Planning Horizon for Production Inventory Models with Production Rate Dependent on Demand and Inventory Level

Abstract

This paper discusses why the selection of a finite planning horizon is preferable to an infinite one for a replenishment policy of production inventory models. In a production inventory model, the production rate is dependent on both the demand rate and the inventory level. When there is an exponentially decreasing demand, the application of an infinite planning horizon model is not suitable. The emphasis of this paper is threefold. First, while pointing out questionable results from a previous study, we propose a corrected infinite planning horizon inventory model for the first replenishment cycle. Second, while investigating the optimal solution for the minimization problem, we found that the infinite planning horizon should not be applied when dealing with an exponentially decreasing demand. Third, we developed a new production inventory model under a finite planning horizon for practitioners. Numerical examples are provided to support our findings.

1. Introduction

Inventory models, in general, can be classified into two categories: infinite and finite planning horizon. For inventory models with the finite planning horizon, the goal is to minimize the total cost. On the other hand, without the present value, that is, not considering the time value of money, the total cost for the entire infinite planning horizon will go to infinity such that researchers are not able to compare the total cost for different inventory policies. The prevailing solution to this dilemma is to minimize the average cost of the first replenishment cycle because of a constant demand that implies an identical replenishment policy for the second replenishment cycle. As a result, the minimization of the average cost for the first replenishment cycle will lead to the optimal solution. The original paper of Wilson’s EOQ model [1] is an example of an infinite planning horizon problem.

It should be noted that practitioners in previous studies seemed to randomly decide whether to use an infinite planning horizon or a finite one. That is, they make their choice either by routine experience or by referencing other studies without explaining or considering the choice that fits the characteristics of date on hand. For examples, under the assumptions of time-vary demand, production, and deterioration rate, Goyal and Giri [2] developed two models by employing different modeling approaches over an infinite planning horizon. On the other hand, Goyal’s model [3] was considered as a finite planning horizon problem over time period [0,𝑇], where the replenishment cycle did not repeat itself in the same manner. It infers that each replenishment cycle within the planning horizon [0,𝑇] has different optimal solution such that the solution finding process requires the minimization of the total cost over the entire time period. For both studies, the reasoning behind the selection of either planning horizons was not explained. The purpose of this paper is to point out that in practice, some inventory models work sensibly over an infinite planning horizon. Managers under a highly competitive environment should be making correct and coherent decisions toward the development of inventory models that fit the pursuit of effective cost control.

Many papers have also discussed production inventory models under different conditions. By viewing the production rate as a variable, Bhunia and Maiti [4] developed two inventory systems. In the first system, the production rate was dependent on the inventory level, while the production rate was dependent upon the demand in the second. Su and Lin [5] combined the two models creating a model where production rate is dependent on both inventory level and demand. Moreover, Su and Lin [5] assumed that shortages were allowed with complete backlog and an exponentially decreasing demand.

We will show that finding the minimum value of the first replenishment cycle is not reasonable with an exponentially decreasing demand since the optimal solution for the production period will go to infinity, implying that the average cost is decreasing to zero. In response, we have developed a finite planning horizon production inventory model.

There are two primary reasons that justify assuming that the demand will decrease exponentially. First, the numerous innovations in the field of technology contribute to the expedited release of new merchandises, tremendously decreasing demand for the existing products in the market. Second, rapid changes in consumer preferences also greatly impact the sales of current merchandise. For instance, less than a year after a new camera cellular phone is introduced, an even newer generation will hit the market, with higher dpi than the previous generation. As a result, the demand for the old cell phone will plunge drastically.

Su and Lin [5] tried to extend the findings of Bhunia and Maiti [4], but their derivation for the differential equations with boundary conditions contained questionable results. Moreover, they could not analyze how many local minimum points exist. Up to now, there have been four published papers that have referred to Su and Lin [5] in their studies, Chu and Chung [6], Alfares et al. [7], Feng and Yamashiro [8], and Kang [9]. However, none of these papers have been made aware of the fundamental flaw in Su and Lin [5].

The derivation of Su and Lin [5] for the inventory level of the third phase contained questionable results such that their findings for relations among decision variables and their objective function also had questionable results. Moreover, we showed that their model is not suitable for infinite planning horizon, and then we studied the inventory model with finite planning horizon. There are two closely related papers, Yang et al. [10] and Lin et al. [11], that are considered for the finite planning horizon. There two models are developed for the EOQ with one decision variable to show that the optimal replenishment policy is independent of the demand type. However, there are four decision variables in our EPQ inventory model. We find two relations among these decision variables of the optimal solution for the infinite planning horizon, then two independent decision variables are left. For the finite planning horizon, we proved that there is only one decision variable left. These two papers have significant contribution for the theoretical development of EOQ inventory models, but their findings cannot be applied to our EPQ inventory model.

There are four phases for an EPQ inventory model, and then we proved that there is an upper bound for the elapse time for the first phase. It is an important finding when we applied a program to locate the optimal solution. For the infinite planning horizon, we showed that four decision variables are related, so only two independent decision variables are left, and we find the relations among decision variables that reduced the tedious computation for the minimum value. For the finite planning horizon, in each replenishment cycle, we proved that there is only one independent decision variable that achieves the efficiency for computation. Our first main contribution is providing an analytical approach to solve the optimal solution such that the result from computer programs is supported by the mathematical theorem. Our second main contribution is to reduce the number of independent decision variables to its minimum such that for obtaining the optimal solution, computer programs can be executed effectively.

2. Notation and Assumptions

To avoid confusion, we will use the same assumptions and notation as Su and Lin [5]:𝜃∶ deterioration rate,𝐼_𝑚: maximum inventory level,𝐼_𝑏: unfilled order backlog,𝐶: setup cost for each new cycle,𝐶_𝑑: the cost of a deteriorated item,𝐶_𝑖: inventory carrying cost per unit time,𝐶_𝑠: shortage cost per unit,𝐾: total average cost of the system.The assumptions below are used.(1) A single item is considered over (a) an infinite planning horizon for the first model and (b) a finite planning horizon of 𝑇 units of time for the second model which is subject to a constant deterioration rate.(2) Demand rate, 𝐷(𝑡), is known and decreases exponentially so that 𝐷(𝑡)=𝐴exp(−𝜆𝑡), where 𝐴 is the initial demand rate and 𝜆 is the decreasing rate of demand, 0≤𝜆≤1.(3)𝐼(𝑡) is the inventory level.(4) Production rate, 𝑃(𝑡), depends on both the demand and the inventory level with 𝑃(𝑡)=𝑎+𝑏𝐷(𝑡)−𝑐𝐼(𝑡), 𝑎>0,0≤𝑏<1, and 0≤𝑐<1.(5) Deterioration of the units is considered only after those units are received and put in inventory.(6) There is no replacement or repair of deteriorated items.(7) Shortages are allowed and fully backordered.(8) Two extra conditions, 𝜆>𝑐+𝜃 and 𝑎≥𝐴, are added (explained in Section 4).

Remark 1. Su and Lin [5] assumed that 𝑇 is a prescribed period of time and denoted by 𝑡₄=𝑇. Note that in the beginning, Su and Lin [5] tried to develop a production inventory model for a finite planning horizon, say [0,𝑇]. However, during their derivation, they considered the problem of minimizing the average cost for the first replenishment cycle in the infinite planning horizon. To clearly distinguish the difference between infinite and finite planning horizon, we will separate the problem into two cases.

Case (a). We minimize the average cost of the first replenishment cycle. It is a minimization problem with an infinite planning horizon.

Case (b). We minimize the total cost over a finite planning horizon of [0,𝑇].

3. A Review of Su and Lin [5]

In Su and Lin [5], the first replenishment cycle can be divided into four phases based on the time interval:(a) the first phase [0,𝑡₁]: the production dominates demand and deterioration, and the inventory level accumulates,(b) the second phase [𝑡₁,𝑡₂]: no production activity takes place. Demand and deterioration dominate, and so the inventory level gradually drops to zero at 𝑡₂,(c)the third phase [𝑡₂,𝑡₃]: no production and no deterioration take place. The shortage accumulates to 𝐼_𝑏 at 𝑡₃,(d)the fourth phase [𝑡₃,𝑡₄]: the production is resumed, shortages accumulated during the third phase are fully backordered, and the inventory level returns to zero at 𝑡₄.The differential equations developed by Su and Lin [5] for governing stock levels over the four different phases during the first replenishment cycle, [0,𝑡₄], can be expressed as follows: 𝑑𝐼(𝑡)(𝑡)(𝑡)(𝑡)()()−()𝐼(𝑡)𝑑𝑡=𝑃−𝐷−𝜃𝐼=𝑎+𝑏−1𝐴exp−𝜆𝑡𝑐+𝜃,0<𝑡<𝑡₁,𝑑𝐼(𝑡)(𝑡)(𝑡)()(𝑡)𝑑𝑡=−𝐷−𝜃𝐼=−𝐴exp−𝜆𝑡−𝜃𝐼,𝑡₁<𝑡<𝑡₂,𝑑𝐼(𝑡)(𝑡)()𝑑𝑡=−𝐷=−𝐴exp−𝜆𝑡,𝑡₂<𝑡<𝑡₃,𝑑𝐼(𝑡)(𝑡)(𝑡)()()(𝑡),𝑡𝑑𝑡=𝑃−𝐷=𝑎+𝑏−1𝐴exp−𝜆𝑡−𝑐𝐼₃<𝑡<𝑡₄. (1) Under the boundary conditions, 𝐼(0)=0,𝐼𝑡₁=𝐼_𝑚,𝐼𝑡₂𝐼𝑡=0,₃=−𝐼_𝑏,𝐼𝑡₄=0, (2) Su and Lin [5] found that 𝐼(𝑡)=𝑎(−()𝑡)+𝐴()𝑐+𝜃1−exp𝑐+𝜃1−𝑏()(−()𝑡,𝜆−𝑐−𝜃exp−𝜆𝑡−exp𝑐+𝜃)0≤𝑡≤𝑡₁,𝐼(𝑡)=𝐴exp(−𝜆𝑡)−()𝑡𝜆−𝜃1−exp𝜆−𝜃₂,𝑡−𝑡₁≤𝑡≤𝑡₂,𝐼(𝑡)=𝐴𝜆exp(−𝜆𝑡)−1,𝑡₂≤𝑡≤𝑡₃,𝐼(𝑡)𝑎=−𝑐exp𝑐𝑡₄−𝐴()−𝑡−11−𝑏()−()𝑡𝜆−𝑐exp−𝜆𝑡exp𝜆−𝑐₄,𝑡−𝑡−1₃≤𝑡≤𝑡₄. (3)(4)(5)(6) However, the result Su and Lin [5] derived in (5) is false. The expression should be revised as 𝐼(𝑡)=𝐴𝜆exp(−𝜆𝑡)−exp−𝜆𝑡₂,𝑡₂≤𝑡≤𝑡₃. (7) Owing to an error in (5) of their derivations for 𝐼_𝑚 and 𝐼_𝑏, the relation between 𝑡₁ and 𝑡₂, say 𝑡₂=𝑅(𝑡₁), and the relation between 𝑡₃ and 𝑡₄, say 𝑡₃=𝑅(𝑡₄), all contain questionable results. It implies that their objective function, 𝐾(𝑡₁,𝑡₂,𝑡₃,𝑡₄), is also false.

Su and Lin [5] derived the expression, 𝐾(𝑡₁,𝑡₂,𝑡₃,𝑡₄)=𝐾(𝑡₁,𝑡₂(𝑡₁),𝑡₃(𝑡₄),𝑡₄), so that the objective function has two independent variables, 𝑡₁ and 𝑡₄. They computed 𝜕𝐾/𝜕𝑡₁=0 and 𝜕𝐾/𝜕𝑡₄=0. However, they could not analyze whether a system that is comprised of 𝜕𝐾/𝜕𝑡₁=0 and 𝜕𝐾/𝜕𝑡₄=0 has solution.

4. Our Improvement for Infinite Planning Horizon Model

It should be pointed out that the result of (3) is based on the condition 𝜆≠𝑐+𝜃. On the other hand, if 𝜆=𝑐+𝜃, (3) should be revised as 𝐼(𝑡)=𝑎𝜆()()()1−exp−𝜆𝑡−𝐴𝑡1−𝑏exp−𝜆𝑡,0≤𝑡≤𝑡₁. (8) Hence, if we try to provide a complete study for the production inventory model of Su and Lin [5], then our model should be divided into seven cases: case (1): 𝑐+𝜃<𝜆, case (2): 𝑐+𝜃=𝜆, case (3): 𝑐<𝜆<𝑐+𝜃, case (4): 𝑐=𝜆, case (5): 𝜃<𝜆<𝑐, case (6): 𝜃=𝜆, and case (7): 𝜃>𝜆.

To focus on the investigation of a production inventory model where the production rate is dependent both on demand and inventory level, demand is exponentially decreasing, and shortages are fully backordered, we add two extra conditions: 𝜆>𝑐+𝜃 and 𝑎≥𝐴.

The reasoning behind the addition of an extra condition, 𝑎≥𝐴, is as follows: when 𝑡=0, the demand rate 𝐷(0)=𝐴, the inventory level 𝐼(0)=0, and the production rate 𝑃(0)=𝑎+𝑏𝐴. For the accumulation of inventory during the first phase [0,𝑡₁], it implies that 𝑎+𝑏𝐴≥𝐴 for 0≤𝑏<1. For the special case of 𝑏=0, we know that 𝑎≥𝐴 is valid. Therefore, we derive that 𝑎≥(1−𝑏)𝐴 when 0<𝑏<1.

If 𝑎<𝐴, then the domain of 𝑏 has a lower bound satisfying the expression, 𝑏≥1−(𝑎/𝐴), such that the domain of 𝑏 should be changed from [0,1) to [1−(𝑎/𝐴),1).

Moreover, Su and Lin [5] assumed in their numerical example that 𝐴=200 and 𝑎=200. Their assumption provides support for our extra condition of 𝑎≥𝐴. They also assumed that 𝜆=0.3, 𝜃=0.05, and 𝑐=0.2 which provides evidence that our condition, 𝜆>𝑐+𝜃, is reasonable. Moreover, the condition of 𝜆>𝑐+𝜃 will focus on the development of a production inventory that is compatible with the numerical examples in Su and Lin [5] and to avoid tedious discussion for different inventory models with different relations among 𝜆,𝑐, and 𝜃.

Based on (3), (4), (6), and (7) and the boundary conditions of (2), we derive that 𝐼_𝑚=𝑎−()𝑡𝑐+𝜃1−exp𝑐+𝜃₁+𝐴(1−𝑏)𝜆−𝑐−𝜃exp−𝜆𝑡₁−exp−(𝑐+𝜃)𝑡₁=𝐴exp−𝜆𝑡₁−()𝑡𝜆−𝜃1−exp𝜆−𝜃₂−𝑡₁,𝐼_𝑏=𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃=𝑎𝑐exp𝑐𝑡₄−𝑡₃+𝐴()−11−𝑏()𝑡𝜆−𝑐exp𝑐−𝜆₄−𝑡₃−1exp−𝜆𝑡₃. (9) From (9), we find the relation between 𝑡₁ and 𝑡₂ and then the relation among 𝑡₂, 𝑡₃, and 𝑡₄: 𝑡₂=1()𝑡𝜃−𝜆lnexp𝜃−𝜆₁−𝑎(𝜆−𝜃)𝐴(𝑐+𝜃)exp𝜃𝑡₁−exp−𝑐𝑡₁−(𝜆−𝜃)(1−𝑏)×()𝑡𝜆−𝑐−𝜃exp𝜃−𝜆₁−exp−𝑐𝑡₁,𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃+𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₃=𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₄exp𝑐𝑡₄−𝑡₃. (10)(11) We will simplify a four-variable problem, 𝑡₁, 𝑡₂, 𝑡₃, and 𝑡₄, to a two-variable problem of 𝑡₁ and 𝑡₃. During [0,𝑡₁], production, demand, and deterioration interact with each other to accumulate items that will be consumed and thus deteriorate during [𝑡₁,𝑡₂] such that, trivially, 𝑡₂−𝑡₁ is dependent on 𝑡₁. We will derive the detailed relation between 𝑡₂−𝑡₁ and 𝑡₁, that is, 𝑡₂ and 𝑡₁. During [𝑡₂,𝑡₃], the shortages will accumulate to be backlogged during [𝑡₃,𝑡₄] so that naturally 𝑡₄−𝑡₃ is dependent on 𝑡₃−𝑡₂. We will derive the detailed relation between 𝑡₄−𝑡₃ and 𝑡₃−𝑡₂, which is the relation of (i) 𝑡₄ and (ii) 𝑡₃ with 𝑡₂. Due to the fact that demand is varied, 𝑡₂ will influence the shortage during [𝑡₂,𝑡₃].

In the following, we will prove that 𝑡₂ can be uniquely decided if 𝑡₁ is given. When𝑡₁ and 𝑡₂ are given, by using the relation in (11), the unique value of 𝑡₄ can be derived if 𝑡₃ is also given. Hence, we will simplify a four-variable problem to a two-variable problem. Let us rewrite (10) as 1−exp−(𝜆−𝜃)𝑡₂−𝑡₁=𝑎(𝜆−𝜃)𝐴(𝑐+𝜃)exp𝜆𝑡₁−exp(𝜆−𝑐−𝜃)𝑡₁+(𝜆−𝜃)(1−𝑏)()𝑡𝜆−𝑐−𝜃1−exp𝜆−𝑐−𝜃₁. (12) We tried to find the condition of 𝑡₁ under which there is a solution to 𝑡₂ with 𝑡₂≥𝑡₁, satisfying (12). For the later discussion, given that 𝑡₁, we denote the unique solution of 𝑡₂ that satisfies (12) as 𝑡₂(𝑡₁). We will prove that the feasible domain of 𝑡₁ is bounded, guaranteeing the existence of 𝑡₂.

Motivated by (12), we assume the following auxiliary function, 𝑔(𝑡₁), to be 𝑔𝑡₁=𝑎(𝜆−𝜃)𝐴(𝑐+𝜃)exp𝜆𝑡₁−exp(𝜆−𝑐−𝜃)𝑡₁+(𝜆−𝜃)(1−𝑏)()𝑡𝜆−𝑐−𝜃1−exp𝜆−𝑐−𝜃₁. (13) Taking the derivative of 𝑔(𝑡₁) with respect to 𝑡₁ yields 𝑑𝑑𝑡₁𝑔𝑡₁=𝑎(𝜆−𝜃)𝐴(𝑐+𝜃)𝜆exp𝜆𝑡₁−(𝜆−𝑐−𝜃)exp(𝜆−𝑐−𝜃)𝑡₁−(𝜆−𝜃)(1−𝑏)exp(𝜆−𝑐−𝜃)𝑡₁. (14) Under the conditions 𝑎≥𝐴 and 𝜆>𝑐+𝜃, it follows that 𝑑𝑑𝑡₁𝑔𝑡₁≥(𝜆−𝜃)(𝑐+𝜃)𝜆exp𝜆𝑡₁−(𝜆−𝑐−𝜃)exp(𝜆−𝑐−𝜃)𝑡₁−(𝜆−𝜃)exp(𝜆−𝑐−𝜃)𝑡₁=𝜆(𝜆−𝜃)𝑐+𝜃exp𝜆𝑡₁−exp(𝜆−𝑐−𝜃)𝑡₁>0, (15) showing that 𝑔(𝑡₁) is an increasing function from 𝑔(0)=0 to lim_𝑡1→∞𝑔(𝑡₁)=∞, since lim_𝑡1→∞𝑔𝑡₁exp(𝜆−𝑐−𝜃)𝑡₁=lim_𝑡1→∞𝑎(𝜆−𝜃)𝐴()()𝑡𝑐+𝜃exp𝑐+𝜃₁+()−1𝜆−𝜃)(1−𝑏−()𝑡𝜆−𝑐−𝜃exp𝜆−𝑐−𝜃₁−1=∞. (16) There is a unique point, say 𝑡^#₁, that satisfies 𝑔(𝑡^#₁)=1.

From (3) and (13), we have 𝜆−𝜃𝐴exp𝜆𝑡^#₁𝐼𝑡^#₁=𝑔𝑡^#₁=1. (17) From 𝜆>𝑐+𝜃, the inequality, 𝜆>𝜃, is held. It follows that 𝐼𝑡^#₁=𝐴𝜆−𝜃exp−𝜆𝑡^#₁. (18) By referring to (12), we obtain 𝑔𝑡^#₁=1=1−exp−(𝜆−𝜃)𝑡₂𝑡^#₁−𝑡^#₁. (19) According to (19), we showed that 𝑡₂(𝑡^#₁)−𝑡^#₁ must go to ∞ so that 𝑡₂(𝑡^#₁) will go to ∞ as well. We will express the result as lim_𝑡1→𝑡^#₁𝑡₂(𝑡₁)=∞ and summarize our findings in the next lemma.

Lemma 2. 𝐼(𝑡^#₁)=(𝐴/(𝜆−𝜃))𝑒^{−𝜆𝑡#1} and  lim_𝑡1→𝑡^#₁𝑡₂(𝑡₁)=∞.

From Lemma 2, we know that the feasible domain of 𝑡₁ should be set as 0≤𝑡₁<𝑡^#₁. (20) Given 𝑡₁, with 𝑡₁<𝑡^#₁, then 𝑔(𝑡₁)<1 so that there is a unique 𝑡₂, say 𝑡₂(𝑡₁), that satisfies 1−exp−(𝜆−𝜃)𝑡₂𝑡₁−𝑡₁=𝑔𝑡₁. (21) We may explicitly express 𝑡₂(𝑡₁) as 𝑡₂𝑡₁=𝑡₁−1𝑎()𝜆−𝜃ln1−𝜆−𝜃𝐴()×𝑐+𝜃exp𝜆𝑡₁−exp(𝜆−𝑐−𝜃)𝑡₁−(𝜆−𝜃)(1−𝑏)()𝑡𝜆−𝑐−𝜃1−exp𝜆−𝑐−𝜃₁. (22) We will summarize our findings in the following lemma.

Lemma 3. If 𝑡₁<𝑡^#₁, then there is a unique 𝑡₂, say 𝑡₂(𝑡₁), as in (22) so that (10) is satisfied.

Next, we consider the relation among 𝑡₂, 𝑡₃, and 𝑡₄ by rewriting (11) as 𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃+𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₃×exp𝑐𝑡₃=𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₄exp𝑐𝑡₄. (23) Motivated by (23), we assume the following auxiliary function: 𝑓𝑡₄=𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₄exp𝑐𝑡₄. (24) We find that 𝑓^′𝑡₄=exp𝑐𝑡₄𝑎−𝐴(1−𝑏)exp−𝜆𝑡₄>𝑎−𝐴(1−𝑏)≥0. (25) Under our assumptions of 𝑎≥𝐴 and 0≤𝑏<1, it can be inferred that 𝑓(𝑡₄) increases from 𝑓(𝑡₃)=(𝑎/𝑐)exp(𝑐𝑡₃)+(𝐴(1−𝑏)/(𝜆−𝑐))exp((𝑐−𝜆)𝑡₃) to lim_𝑡4→∞𝑓(𝑡₄)=∞.

The relation below, 𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃+𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₃×exp𝑐𝑡₃≥𝑎𝑐exp𝑐𝑡₃+𝐴(1−𝑏)()𝑡𝜆−𝑐exp𝑐−𝜆₃=𝑓𝑡₃, (26) holds since 𝑡₂≤𝑡₃. Therefore, if 𝑡₁ and 𝑡₃ are decided, with the restriction 𝑡₁<𝑡^#₁, then there is a unique 𝑡₂(𝑡₁) that satisfies (10). Also, from (26) and the increasing function 𝑓(𝑡₄), we know that for a given 𝑡₃ under the condition, 𝑡₂(𝑡₁)≤𝑡₃, there is a unique point explicitly denoted as 𝑡₄(𝑡₁,𝑡₂(𝑡₁),𝑡₃), simply say 𝑡₄, that satisfies the condition, 𝑡₄≥𝑡₃, such that the following expression, 𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃+𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₃×exp𝑐𝑡₃=𝑓𝑡₄, (27) satisfies (23). We will summarize our results in the next lemma.

Lemma 4. If 𝑡₁ and 𝑡₃ are given with 𝑡₁<𝑡^#₁ and 𝑡₂(𝑡₁)≤𝑡₃, then there is a unique 𝑡₄, denoted as 𝑡₄(𝑡₁,𝑡₂(𝑡₁),𝑡₃) that satisfies (11).

Up to this point, the corrected objective function below can be provided as𝐾𝑡₁,𝑡₂,𝑡₃,𝑡₄=𝑎𝑡𝑐+𝜃¹+exp−(𝑐+𝜃)𝑡₁−1+𝐴()𝑐+𝜃1−𝑏×−()𝑡𝜆−𝑐−𝜃exp𝑐+𝜃₁−1−𝑐+𝜃exp−𝜆𝑡₁−1𝜆+𝐴𝜆−𝜃exp−𝜆𝑡₁−exp−𝜆𝑡₂𝜆+exp−(𝜆−𝜃)𝑡₂×exp−𝜃𝑡₂−exp−𝜃𝑡₁𝜃𝜃𝐶_𝑑+𝐶_𝑖𝑡₄+𝐶_𝑠𝑡₄𝐴𝜆²exp−𝜆𝑡₃−exp−𝜆𝑡₂+𝐴𝜆exp−𝜆𝑡₂𝑡₃−𝑡₂−𝑎𝑐²1−exp𝑐𝑡₄−𝑡₃−𝑎𝑐𝑡₄−𝑡₃+𝐴(1−𝑏)×−()𝑡𝜆−𝑐exp𝜆−𝑐₄exp−𝑐𝑡₃−exp−𝑐𝑡₄𝑐+exp−𝜆𝑡₄−exp−𝜆𝑡₃𝜆+𝐶𝑡₄, (28) with the conditions 0<𝑡₁≤𝑡₂≤𝑡₃≤𝑡₄.

Given 𝑡₁, with 𝑡₁<𝑡^#₁ and (22), 𝑡₂(𝑡₁) can be derived. Given a 𝑡₃ that satisfies 𝑡₂(𝑡₁)≤𝑡₃, then, by (23), 𝑡₄ can also be obtained. We have learned from above discussion that only 𝑡₁ and 𝑡₃ are independent variables.

Hence, the problem becomes the minimization of 𝐾(𝑡₁,𝑡₂(𝑡₁),𝑡₃,𝑡₄(𝑡₁,𝑡₂(𝑡₁),𝑡₃)) under two restrictions: 𝑡₁<𝑡^#₁,𝑡₂𝑡₁≤𝑡₃. (29) We have derived a two-variable minimum problem of 𝑡₁ and 𝑡₃ under the conditions of (29), for the infinite horizon minimum cost inventory model. The findings are concluded in the next theorem.

Theorem 5. For the production inventory model with infinite planning horizon, if one minimizes the average cost for the first replenishment cycle, then there are two necessary conditions, 𝑡₁<𝑡^#₁ and 𝑡₂(𝑡₁)≤𝑡₃, for the production inventory model of Su and Lin [5].

5. Numerical Examples for Infinite Planning Horizon Inventory Model

We will employ the same numerical examples as Su and Lin [5] for comparison purposes where 𝐴=200, 𝜆=0.3, 𝜃=0.05, 𝐶=100, 𝐶_𝑑=3, 𝐶_𝑠=10, 𝐶_𝑖=1, 𝑎=200, 𝑏=0.2, and 𝑐=0.2. Some computation results are showed in Table 1 with 𝑡^#₁=3.6699 arranged according to a sequence of different values of 𝑡₃.

Table 1 

The results for average cost of the first cycle for infinite planning horizon.

From the numerical examples in Table 1, it reveals that if we prolong the replenishment cycle, then the average cost will eventually decrease. The rationale is that with a negatively exponential demand function, the market demand will dramatically decrease, especially in a longer inventory horizon, which will in turn significantly bring down the corresponding average holding and shortage costs. On the other hand, when we prolong the shortage phase with 𝑡₃=19998.2832 (the personal computer’s computational limit) in order to reduce the average cost, the ordinary customers may lose patience when waiting for the backorder. Hence, a full backorder cannot be performed. In other words, it is impossible to simultaneously achieve full backorder and minimize average inventory cost. We may conclude that for the negatively exponential demand, 𝐷(𝑡)=𝐴exp(−𝜆𝑡), the infinite planning horizon production inventory model is not adequate. Therefore, we stop the discussion of case (a) in the infinite planning horizon.

6. Our Proposed Production Inventory Model with Finite Planning Horizon

Next, we consider case (b) with a finite planning horizon, denoted by [0,𝑇]. To simplify the discussion, we assume that there is one replenishment cycle during the finite planning horizon. Our results can be easily extended to several replenishment cycles. In this setting, (28) should be revised as follows: 𝐾𝑡₁,𝑡₂,𝑡₃,𝑡₄=𝑎=𝑇𝑡𝑐+𝜃¹+exp−(𝑐+𝜃)𝑡₁−1+𝐴()𝑐+𝜃1−𝑏×−()𝑡𝜆−𝑐−𝜃exp𝑐+𝜃₁−1−𝑐+𝜃exp−𝜆𝑡₁−1𝜆+𝐴𝜆−𝜃exp−𝜆𝑡₁−exp−𝜆𝑡₂𝜆+exp−(𝜆−𝜃)𝑡₂×exp−𝜃𝑡₂−exp−𝜃𝑡₁𝜃𝜃𝐶_𝑑+𝐶_𝑖𝑇+𝐶_𝑠𝑇𝐴𝜆²exp−𝜆𝑡₃−exp−𝜆𝑡₂+𝐴𝜆exp−𝜆𝑡₂𝑡₃−𝑡₂−𝑎𝑐²1−exp𝑐𝑇−𝑡₃−𝑎𝑐𝑇−𝑡₃+𝐴(1−𝑏)×(−()𝑇𝜆−𝑐exp𝜆−𝑐)exp−𝑐𝑡₃−exp(−𝑐𝑇)𝑐+exp(−𝜆𝑇)−exp−𝜆𝑡₃𝜆+𝐶𝑇. (30) Here, we will derive a stronger condition than 𝑡₁<𝑡^#₁ for the feasible domain of 𝑡₁. For a given 𝑡₄, from (10), since 𝑡₂(𝑡₁) is an increasing function of 𝑡₁, there is a unique point, say 𝑡^∧₁(𝑡₄) with 𝑡₂(𝑡^∧₁(𝑡₄))=𝑡₄. Under the condition 𝑡₁<𝑡^∧₁𝑡₄, (31) the desired result 𝑡₂(𝑡₁)<𝑡₄ is achieved, since 𝑡₂(𝑡₁) is an increasing function of 𝑡₁.

Lemma 6. For a given 𝑡₄, the feasible domain of 𝑡₁ is [0,𝑡^∧₁(𝑡₄)), implying that 𝑡₂(𝑡₁)<𝑡₄ with 𝑡₂(𝑡^∧₁(𝑡₄))=𝑡₄.

In the following, when 𝑡₄ is given, if we take a 𝑡₁ that satisfies (31), then we will prove that there is a unique 𝑡₃ that satisfies (23).

Based on (23), let us assume other two auxiliary functions, 𝑘(𝑡) and ℎ(𝑡₃), where 𝑘(𝑡)=𝑎𝑐()+𝐴()exp𝑐𝑡1−𝑏(−()𝑡)𝜆−𝑐exp𝜆−𝑐,for𝑡₂≤𝑡≤𝑇. (32) With a restricted domain, 𝑘(𝑡) is related to our previous auxiliary function 𝑓(𝑡₄) of (24) and ℎ𝑡₃=𝐴𝜆exp−𝜆𝑡₂−exp−𝜆𝑡₃+𝑎𝑐+𝐴(1−𝑏)𝜆−𝑐exp−𝜆𝑡₃exp𝑐𝑡₃,for𝑡₂≤𝑡₃≤𝑇. (33) From (𝑑/𝑑𝑡)𝑘(𝑡)=𝑎exp(𝑐𝑡)−𝐴(1−𝑏)exp(−(𝜆−𝑐)𝑡)>0, under the conditions 𝑎≥𝐴 and 0≤𝑏<1, 𝑘(𝑡) is an increasing function which implies that 𝑘𝑡₂=𝑎𝑐exp𝑐𝑡₂+𝐴(1−𝑏)−()𝑡𝜆−𝑐exp𝜆−𝑐₂(𝑇)=𝑎<𝑘𝑐()+𝐴()exp𝑐𝑇1−𝑏(−()𝑇).𝜆−𝑐exp𝜆−𝑐 (34) On the other hand, the expression 𝑑𝑑𝑡₃ℎ𝑡₃=𝐴𝜆𝜆exp−(𝜆−𝑐)𝑡₃+𝑐exp−𝑐𝑡₂−exp−𝜆𝑡₃exp𝑐𝑡₃+𝑎exp𝑐𝑡₃−𝐴(1−𝑏)exp(𝑐−𝜆)𝑡₃>0 (35) shows that ℎ(𝑡₃) is an increasing function. If we apply (33) and (34), then ℎ(𝑡₃) increases from ℎ𝑡₂=𝑘𝑡₂<𝑘(𝑇), (36) to ℎ(𝑇)=𝐴𝜆exp−𝜆𝑡₂−exp(−𝜆𝑇)+𝑘(𝑇)>𝑘(𝑇). (37) Therefore, there is a unique point, say 𝑡₃(𝑡₂), that satisfies ℎ𝑡₃𝑡₂=𝑘(𝑇). (38) When 𝑡₄=𝑇 is given, based on the previous discussion, if 𝑡₁ is given with 𝑡₁<𝑡^∧₁(𝑡₄), then we have 𝑡₂(𝑡₁)<𝑡₄. From (38), there exists a unique point, 𝑡₃(𝑡₂), with ℎ(𝑡₃(𝑡₂(𝑡₁)))=𝐾(𝑇) such that 𝑡₂(𝑡₁), 𝑡₃(𝑡₂(𝑡₁)), and 𝑡₄ satisfy (23). Hence, for a finite-horizon minimum cost inventory model, we have simplified a four-variable problem to a one-variable problem.

Hence, in the following, if we only consider those 𝑡₁s that satisfy the condition of (31), then 𝑡₂𝑡₁<𝑡₄. (39) By (23) and (38), the relation, 𝑡₃=𝑡₃(𝑡₂), implies that 𝑡₄𝑡₁,𝑡₂𝑡₁,𝑡₃𝑡₂𝑡₁=𝑡₄, (40) where 𝑡₄(𝑡₁,𝑡₂(𝑡₁),𝑡₃), defined in Lemma 4, satisfies (11).

The objective function becomes a one-variable problem 𝐾𝑡₁=𝐾𝑡₁,𝑡₂𝑡₁,𝑡₃𝑡₂𝑡₁,𝑡₄=𝐶=𝑇𝑇+𝜃𝐶_𝑑+𝐶_𝑖𝑇×𝑎𝑡𝑐+𝜃¹+exp−(𝑐+𝜃)𝑡₁−1+𝐴()𝑐+𝜃1−𝑏−()𝑡𝜆−𝑐−𝜃exp𝑐+𝜃₁−1−𝑐+𝜃exp−𝜆𝑡₁−1𝜆+𝐴𝜆−𝜃exp−𝜆𝑡₁−exp−𝜆𝑡₂𝜆+exp−(𝜆−𝜃)𝑡₂×exp−𝜃𝑡₂−exp−𝜃𝑡₁𝜃+𝐶₃𝑇𝐴𝜆²exp−𝜆𝑡₃−exp−𝜆𝑡₂+𝐴𝜆exp−𝜆𝑡₂𝑡₃−𝑡₂−𝑎𝑐²1−exp𝑐𝑇−𝑡₃−𝑎𝑐𝑇−𝑡₃+𝐴(1−𝑏)(−()𝑇)×𝜆−𝑐exp𝜆−𝑐exp−𝑐𝑡₃−exp(−𝑐𝑇)𝑐+exp(−𝜆𝑇)−exp−𝜆𝑡₃𝜆. (41) We will summarize our findings in the next theorem.

Theorem 7. For the production inventory model with the finite planning horizon, [0,𝑡₄], if one only considers one replenishment cycle, there is a natural restriction 𝑡₁<𝑡^∧₁(𝑡₄), creating a one-variable minimum problem.

From Theorem 7, computer program as MathCAD can be adopted to locate the optimal solution. We may point out that the benefits to the simplified production inventory model that we have proposed include (a) easy to use for decision makers, (b) reduction of the solution space (computation time) in determining the parameter setting, and (c) reduction of the model complexity.

7. Numerical Example for the Finite Planning Horizon

For the finite planning horizon production inventory model with the same data, 𝐴=200, 𝜆=0.3, 𝜃=0.05, 𝐶=100, 𝐶_𝑑=3, 𝐶_𝑠=10, 𝐶_𝑖=1, 𝑎=200, 𝑏=0.2, 𝑐=0.2, and 𝑇=2, we find the optimal solution, 𝑡^∗₁=1.2742. With (12), it shows that 𝑡^∗₂=1.8620. With (38), it shows that 𝑡^∗₃=1.9306. Finally, with (41), we find that the minimum cost is 𝐾(𝑡^∗₁,𝑡^∗₂,𝑡^∗₃,𝑡^∗₄)=89.7151. The above discussion is based on the preset condition that there is one replenishment cycle. However, under the multiple replenishment cycles, the total setup cost will be at least 200, and then the average cost during [0,2] is more than one hundred that is larger than the result of one replenishment cycle. It specifies that the average cost for multiple replenishment cycles is much larger than that of one replenishment cycle. Hence, for this numerical example, we only consider one replenishment cycle.

Particle swarm optimization is applied to check our findings. Both approaches have the same optimal solution.

8. Conclusion and Further Direction

We have shown that with an exponentially decreasing demand, the goals of simultaneously minimizing the average cost for the first replenishment cycle and fully backordering the shortage items cannot be applied for an infinite-horizon minimum cost inventory model. The result of our investigation explicitly reveals that using the infinite planning horizon model is inappropriate in practice. For the finite planning horizon, we have shown that the four-phase production inventory model can be converted to a single variable problem in order to find the minimum solution. Our study not only provides a sound operational formulation but also offers a practical and efficient approach in the location of the optimal solution.

The study we have carried out can probably be viewed as the first attempt to solve a finite planning horizon production inventory model. In the future, it would be interesting to show that our objective function is convex to ensure the existence of a local minimum. Moreover, the issues of how to decide the optimal solution under several replenishment cycles and how to verify the convexity of the minimum value under multiple replenishment cycles deserve further study.

Acknowledgments

This paper is partially supported by the National Science Council of Taiwan, Taiwan, with Grant NSC 101-2410-H-015-02. The authors want to express their gratitude to Sophia Liu ([email protected]) for her English revisions of their paper and Professor Shih-Wei Lin for his help with particle swarm optimization.