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

Abstract

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.

1. 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.

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.

Table 1 

Major characteristic of inventory models on selected articles.

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.

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 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.(4)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).(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. [4].(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: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],𝑐_𝑣(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.

3. Mathematical Model

For simplicity, we use the same inventory model as in Teng and Yang [11], which is shown in Figure 1.

Figure 1 

Graphical representation of inventory model.

As a result, we obtain the time-weighted inventory during the 𝑖th cycle as 𝐼_𝑖=1𝜃^𝑠𝑖_𝑡𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}−1𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛. (3.1) Similarly, the time-weighted backorders due to shortages during the 𝑖th cycle is𝐵_𝑖=^𝑡𝑖_𝑠𝑖−1𝑡_𝑖−𝑡𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛, (3.2) and the total number of lost sales due to shortages during the 𝑖th cycle is𝐿_𝑖=^𝑡𝑖_𝑠𝑖−11−𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛. (3.3) The order quantity and unit sold at 𝑡_𝑖 in the 𝑖th replenishment cycle is𝑄_𝑖=^𝑡𝑖_𝑠𝑖−1𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛, (3.4) and unit sold at 𝑡_𝑖 in the 𝑖th replenishment cycle is𝑆_𝑖=^𝑡𝑖_𝑠𝑖−1𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛. (3.5) Therefore, the purchasing cost during the 𝑖th replenishment cycle is𝑃_𝑖=𝑐_𝑓+𝑐_𝑣𝑡_𝑖𝑄_𝑖=𝑐_𝑓+𝑐_𝑣𝑡_𝑖^𝑡𝑖_𝑠𝑖−1𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)𝑑𝑡, (3.6) and the revenue realized in the 𝑖th replenishment cycle is𝑅_𝑖=𝑝𝑡_𝑖^𝑡𝑖_𝑠𝑖−1𝛽𝑡_𝑖−𝑡𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑓(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛. (3.7) Hence, if 𝑛 replenishment orders are placed in [0, 𝐻], then the total realized profit of the inventory system during the planning horizon 𝐻 is𝑇𝑃𝑛,𝑠_𝑖,𝑡_𝑖=^𝑛_𝑖=1𝑅_𝑖−𝑃_𝑖−𝑐_ℎ𝐼_𝑖−𝑐_𝑏𝐵_𝑖−𝑐_𝑙𝐿_𝑖=^𝑛_𝑖=1^𝑡𝑖_𝑠𝑖−1𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑡_𝑖−𝑡+𝑐_𝑙𝛽𝑡_𝑖−𝑡−𝑐_𝑙𝑓(𝑡)𝑑𝑡−𝑛𝑐_𝑓+^𝑛_𝑖=1^𝑠𝑖_𝑡𝑖𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖−𝑐_ℎ𝜃+𝑐_𝑣𝑡_𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}−1𝑓(𝑡)𝑑𝑡, (3.8) with 0=𝑠₀<𝑡₁ 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.

4. Theoretical Results and Solution

For a fixed value of 𝑛, the necessary conditions for 𝑇𝑃(𝑛,{𝑠_𝑖},{𝑡_𝑖}) to be maximized are: 𝜕𝑇𝑃(𝑛,{𝑠_𝑖},{𝑡_𝑖})/𝜕𝑡_𝑖=0 for 𝑖=1,2,…,𝑛, and 𝜕𝑇𝑃(𝑛,{𝑠_𝑖},{𝑡_𝑖})/𝜕𝑠_𝑖=0, for. 𝑖=1,2,…,𝑛−1. Consequently, we obtain^𝑠𝑖_𝑡𝑖𝑝^′𝑡_𝑖+𝑐_ℎ+𝜃𝑐_𝑣𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)𝑑𝑡=−^𝑡𝑖_𝑠𝑖−1𝑝^′𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖−𝑐_𝑏𝛽𝑡_𝑖−𝑡+𝑝𝑡_𝑖+𝑐_𝑙−𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑡_𝑖−𝑡𝛽^′𝑡_𝑖𝑓(𝑡)𝑝𝑡−𝑡𝑑𝑡,_𝑖+1−𝑐_𝑣𝑡_𝑖+1−𝑐_𝑏𝑡_𝑖+1−𝑠_𝑖𝛽𝑡_𝑖+1−𝑠_𝑖−𝑐_𝑙1−𝛽𝑡_𝑖+1−𝑠_𝑖=𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖−𝑐_ℎ𝜃+𝑐_𝑣𝑡_𝑖𝑒^{𝜃(𝑠𝑖−𝑡𝑖)}−1, (4.1)(4.2) respectively. Note that (4.1) and (4.2) are coincident with the following articles’(1)Equations (12) and (11) in Teng and Yang [11], if 𝑝(𝑡)=0,(2)Equations (15) and (14) in Teng et al. [4], if 𝑝(𝑡)=0 and 𝑐_𝑣(𝑡)=𝑐_𝑣,(3)Equations (11) and (10) in Chang et al. [5], if 𝑝(𝑡)=𝑝 and 𝑐_𝑣(𝑡)=𝑐_𝑣.

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 be𝑅(𝑡,𝑢)=𝑝(𝑡)−𝑐_𝑣(𝑡)−𝑐_ℎ𝜃+𝑐_𝑣(𝑡)𝑒^{𝜃(𝑢−𝑡)}−1, (4.3) with 𝑡≤𝑢, and𝑃(𝑠,𝑡)=𝑝(𝑡)−𝑐_𝑣(𝑡)−𝑐_𝑏(𝑡−𝑠)𝛽(𝑡−𝑠)−𝑐_𝑙1−𝛽(𝑡−𝑠), (4.4) with 𝑠≤𝑡, respectively. Taking the partial derivative of 𝑅(𝑡,𝑢) and 𝑃(𝑠,𝑡) with respect to 𝑡 respectively, we obtained the following results:𝑅_𝑡(𝑡,𝑢)=𝑝^′(𝑡)+𝑐_ℎ+𝜃𝑐_𝑣(𝑡)−𝑐^′_𝑣(𝑡)𝑒^{𝜃(𝑢−𝑡)}𝑃_𝑡,(𝑠,𝑡)=𝑝^′(𝑡)−𝑐^′_𝑣(𝑡)−𝑐_𝑏𝛽(𝑡−𝑠)+𝑝(𝑡)+𝑐_𝑙−𝑐_𝑣(𝑡)−𝑐_𝑏(𝑡−𝑠)𝛽^′(𝑡−𝑠). (4.5)(4.6) 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.

Lemma 4.1. For any given n, if 𝑅_𝑡𝑖(𝑡_𝑖,𝑡)≤0, with 𝑡≥𝑡_𝑖, and 𝑃_𝑡𝑖(𝑡,𝑡_𝑖)<0, with 𝑡≤𝑡_𝑖,𝑖=1,2,…,𝑛, then the optimal solution is 𝑛^∗=1 and 𝑡^∗₁=0 (i.e., purchase at the beginning).

Proof. See Appendix A.

The results in Lemma 4.1 can be interpreted as follows. The condition 𝑅_𝑡𝑖(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_𝑡𝑖(𝑡_𝑖,𝑡)>0, with 𝑡≥𝑡_𝑖, and 𝑃_𝑡𝑖(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.

Proof. See Appendix B.

The result in Theorem 4.2 reduces the 2𝑛-dimensional problem of finding {𝑠^∗_𝑖} and {𝑡^∗_𝑖} to a one-dimensional problem. Since 𝑠₀=0, we only need to find 𝑡^∗₁ to generate 𝑠^∗₁ by (4.1), 𝑡^∗₂ by (4.2), and then the rest of {𝑠^∗_𝑖} and {𝑡^∗_𝑖} uniquely by repeatedly using (4.1) and (4.2). For any chosen 𝑡^∗₁, if 𝑠^∗_𝑛=𝐻, then 𝑡^∗₁ is chosen correctly. Otherwise, we can easily find the optimal 𝑡^∗₁ by standard search techniques.

Having calculated the second partial derivatives of the function 𝑇𝑃(𝑛,{𝑠_𝑖},{𝑡_𝑖}) shows that the Hessian matrix is negative definite if𝜕²𝑇𝑃𝜕𝑡²_𝑖𝜕≤−²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖+𝜕²𝑇𝑃𝜕𝑡_𝑖𝜕𝑠_𝑖−1𝜕<0,for𝑖=1,2,…,𝑛,₂𝑇𝑃𝜕𝑠²_𝑖𝜕≤−²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖+𝜕²𝑇𝑃𝜕𝑡_𝑖+1𝜕𝑠_𝑖<0,for𝑖=1,2,…,𝑛. (4.7)(4.8)

Theorem 4.3. For any given n, if 𝑅_𝑡𝑖(𝑡_𝑖,𝑡)>0, with 𝑡≥𝑡_𝑖, and 𝑃_𝑡𝑖(𝑡,𝑡_𝑖)<0, with 𝑡≤𝑡_𝑖, 𝑖=1,2,…,𝑛, under conditions (4.7)-(4.8), then the solution that satisfies the system of (4.1) and (4.2) is a global maximum solution.

Proof. See Appendix C.

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, let𝑇𝑃(𝑛)=𝑃𝑛,𝑠^∗_𝑖,𝑡^∗_𝑖. (4.9) By applying Bellman’s principle of optimality [12], we have the following theorem:

Theorem 4.4. 𝑇𝑃(𝑛) is concave in 𝑛.

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 as𝑛₁𝑐=roundedintegerof^ℎ+𝜃𝑐_𝑣𝑐_𝑏𝛽+𝑐_𝑙−𝑐_𝑣+𝑝1−𝛽𝐻𝑄(𝐻)2𝑐_𝑓𝑐_ℎ+𝜃𝑐_𝑣+𝑐_𝑏𝛽+𝑐_𝑙−𝑐_𝑣+𝑝1−𝛽^1/2, (4.10) where 𝑄(𝐻)=∫^𝐻₀𝑓(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 2. If 𝑇𝑃(𝑛)≥𝑇𝑃(𝑛−1), then compute 𝑇𝑃(𝑛+1), 𝑇𝑃(𝑛+2), …, until we find 𝑇𝑃(𝑘)>𝑇𝑃(𝑘+1). Set 𝑛^∗=𝑘 and stop.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.

5. Numerical Examples

Example 5.1. Let 𝑓(𝑡)=200+20𝑡, 𝐻=3, 𝑝(𝑡)=200+30𝑡, 𝑐_𝑣(𝑡)=150+10𝑡, 𝑐_𝑓=250, 𝑐_ℎ=40, 𝑐_𝑏=50, 𝑐_𝑙=200,𝜃=0.08, 𝛽(𝑡)=𝑒^−0.4𝑡 in appropriate units. After calculation, we have 𝑝=245, 𝑐_𝑣=165, and 𝛽=0.582. By (4.10), we obtain the estimate number of replenishments 𝑛₁=12. From computational results, we have 𝑇𝑃(13)=49021.79,𝑇𝑃(14)=49044.31, and 𝑇𝑃(15)=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.

Table 2 

The optimal replenishment schedule for Example 5.1.

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.
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 𝑛₁ is very close to the optimal replenishment number 𝑛^∗, no matter what magnitude of the parameters changed.

Table 3 

Sensitivity analysis on parameters changed for Example 5.2.

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.(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 𝑛₁ is also close to the optimal replenishment number 𝑛^∗.(4)The optimal replenishment number is slightly sensitive to the change of rate of change.

Table 4 

Sensitivity analysis on rate of change for Example 5.3.

6. 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.

Appendices

A. Proof of Lemma 4.1

Let𝑇𝑃_𝑖𝑠_𝑖−1,𝑡_𝑖,𝑠_𝑖=^𝑡𝑖_𝑠𝑖−1𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑡_𝑖−𝑡+𝑐_𝑙𝛽𝑡_𝑖−𝑡−𝑐_𝑙𝑓(𝑡)+𝑑𝑡_{𝑠𝑖𝑡𝑖}𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖−𝑐_ℎ𝜃+𝑐_𝑣𝑡_𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)=−1𝑑𝑡_{𝑡𝑖𝑠𝑖−1}𝑃𝑡,𝑡_𝑖𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑅𝑡_𝑖,𝑡𝑓(𝑡)𝑑𝑡. (A.1) We then have𝜕𝑇𝑃_𝑖𝜕𝑡_𝑖=^𝑡𝑖_𝑠𝑖−1𝑃_𝑡𝑖𝑡,𝑡_𝑖𝑓(𝑡)𝑑𝑡+^𝑠𝑖_𝑡𝑖𝑅_𝑡𝑖𝑡_𝑖,𝑡𝑓(𝑡)𝑑𝑡. (A.2) If 𝑅_𝑡𝑖(𝑡_𝑖,𝑡)≤0, then we know from (A.2) that 𝜕𝑇𝑃_𝑖/𝜕𝑡_𝑖≤0. Therefore, for any given 𝑖, 𝑇𝑃_𝑖 is decreasing with 𝑡_𝑖. This implies that 𝑇𝑃_𝑖(𝑠_𝑖−1,𝑠_𝑖−1,𝑠_𝑖)≥𝑇𝑃_𝑖(𝑠_𝑖−1,𝑡_𝑖,𝑠_𝑖) for any fixed 𝑖. Consequently, we obtain𝑇𝑃𝑛,𝑠_𝑖,𝑡_𝑖=^𝑛_𝑖=1𝑇𝑃_𝑖𝑠_𝑖−1,𝑡_𝑖,𝑠_𝑖−𝑛𝑐_𝑓≤^𝑛_𝑖=1𝑇𝑃_𝑖𝑠_𝑖−1,𝑠_𝑖−1,𝑠_𝑖−𝑛𝑐_𝑓=^𝑛_𝑖=1^𝑠𝑖_𝑠𝑖−1𝑅𝑠_𝑖−1,𝑡𝑓(𝑡)𝑑𝑡−𝑛𝑐_𝑓≤^𝑛_𝑖=1^𝑠𝑖_𝑠𝑖−1𝑅(0,𝑡)𝑓(𝑡)𝑑𝑡−𝑛𝑐_𝑓since𝑅_𝑡𝑖𝑡_𝑖≤,𝑡≤0_𝐻0𝑅(0,𝑡)𝑓(𝑡)𝑑𝑡−𝑐_𝑓. (A.3) This completes the proof.

B. Proof of Theorem 4.2

For any given 𝑠_𝑖−1 and 𝑡_𝑖, from (4.1), we set𝐹(𝑥)=^𝑥_𝑡𝑖𝑝^′𝑡_𝑖+𝑐_ℎ+𝜃𝑐_𝑣𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)+𝑑𝑡_{𝑡𝑖𝑠𝑖−1}𝑝^′𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖−𝑐_𝑏𝛽𝑡_𝑖−𝑡+𝑝𝑡_𝑖+𝑐_𝑙−𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑡_𝑖−𝑡𝛽^′𝑡_𝑖𝑓(𝑡)=−𝑡𝑑𝑡_𝑥𝑡𝑖𝑅_𝑡𝑖𝑡_𝑖,𝑡𝑓(𝑡)𝑑𝑡+^𝑡𝑖_𝑠𝑖−1𝑃_𝑡𝑖𝑡,𝑡_𝑖𝑓(𝑡)𝑑𝑡,with𝑥≥𝑡_𝑖≥𝑠_𝑖−1. (B.1) We then have𝐹𝑡_𝑖=^𝑡𝑖_𝑠𝑖−1𝑃_𝑡𝑖𝑡,𝑡_𝑖𝑓(𝑡)𝑑𝑡<0, (B.2) and lim_𝑥→∞𝐹(𝑥)>0. Taking the first derivatives of 𝐹(𝑥) with respect to 𝑥, we obtain𝐹^′(𝑥)=𝑅_𝑡𝑖𝑡_𝑖,𝑥𝑓(𝑥)>0. (B.3) 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 set𝐺(𝑥)=𝑝(𝑥)−𝑐_𝑣(𝑥)−𝑐_𝑏𝑥−𝑠_𝑖+𝑐_𝑙𝛽𝑥−𝑠_𝑖−𝑝𝑡_𝑖−𝑐_𝑣𝑡_𝑖+𝑐_𝑙+𝑐_ℎ𝜃+𝑐_𝑣𝑡_𝑖𝑒^{𝜃(𝑠𝑖−𝑡𝑖)}−1=𝑃𝑠_𝑖,𝑥−𝑅𝑡_𝑖,𝑠_𝑖,with𝑥≥𝑠_𝑖≥𝑡_𝑖. (B.4) We then have 𝐺𝑠_𝑖=𝑝𝑠_𝑖−𝑐_𝑣𝑠_𝑖−𝑅𝑡_𝑖,𝑠_𝑖=𝑅𝑠_𝑖,𝑠_𝑖−𝑅𝑡_𝑖,𝑠_𝑖>0, (B.5) since 𝑅_𝑡𝑖(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 obtain𝐺^′(𝑥)=𝑃_𝑥𝑠_𝑖,𝑥<0. (B.6) 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 𝜕²𝑇𝑃𝜕𝑡²_𝑖=𝑝𝑡_𝑖+𝑐_𝑙−𝑐_𝑣𝑡_𝑖𝛽^′(0)−𝑐_ℎ−𝜃𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑓𝑡_𝑖+^𝑡𝑖_𝑠𝑖−1𝑝^′′𝑡_𝑖−𝑐_𝑣^′′𝑡_𝑖𝛽𝑡_𝑖−𝑡+2𝑝^′𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖−𝑐_𝑏𝛽^′𝑡_𝑖+𝑝𝑡−𝑡_𝑖+𝑐_𝑙−𝑐_𝑣𝑡_𝑖−𝑐_𝑏𝑡_𝑖−𝑡𝛽^′′𝑡_𝑖𝑓(𝑡)+−𝑡𝑑𝑡_{𝑠𝑖𝑡𝑖}𝑝^′′𝑡_𝑖+𝜃𝑐^′_𝑣𝑡_𝑖−𝑐_𝑣^′′𝑡_𝑖−𝜃𝑐_ℎ+𝜃𝑐_𝑣𝑡_𝑖−𝑐^′_𝑣𝑡_𝑖𝑒^{𝜃(𝑡−𝑡𝑖)}𝑓(𝑡)𝜕𝑑𝑡,₂𝑇𝑃𝜕𝑠_𝑖−1𝜕𝑡_𝑖=−𝑃_𝑡𝑖𝑠_𝑖−1,𝑡_𝑖𝑓𝑠_𝑖−1𝜕>0,₂𝑇𝑃𝜕𝑡_𝑖𝜕𝑠_𝑖=𝑅_𝑡𝑖𝑡_𝑖,𝑠_𝑖𝑓𝑠_𝑖𝜕>0,₂𝑇𝑃𝜕𝑠²_𝑖=𝑝𝑡_𝑖+1+𝑐_𝑙−𝑐_𝑣𝑡_𝑖+1−𝑐_𝑏𝑡_𝑖+1−𝑠_𝑖𝛽^′𝑡_𝑖+1−𝑠_𝑖−𝑐_𝑏𝛽𝑡_𝑖+1−𝑠_𝑖−𝑐_ℎ+𝜃𝑐_𝑣𝑡_𝑖𝑒^{𝜃(𝑠𝑖−𝑡𝑖)}𝑓𝑠_𝑖. (C.1) Let Δ_𝑘 be the principal minor of order 𝑘, then, under condition (4.7)-(4.8), it is clear that Δ₁=𝜕²𝑇𝑃𝜕𝑡²₁𝜕≤−²𝑇𝑃𝜕𝑠₁𝜕𝑡₁=−𝑅_𝑡1𝑡₁,𝑠₁𝑓𝑠₁<0, (C.2)

which implies that Δ₁+(𝜕²𝑇𝑃/𝜕𝑠₁𝜕𝑡₁)<0:Δ₂=𝜕²𝑇𝑃𝜕𝑡²₁𝜕²𝑇𝑃𝜕𝑠²₁−𝜕²𝑇𝑃𝜕𝑠₁𝜕𝑡₁𝜕²𝑇𝑃𝜕𝑡₁𝜕𝑠₁𝜕≥−²𝑇𝑃𝜕𝑡₂𝜕𝑠₁Δ₁−𝜕²𝑇𝑃𝜕𝑡₁𝜕𝑠₁Δ₁+𝜕²𝑇𝑃𝜕𝑡₁𝜕𝑠₁>0, (C.3) which implies that Δ₂+(𝜕²𝑇𝑃/𝜕𝑡₂𝜕𝑠₁)Δ₁>0. For principal minor of higher order, 𝑖=2,3,…, it is not difficult to show that they satisfy the following recursive relation:Δ_2𝑖−1=𝜕²𝑇𝑃𝜕𝑡²_𝑖Δ_2𝑖−2−𝜕²𝑇𝑃𝜕𝑡_𝑖𝜕𝑠_𝑖−1²Δ_2𝑖−3,Δ_2𝑖=𝜕²𝑇𝑃𝜕𝑠²_𝑖Δ_2𝑖−1−𝜕²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖²Δ_2𝑖−2, (C.4) with the initial Δ₀=1. From (4.7)-(4.8) and the relation between second-order partial derivatives, we haveΔ_2𝑖−1𝜕≤−²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖Δ_2𝑖−2−𝜕²𝑇𝑃𝜕𝑡_𝑖𝜕𝑠_𝑖−1Δ_2𝑖−2+𝜕²𝑇𝑃𝜕𝑡_𝑖𝜕𝑠_𝑖−1Δ_2𝑖−3,Δ_2𝑖𝜕≥−²𝑇𝑃𝜕𝑡_𝑖+1𝜕𝑠_𝑖Δ_2𝑖−1−𝜕²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖Δ_2𝑖−1+𝜕²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖Δ_2𝑖−2. (C.5) For 𝑖=2 in (C.5), we obtainΔ₃+𝜕²𝑇𝑃𝜕𝑠₂𝜕𝑡₂Δ₂𝜕≤−²𝑇𝑃𝜕𝑡₂𝜕𝑠₁Δ₂+𝜕²𝑇𝑃𝜕𝑡₂𝜕𝑠₁Δ₁Δ<0,₄+𝜕²𝑇𝑃𝜕𝑡₃𝜕𝑠₂Δ₃𝜕≥−²𝑇𝑃𝜕𝑠₂𝜕𝑡₂Δ₃+𝜕²𝑇𝑃𝜕𝑠₂𝜕𝑡₂Δ₂>0. (C.6) Thus,Δ₃𝜕<−²𝑇𝑃𝜕𝑠₂𝜕𝑡₂Δ₂<0,Δ₄𝜕>−²𝑇𝑃𝜕𝑡₃𝜕𝑠₂Δ₃>0. (C.7) Proceeding inductively, we haveΔ_2𝑖−1+𝜕²𝑇𝑃𝜕𝑠_𝑖𝜕𝑡_𝑖Δ_2𝑖−2<0,Δ_2𝑖+𝜕²𝑇𝑃𝜕𝑡_𝑖+1𝜕𝑠_𝑖Δ_2𝑖−1>0. (C.8) Therefore, Δ_2𝑖−1<0 and Δ_2𝑖>0, for 𝑖=2,3,…. This completes the proof.

Acknowledgments

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.