗ is weakly positively sensitive to changes in parameters 𝐴, 𝑐₁, 𝑐₂, 𝑐₃, and 𝜃, whereas 𝑝^∗ is highly positively sensitive to changes in parameter 𝑐. It is reasonable that the purchase cost has a strong and positive effect upon the optimal selling price.(b)The optimal length of time in which there is no inventory shortage 𝑡^∗₁ increases with increased values of parameters 𝐴, 𝑐₂, and 𝑐₃ while it decreases as the values of parameters 𝑐₁ and 𝜃 increase. From an economic viewpoint, this means that the retailer will avoid shortages when the order cost, shortage cost, and cost of lost sales are high. (c)The optimal length of the replenishment cycle 𝑇^∗ increases with an increase in the value of parameter 𝐴, while it decreases as the values of parameters 𝑐₁, 𝑐₂, 𝑐₃, and 𝜃 increase. This implies that the higher the order cost the longer the length of the replenishment cycle, while the lower the holding cost, shortage cost, cost of lost sales, and deteriorating rate, the longer the length of the replenishment cycle.(d)The optimal order quantity 𝑄^∗ increases with an increase in the value of parameter 𝐴 and decreases with an increase in the values of parameters 𝑐, 𝑐₁, 𝑐₂, 𝑐₃, and 𝜃. The corresponding managerial insight is that as the order cost increases, the order quantity increases. On the other hand, as the purchasing cost, holding cost, shortage cost, cost of lost sales, and deterioration rate increase, the order quantity decreases.(e)The optimal total profit per unit time 𝑇𝑃^∗ decreases with an increase in the values of parameters 𝐴, 𝑐, 𝑐₁, 𝑐₂, 𝑐₃, and 𝜃. This implies that increases in costs and the deterioration rate have a negative effect upon the total profit per unit time.

6. Conclusions

The problem of determining the optimal replenishment policy for non-instantaneous deteriorating items with price-dependent demand is considered in this study. A model is developed in which shortages are allowed and the backlogging rate is variable and dependent on the waiting time for the next replenishment. There are two possible scenarios in this study: (1) the length of time in which there is no shortage is larger than or equal to the length of time in which the product exhibits no deterioration (i.e., 𝑡₁≥𝑡_𝑑) and (2) the length of time in which there is no shortage is shorter than or equal to the length of time in which the product exhibits no deterioration (𝑡₁≤𝑡_𝑑). Through theoretical analysis several useful theorems are developed and an algorithm is provided to determine the optimal selling price, the optimal length of time in which there is no inventory shortage, and the optimal replenishment cycle time for various situations. Several numerical examples are provided to illustrate the theoretical results under various situations and a sensitivity analysis of the optimal solution with respect to major parameters is also carried out. This paper contributes to existing methodology in several ways. Firstly, it addresses the problem of non-instantaneous deteriorating items under the circumstances in which the demand rate is price sensitive and there is partial backlogging, hitherto not treated in the literature. Secondly, it develops several useful theoretical results and provides an algorithm to determine the optimal selling price and length of replenishment cycle. Finally, from the theoretical results, it can be seen that the retailer may determine the optimal order quantity and selling price by considering whether to sell his/her stocks before or after the products begin to deteriorate in the case of non-instantaneous deteriorating items.

In the future it is hoped that the model will be further developed to incorporate other realistic circumstances such as capital investment in storehouse equipment to reduce the deterioration rate of items, stochastic demand, and a finite replenishment rate.

Appendices

A. Proof of Lemma 4.1

Proof. It can be seen that F(x) is a strictly decreasing function in 𝑥∈[𝑡_𝑑,𝑡^𝑏₁) and lim_𝑥→𝑡^𝑏₁⁻𝐹(𝑥)=−∞. Therefore, if Δ(𝑝)≡𝐹(𝑡_𝑑)≥0, then by using the Intermediate Value Theorem, there exists a unique value of 𝑡₁ (say 𝑡₁₁) such that 𝐹(𝑡₁₁)=0; that is, 𝑡₁₁ is the unique solution of (4.4). Once the value 𝑡₁₁ is found, then the value of 𝑇 (denoted by 𝑇₁) can be found from (4.3) and is given by 𝑇₁=𝑡₁₁+((𝑁[𝑒^{𝜃(𝑡11−𝑡𝑑)}−1]+𝑐₁𝑡_𝑑)/𝛿{𝑝−𝑐+𝑀−𝑁[𝑒^{𝜃(𝑡11−𝑡𝑑)}−1]−𝑐₁𝑡_𝑑}).

Proof. If Δ(𝑝)≡𝐹(𝑡_𝑑)<0, then from 𝐹(𝑥) is a strictly decreasing function of 𝑥∈[𝑡_𝑑,𝑡^𝑏₁), which implies 𝐹(𝑥)<0 for all 𝑥∈[𝑡_𝑑,𝑡^𝑏₁). Thus, a value 𝑡₁∈[𝑡_𝑑,𝑡^𝑏₁) cannot be found such that 𝐹(𝑡₁)=0. This completes the proof.

B. Proof of Lemma 4.2

Proof. For any given p, taking the second derivatives of 𝑇𝑃₁(𝑡₁,𝑇,𝑝) with respect to 𝑡₁ and 𝑇 and then finding the values of these functions at point (𝑡₁,𝑇)=(𝑡₁₁,𝑇₁) give 𝜕²𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑡²₁|_{(𝑡11,𝑇1)}=𝐷(𝑝)𝑇₁{−𝛿(𝑝−𝑐+𝑀)[1+𝛿(𝑇₁−𝑡₁₁)]²−𝑁𝑒^{𝜃(𝑡11−𝑡𝑑)}𝜕}<0,₂𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑇²|_{(𝑡11,𝑇1)}=𝐷(𝑝)𝑇₁{−𝛿(𝑝−𝑐+𝑀)[1+𝛿(𝑇₁−𝑡₁₁)]²𝜕}<0,₂𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑡₁|𝜕𝑇_{(𝑡11,𝑇1)}=𝐷(𝑝)𝑇₁{𝛿(𝑝−𝑐+𝑀)[1+𝛿(𝑇₁−𝑡₁₁)]²𝜕},₂𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑡²₁|_{(𝑡11,𝑇1)}×𝜕²𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑇²|_{(𝑡11,𝑇1)}𝜕−[²𝑇𝑃₁(𝑝,𝑡₁,𝑇)𝜕𝑡₁|𝜕𝑇_{(𝑡11,𝑇1)}]²=(𝐷(𝑝)𝑇₁)²{𝛿(𝑝−𝑐+𝑀)𝑁𝑒^{𝜃(𝑡11−𝑡𝑑)}[1+𝛿(𝑇₁−𝑡₁₁)]⁴}>0. (B.1) Because (𝑡₁₁,𝑇₁) is the unique solution of (4.1) if Δ(𝑝)≥0, therefore, for any given p, (𝑡₁₁,𝑇₁) is the global maximum point of 𝑇𝑃₁(𝑡₁,𝑇,𝑝).

Proof. For any given p, if Δ(𝑝)<0, then it is known that 𝐹(𝑥)<0, for all 𝑥∈[𝑡_𝑑,𝑡^𝑏₁). Thus, 𝑑𝑇𝑃₁(𝑝,𝑡₁,𝑇)=𝑑𝑇𝐷(𝑝)𝑇²×{−{𝑁[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝑐₁𝑡_𝑑}𝑡₁+𝑁[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]+𝑐₁𝑡_𝑑𝛿−(𝑝−𝑐+𝑀)𝛿ln[𝑝−𝑐+𝑀𝑝−𝑐+𝑀−𝑁[𝑒^{𝜃(𝑡1−𝑡𝑑)}−1]−𝑐₁𝑡_𝑑]+𝑁𝜃[𝑒^{𝜃(𝑡1−𝑡𝑑)}−𝜃(𝑡₁−𝑡_𝑑)−1]+𝑐₁𝑡_𝑑𝑡₁−𝑐₁𝑡²_𝑑2+𝐴}=𝐷(𝑝)𝐷(𝑝)𝐹(𝑡₁)𝑇²<0,∀𝑡₁∈[𝑡_𝑑,𝑡^𝑏₁), (B.2) which implies that 𝑇𝑃₁(𝑡₁,𝑇,𝑝) is a strictly decreasing function of 𝑇. Thus, 𝑇𝑃₁(𝑡₁,𝑇,𝑝) has a maximum value when 𝑇 is minimum. On the other hand, from (4.3), it can be seen that 𝑇 has a minimum value of 𝑡_𝑑+(𝑐₁𝑡_𝑑/(𝛿(𝑝−𝑐+𝑀−𝑐₁𝑡_𝑑)))as 𝑡₁=𝑡_𝑑. Therefore, 𝑇𝑃₁(𝑡₁,𝑇,𝑝) has a maximum value at the point(𝑡₁₁,𝑇₁), where 𝑡₁₁=𝑡_𝑑 and 𝑇₁=𝑡_𝑑+(𝑐₁𝑡_𝑑/(𝛿(𝑝−𝑐+𝑀−𝑐₁𝑡_𝑑))). This completes the proof.

Acknowledgment

The authors greatly appreciate the anonymous referees for their valuable and helpful suggestions regarding earlier version of the paper.