Order Level Inventory Models for Deteriorating Seasonable/Fashionable Products with Time Dependent Demand and Shortages

Abstract

An order level inventory model for seasonable/fashionable products subject to a period of increasing demand followed by a period of level demand and then by a period of decreasing demand rate (three branches ramp type demand rate) is considered. The unsatisfied demand is partially backlogged with a time dependent backlogging rate. In addition, the product deteriorates with a time dependent, namely, Weibull, deterioration rate. The model is studied under the following different replenishment policies: (a) starting with no shortages and (b) starting with shortages. The optimal replenishment policy for the model is derived for both the above mentioned policies.

1. Introduction

It is observed that the life cycle of many seasonal products, over the entire time horizon, can be portrayed as a period of growth, followed by a period of relatively level demand and finishing with a period of decline. So researchers commonly use a time-varying demand pattern to reflect sales in different phases of product life cycle. Resh et al. [1] and Donaldson [2] are the first researchers who considered an inventory model with a linear trend in demand. Thereafter, numerous research works have been carried out incorporating time-varying demand patterns into inventory models. The time dependent demand patterns, mainly, used in literature are, (i) linearly time dependent and, (ii) exponentially time dependent (Dave and Patel [3], Goyal [4], Hariga [5], Hariga and Benkherouf [6], Yang et al. [7]). The time dependent demand patterns reported above are unidirectional, that is, increase continuously or decrease continuously. Hill [8] proposed a time dependent demand pattern by considering it as the combination of two different types of demand in two successive time periods over the entire time horizon and termed it as “ramp-type” time dependent demand pattern. Then, inventory models with ramp type demand rate also are studied by Mandal and Pal [9], Wu et al. [10] and Wu and Ouyang [11], Wu [12], Giri et al. [13], and Manna and Chaudhuri [14]. In these papers, the determination of the optimal replenishment policy requires the determination of the time point, when the inventory level falls to zero. So the following two cases should be examined: (1) this time point occurs before the point, where the demand is stabilized, and (2) this time point occurs after the point, where the demand is stabilized. Almost all of the researchers examine only the first case. Deng et al. [15] reconsidered the inventory model of Mandal and Pal [9] and Wu and Ouyang [11] and studied it exploring these two cases. Skouri et al. [16] extend the work of Deng et al. [15] by introducing a general ramp type demand rate and considering Weibull distributed deterioration rate.

The assumption that the goods in inventory always preserve their physicalcharacteristics is not true in general. There are some items, which are subject to risks of breakage, deterioration, evaporation, obsolescence, and so forth. Food items, pharmaceuticals, photographic film, chemicals, and radioactivesubstances arefew items in which appreciable deterioration can take place during the normal storage of the units. A model with exponentially decaying inventory was initially proposed by Ghare and Schrader [17]. Covert and Phillip [18] and Tadikamalla [19] developed an economic order quantity model with Weibull and Gamma distributed deterioration rates, respectively. Thereafter, a great deal of research efforts have been devoted to inventory models of deteriorating items, the details can be found in the review articles by Raafat [20], and Goyal and Giri [21].

In most of the above-mentioned papers, the demand during stockout period is totally backlogged. In practice, there are customers who are willing to wait and receive their orders at the end of stockout period, while others are not. In the last few years, considerable attention has been paid to inventory models with partial backlogging. The backlogging rate can be modelled taking into account the customers’ behavior. The first paper in which customers’ impatience functions are proposed seems to be that by Abad [22]. Chang and Dye [23] developed a finite horizon inventory model using Abad’s reciprocal backlogging rate. Skouri and Papachristos [24] studied a multiperiod inventory model using the negative exponential backlogging rate proposed by Abad [22]. Teng et al. [25] extended Chang and Dye ’s [23] and Skouri and Papachristos’ [24] models, assuming as backlogging rate any decreasing function of the waiting time up to the next replenishment. Research on models with partial backlogging continues with Wang [26] and San Jose et al. [27] and [28].

Manna and Chaudhuri [14] noted that ramp type demand pattern is generally followed by new brand of consumer goods coming to the market. But for fashionable products as well as for seasonal products, the steady demand will never be continued indefinitely. Rather it would be followed by decrement with respect to time after a period of time and becomes asymptotic in nature. Thus the demand may be illustrated by three successive time periods that classified time dependent ramp-type function, in which in the first phase the demand increases with time and after that it becomes steady, and towards the end in the final phase it decreases and becomes asymptotic. Chen et al. [29] proposed a search procedure based on Nelder-Mead algorithm to find a solution for the case of inventory systems with shortage allowance and nonlinear demand pattern. Also, Chen et al. [30] proposed a net present value approach for the previous inventory system without shortages. For both models, the demand rate is a revised version of the Beta distribution function and so is a differentiable with respect to time.

The purpose of the present paper is to study an order level inventory model when the demand is described by a three successive time periods that classified time dependent ramp-type function. Any such function has points, at least one, where differentiation is not possible, and this introduces extra complexity in the analysis of the relevant models. The unsatisfied demand is partially backlogged with time dependent backlogging rate, and units in inventory are subject to deterioration with Weibull-distributed deterioration rate.

The rest of the paper is organized as follows. In the next section the assumptions and notations for the development of the model are provided. The model starting with no shortages is studied in Section 3, and the corresponding one starting with shortages is studied in Section 4. For each model the optimal policy is obtained. Numerical examples highlighting the results obtained are given in Section 5. The paper closes with concluding remarks in Section 6.

2. Notation and Assumptions

The following notations and assumptions are used in developing the model:

Notations
𝑇The constant scheduling period (cycle)𝑡₁ The time when the inventory level reaches zero𝑆The maximum inventory level at each scheduling period (cycle)𝑐₁The inventory holding cost per unit per unit time𝑐₂The shortage cost per unit per unit time𝑐₃The cost incurred from the deterioration of one unit𝑐₄The per unit opportunity cost due to the lost sales𝜇The time point that increasing demand becomes steady𝛾The time point, after 𝜇, until the demand is steady and then decreases𝐼(𝑡)The inventory level at time 𝑡∈[0,𝑇].

Assumptions
(1)The ordering quantity brings the inventory level up to the order level 𝑆. Replenishment rate is infinite.(2)Shortages are backlogged at a rate 𝛽(𝑥), which is a nonincreasing function of 𝑥(𝛽^′(𝑥)≤0) with 0≤𝛽(𝑥)≤1, 𝛽(0)=1, and 𝑥 is the waiting time up to the next replenishment. Moreover it is assumed that 𝛽(𝑥) satisfies the relation 𝛽(𝑥)+𝑇𝛽^′(𝑥)≥0, where 𝛽^′(𝑥) is the derivate of 𝛽(𝑥). The cases with 𝛽(𝑥)=1 (or 0) correspond to complete backlogging (or complete lost sales) models.(3)The time to deterioration of the item is distributed as Weibull (𝑎,𝑏); that is, the deterioration rate is 𝜃(𝑡)=𝑎𝑏𝑡^𝑏−1 (𝑎>0,𝑏>0,𝑡>0). There is no replacement or repair of deteriorated units during the period 𝑇. For 𝑏=1, 𝜃(𝑡) becomes constant, which corresponds to exponentially decaying case.(4)The demand rate 𝐷(𝑡) is a time dependent ramp-type function and is of the following form:𝐷(𝑡)=𝑓(𝑡),0<𝑡<𝜇,𝑓(𝜇)=𝑔(𝛾),𝜇≤𝑡≤𝛾,𝑔(𝑡),𝛾<𝑡, (2.1) where 𝑓(𝑡) is a positive, continuous, and increasing function of 𝑡, and 𝑔(𝑡) is a positive, continuous and decreasing function of 𝑡.

3. The Mathematical Formulation of the Model Starting with No Shortages

The replenishment at the beginning of the cycle brings the inventory level up to 𝑆. Due to demand and deterioration, the inventory level gradually depletes during the period (0,𝑡₁) and falls to zero at 𝑡=𝑡₁. Thereafter shortages occur during the period (𝑡₁,𝑇), which are partially backlogged. Consequently, the inventory level, 𝐼(𝑡), during the time interval 0≤𝑡≤𝑇, satisfies the following differential equations:

𝑑𝐼(𝑡)𝑑𝑡+𝜃(𝑡)𝐼(𝑡)=−𝐷(𝑡),0≤𝑡≤𝑡₁,𝐼(𝑡₁)=0,𝑑𝐼(𝑡)𝑑𝑡=−𝐷(𝑡)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝑇,𝐼(𝑡₁)=0. (3.1)(3.2)

The solutions of these differential equations are affected from the relation between 𝑡₁, 𝜇, and 𝛾 through the demand rate function. Since the demand has three components in three successive time periods, the following cases: (i)𝑡₁<𝜇<𝛾<𝑇, (ii) 𝜇<𝑡₁<𝛾<𝑇, and (iii) 𝜇<𝛾<𝑡₁<𝑇 must be considered to determine the total cost and then the optimal replenishment policy.

Case (𝑡₁<𝜇<𝛾<𝑇). In this case, (3.1) becomes 𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝑡),0≤𝑡≤𝑡₁,𝐼(𝑡₁)=0. (3.3) Equation (3.2) leads to the following three: 𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝑡)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝜇,𝐼(𝑡₁)=0,𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝜇)𝛽(𝑇−𝑡),𝜇≤𝑡≤𝛾,𝐼(𝜇₋)=𝐼(𝜇₊),𝑑𝐼(𝑡)𝑑𝑡=−𝑔(𝑡)𝛽(𝑇−𝑡),𝛾≤𝑡≤𝑇. (3.4)(3.5)(3.6) The solutions of (3.3), (3.4), (3.5), and (3.6), are, respectively, 𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏}∫^𝑡1_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥,0≤𝑡≤𝑡₁,𝐼(𝑡)=−^𝑡_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥,𝑡₁≤𝑡≤𝜇,𝐼(𝑡)=−𝑓(𝜇)^𝑡_𝜇𝛽(𝑇−𝑥)𝑑𝑥−^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥,𝜇≤𝑡≤𝛾,𝐼(𝑡)=−^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥−^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥−𝑓(𝜇)^𝛾_𝜇𝛽(𝑇−𝑥)𝑑𝑥,𝛾≤𝑡≤𝑇. (3.7)(3.8)(3.9)(3.10) The total amount of deteriorated items during [0,𝑡₁] is 𝐷=^𝑡1₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡−^𝑡1₀𝑓(𝑡)𝑑𝑡. (3.11) The cumulative inventory carried in the interval [0,𝑡₁] is found from (3.7) and is 𝐼₁=^𝑡1₀𝐼(𝑡)𝑑𝑡=^𝑡1₀𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡. (3.12) Due to (3.8), (3.9), and (3.10), the time-weighted backorders during the interval [𝑡₁,𝑇] are 𝐼₂=^𝑇_𝑡1=[−𝐼(𝑡)]𝑑𝑡_𝜇𝑡1[−𝐼(𝑡)]𝑑𝑡+^𝛾_𝜇[−𝐼(𝑡)]𝑑𝑡+^𝑇_𝛾=[−𝐼(𝑡)]𝑑𝑡_𝜇𝑡1(𝜇−𝑡)𝑓(𝑡)𝛽(𝑇−𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[^𝑡_𝜇𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+^𝛾_𝜇[^𝜇_𝑡1+𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡_𝑇𝛾[^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+^𝑇_𝛾[^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+𝑓(𝜇)^𝑇_𝛾[^𝛾_𝜇𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡. (3.13) The amount of lost sales during [𝑡₁,𝑇] is 𝐿=^𝜇_𝑡1(1−𝛽(𝑇−𝑡))𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇(1−𝛽(𝑇−𝑡))𝑑𝑡+^𝑇_𝛾(1−𝛽(𝑇−𝑡))𝑔(𝑡)𝑑𝑡. (3.14)
The total cost in the time interval [0,𝑇] is the sum of holding, shortage, deterioration, and opportunity costs and is given by 𝑇𝐶₁(𝑡₁)=𝑐₁𝐼₁+𝑐₂𝐼₂+𝑐₃𝐷+𝑐₄𝐿=𝑐₁{^𝑡1₀𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡}+𝑐₃{^𝑡1₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡−^𝑡1₀𝑓(𝑡)𝑑𝑡}+𝑐₂{^𝜇_𝑡1(𝜇−𝑡)𝑓(𝑡)𝛽(𝑇−𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[^𝑡_𝜇+𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡_𝛾𝜇[^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+^𝑇_𝛾[^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+^𝑇_𝛾[^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡+𝑓(𝜇)^𝑇_𝛾[^𝛾_𝜇𝛽(𝑇−𝑥)𝑑𝑥]𝑑𝑡}+𝑐₄{^𝜇_𝑡1(1−𝛽(𝑇−𝑡))𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇(1−𝛽(𝑇−𝑡))𝑑𝑡+^𝑇_𝛾(1−𝛽(𝑇−𝑡))𝑔(𝑡)𝑑𝑡}. (3.15)

Case 2 (𝜇<𝑡₁<𝛾<𝑇). In this case, (3.1) reduces to the following two: 𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝑡),0≤𝑡≤𝜇,𝐼(𝜇⁻)=𝐼(𝜇⁺),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝜇),𝜇≤𝑡≤𝑡₁,𝐼(𝑡₁)=0. (3.16) Equation (3.2) leads to the following two: 𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝜇)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝛾,𝐼(𝑡₁)=0,𝑑𝐼(𝑡)𝑑𝑡=−𝑔(𝑡)𝛽(𝑇−𝑡),𝛾≤𝑡≤𝑇,𝐼(𝛾⁻)=𝐼(𝛾⁺). (3.17) Their solutions are, respectively, 𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝑡1_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥],0≤𝑡≤𝜇,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏}𝑓(𝜇)^𝑡1_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥,𝜇≤𝑡≤𝑡₁,𝐼(𝑡)=−𝑓(𝜇)^𝑡_𝑡1𝛽(𝑇−𝑥)𝑑𝑥,𝑡₁≤𝑡≤𝛾,𝐼(𝑡)=−^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥−𝑓(𝜇)^𝛾_𝑡1𝛽(𝑇−𝑥)𝑑𝑥,𝛾≤𝑡≤𝑇. (3.18) The total amount of deteriorated items during [0,𝑡₁] is 𝐷=𝐼(0)−^𝑡1₀𝐷(𝑡)𝑑𝑡=^𝜇₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^𝑎𝑡𝑏𝑑𝑡−^𝜇₀𝑓(𝑡)𝑑𝑡−𝑓(𝜇)(𝑡₁−𝜇). (3.19) The total inventory carried during the interval [0,𝑡₁] is 𝐼₁=^𝑡1₀𝐼(𝑡)𝑑𝑡=^𝜇₀𝐼(𝑡)𝑑𝑡+^𝑡1_𝜇=𝐼(𝑡)𝑑𝑡_𝜇0𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝑡1_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡. (3.20) The time-weighted backorders during the interval [𝑡₁,𝑇] are 𝐼₂=^𝑇_𝑡1[−𝐼(𝑡)]𝑑𝑡=^𝛾_𝑡1[−𝐼(𝑡)]𝑑𝑡+^𝑇_𝛾=[−𝐼(𝑡)]𝑑𝑡_𝛾𝑡1𝑓(𝜇)^𝑡_𝑡1𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+^𝑇_𝛾^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+𝑓(𝜇)^𝑇_𝛾^𝛾_𝑡1𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡. (3.21) The lost sales in the interval [𝑡₁,𝑇] are 𝐿=𝑓(𝜇)^𝛾_𝑡1[1−𝛽(𝑇−𝑡)]𝑑𝑡+^𝑇_𝛾[1−𝛽(𝑇−𝑡)]𝑔(𝑡)𝑑𝑡. (3.22)
The inventory cost for this case is 𝑇𝐶₂(𝑡₁)=𝑐₁𝐼₁+𝑐₂𝐼₂+𝑐₃𝐷+𝑐₄𝐿=𝑐₁{^𝜇₀𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝑡1_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡}+𝑐₂{^𝛾_𝑡1𝑓(𝜇)^𝑡_𝑡1𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+^𝑇_𝛾^𝑡_𝛾𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+𝑓(𝜇)^𝑇_𝛾^𝛾_𝑡1𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡}+𝑐₃{^𝜇₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^𝑎𝑡𝑏𝑑𝑡−^𝜇₀𝑓(𝑡)𝑑𝑡−𝑓(𝜇)(𝑡₁−𝜇)}+𝑐₄{𝑓(𝜇)^𝛾_𝑡1[1−𝛽(𝑇−𝑡)]𝑑𝑡+^𝑇_𝛾[1−𝛽(𝑇−𝑡)]𝑔(𝑡)𝑑𝑡}. (3.23)

Case 3 (𝜇<𝛾<𝑡₁<𝑇). In this case, (3.1) reduces to the following three: 𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝑡),0≤𝑡≤𝜇,𝐼(𝜇⁻)=𝐼(𝜇⁺),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝜇),𝜇≤𝑡≤𝛾,𝐼(𝛾⁻)=𝐼(𝛾⁺),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑔(𝑡),𝛾≤𝑡≤𝑡₁,𝐼(𝑡₁)=0. (3.24) Equation (3.2) leads to the following: 𝑑𝐼(𝑡)𝑑𝑡=−𝑔(𝑡)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝑇,𝐼(𝑡₁)=0. (3.25) Their solutions are, respectively, 𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥],0≤𝑡≤𝜇,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥],𝜇≤𝑡≤𝛾,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏𝑡1}_𝑡𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥,𝛾≤𝑡≤𝑡₁,𝐼(𝑡)=−^𝑡_𝑡1𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥,𝑡₁≤𝑡≤𝑇. (3.26)(3.27)(3.28)(3.29) The total amount of deteriorated items during [0,𝑡₁] is 𝐷=𝐼(0)−^𝑡1₀=𝐷(𝑡)𝑑𝑡_𝜇0𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏−𝑑𝑥_𝜇0𝑓(𝑡)𝑑𝑡−𝑓(𝜇)(𝛾−𝜇)−^𝑡1_𝛾𝑔(𝑡)𝑑𝑡. (3.30) The total inventory carried during the interval [0,𝑡₁], using (3.26), (3.27), and (3.28) is 𝐼₁=^𝑡1₀𝐼(𝑡)𝑑𝑡=^𝜇₀𝐼(𝑡)𝑑𝑡+^𝛾_𝜇𝐼(𝑡)𝑑𝑡+^𝑡1_𝛾=𝐼(𝑡)𝑑𝑡_𝜇0𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏+𝑑𝑥]𝑑𝑡_𝛾𝜇𝑒^{−𝑎𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡+^𝑡1_𝛾𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡. (3.31) The time-weighted backorders during the interval [𝑡₁,𝑇] are 𝐼₂=^𝑇_𝑡1[−𝐼(𝑡)]𝑑𝑡=^𝑇_𝑡1^𝑡_𝑡1𝑔(𝑥)𝛽(𝑇−𝑥)𝑑𝑥. (3.32) The lost sales in the interval [𝑡₁,𝑇] are 𝐿=^𝑇_𝑡1[1−𝛽(𝑇−𝑡)]𝑔(𝑡)𝑑𝑡. (3.33)
The inventory cost for this case is 𝑇𝐶₃(𝑡₁)=𝑐₁𝐼₁+𝑐₂𝐼₂+𝑐₃𝐷+𝑐₄𝐿=𝑐₁{^𝜇₀𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏+𝑑𝑥]𝑑𝑡_𝛾𝜇𝑒^{−𝑎𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡+^𝑡1_𝛾𝑒^{−𝑎𝑡𝑏}[^𝑡1_𝑡𝑔(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥]𝑑𝑡}+𝑐₂{^𝑇_𝑡1^𝑡_𝑡1𝑔(𝑥)𝛽(𝑇−𝑡)𝑑𝑥}+𝑐₄{^𝑇_𝑡1[1−𝛽(𝑇−𝑡)]𝑔(𝑡)𝑑𝑡}+𝑐₃{^𝜇₀𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑡1_𝛾𝑔(𝑥)𝑒^𝑎𝑥𝑏−𝑑𝑥_𝜇0𝑓(𝑡)𝑑𝑡−𝑓(𝜇)(𝛾−𝜇)−^𝑡1_𝛾𝑔(𝑡)𝑑𝑡}. (3.34) Finally the total cost function of the system over [0,𝑇] takes the following form: 𝑇𝐶(𝑡₁)=𝑇𝐶₁(𝑡₁),if𝑡₁≤𝜇,𝑇𝐶₂(𝑡₁),if𝜇<𝑡₁<𝛾,𝑇𝐶₃(𝑡₁),if𝛾≤𝑡₁. (3.35) It is easy to check that this function is continuous at 𝜇 and 𝛾. The problem now is the minimization of this, three branches, function 𝑇𝐶(𝑡₁). This requires, separately, studying each of these branches and then combining the results to state the algorithm giving the optimal policy.

3.1. The Optimal Replenishment Policy

In this subsection we present the results, which ensure the existence of a unique optimal value for 𝑡₁, say 𝑡^∗₁, which minimizes the total cost function. Although the optimality procedure requires the constrained optimization of the functions 𝑇𝐶₁(𝑡₁),𝑇𝐶₂(𝑡₁), and 𝑇𝐶₃(𝑡₁), we will, firstly, search for their unconstrained minimum. The first- and second-order derivatives of 𝑇𝐶₁(𝑡₁), 𝑇𝐶₂(𝑡₁), and 𝑇𝐶₃(𝑡₁) are, respectively,

𝑑𝑇𝐶₁(𝑡₁)𝑑𝑡₁=𝑓(𝑡₁)ℎ(𝑡₁𝑑),₂𝑇𝐶₁(𝑡₁)𝑑𝑡₁²=𝑑𝑓(𝑡₁)𝑑𝑡₁ℎ(𝑡₁)+𝑓(𝑡₁)𝑑ℎ(𝑡₁)𝑑𝑡₁,𝑑𝑇𝐶₂(𝑡₁)𝑑𝑡₁=𝑓(𝜇)ℎ(𝑡₁𝑑),₂𝑇𝐶₂(𝑡₁)𝑑𝑡₁²=𝑓(𝜇)𝑑ℎ(𝑡₁)𝑑𝑡₁,𝑑𝑇𝐶₃(𝑡₁)𝑑𝑡₁=𝑔(𝑡₁)ℎ(𝑡₁𝑑),₂𝑇𝐶₃(𝑡₁)𝑑𝑡₁²=𝑑𝑔(𝑡₁)𝑑𝑡₁ℎ(𝑡₁)+𝑔(𝑡₁)𝑑ℎ(𝑡₁)𝑑𝑡₁, (3.36) where

ℎ(𝑡₁)=𝑐₁𝑒^{𝑎𝑡1𝑏𝑡1}₀𝑒^{−𝑎𝑡𝑏}𝑑𝑡+𝑐₃(𝑒^{𝑎𝑡1𝑏}−1)−𝑐₂(𝑇−𝑡₁)𝛽(𝑇−𝑡₁)−𝑐₄(1−𝛽(𝑇−𝑡₁)). (3.37) Equation (3.37) is the same as (16) of the paper of Skouri et al. [16]. So, following the methodology proposed by Skouri et al. [16], the algorithm, which gives the optimal replenishment policy, is as follows.

Step 1. Compute 𝑡^∗₁ from ℎ(𝑡₁)=0.

Step 2. If 𝑡^∗₁≤𝜇, then the optimal order quantity is given by 𝑄^∗=^𝑡∗1₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+^𝜇_𝑡∗1𝛽(𝑇−𝑡)𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇𝛽(𝑇−𝑡)𝑑𝑡+^𝑇_𝛾𝛽(𝑇−𝑡)𝑔(𝑡)𝑑𝑡 (3.38) and the total cost is given by 𝑇𝐶₁(𝑡^∗₁).
If 𝜇<𝑡^∗₁<𝛾, then the optimal order quantity is given by 𝑄^∗=^𝜇₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+𝑓(𝜇)^𝑡∗1_𝜇𝑒^𝑎𝑡𝑏𝑑𝑡+𝑓(𝜇)^𝛾_𝑡∗1𝛽(𝑇−𝑡)𝑑𝑡+^𝑇_𝛾𝑔(𝑡)𝛽(𝑇−𝑡)𝑑𝑡 (3.39) and the total cost is given by 𝑇𝐶₂(𝑡^∗₁).
If 𝛾<𝑡^∗₁<𝑇, then the optimal order quantity is given by 𝑄^∗=^𝜇₀𝑓(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+𝑓(𝜇)^𝛾_𝜇𝑒^𝑎𝑡𝑏𝑑𝑡+^𝑡∗1_𝛾𝑔(𝑡)𝑒^𝑎𝑡𝑏𝑑𝑡+^𝑇_𝑡∗1𝑔(𝑡)𝛽(𝑇−𝑡)𝑑𝑡 (3.40) and the total cost is given by 𝑇𝐶₃(𝑡^∗₁).

Remark 3.1. The previous analysis shows that 𝑡^∗₁ is independent from the demand rate 𝐷(𝑡). This very interesting result agrees with the classical result, in many order level inventory systems, that the point 𝑡^∗₁ is independent from the demand rate (Naddor [31, page 67]).

3.2. The Special Case 𝛽(𝑥)=1 and 𝛼=0

If we are considering the case that there is no deterioration of the product (𝑎=0) and unsatisfied demand is complete backlogged (𝛽(𝑥)=1), then the total cost function of the model starting with no shortages over [0,𝑇] takes the following form:

𝑇𝐶(𝑡₁𝑐)=^{1𝑡10𝑡1𝑡}𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑐₂^𝜇_𝑡1^𝑡_𝑡1𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑐₂𝑓(𝜇)^𝛾_𝜇(𝑡−𝜇)𝑑𝑡+𝑐₂^𝛾_𝜇^𝜇_𝑡1𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑐₂^𝑇_𝛾^𝑡_𝛾𝑔(𝑥)𝑑𝑥𝑑𝑡𝑑𝑡+𝑐₂𝑓(𝜇)(𝛾−𝜇)(𝑇−𝛾)+𝑐₂^𝑇_𝛾^𝜇_𝑡1𝑡𝑓(𝑥)𝑑𝑥𝑑𝑡,₁𝑐<𝜇<𝛾<𝑇,_{1𝜇0𝜇𝑡}𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑐₁^𝜇₀^𝑡1_𝜇𝑓(𝜇)𝑑𝑥𝑑𝑡+𝑐₁^𝑡1_𝜇^𝑡1_𝑡𝑓(𝜇)𝑑𝑥𝑑𝑡+𝑐₂^𝛾_𝑡1^𝑡_𝑡1𝑓(𝜇)𝑑𝑥𝑑𝑡+𝑐₂^𝑇_𝛾^𝑡_𝛾𝑔(𝑥)𝑑𝑥𝑑𝑡+𝑐₂^𝑇_𝛾^𝛾_𝑡1𝑓(𝜇)𝑑𝑥𝑑𝑡,𝜇<𝑡₁𝑐<𝛾<𝑇,_{1𝜇0𝜇𝑡}𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑐₁^𝜇₀^𝛾_𝜇𝑓(𝜇)𝑑𝑥𝑑𝑡+𝑐₁^𝜇₀^𝑡1_𝛾𝑔(𝑥)𝑑𝑥𝑑𝑡+𝑐₁^𝛾_𝜇^𝛾_𝑡𝑓(𝜇)𝑑𝑥𝑑𝑡+𝑐₁^𝛾_𝜇^𝑡1_𝛾𝑔(𝑥)𝑑𝑥𝑑𝑡+𝑐₁^𝑡1_𝛾^𝑡1_𝑡𝑔(𝑥)𝑑𝑥𝑑𝑡+𝑐₂^𝑇_𝑡1^𝑡_𝑡1𝑔(𝑥)𝑑𝑥𝑑𝑡,𝜇<𝛾<𝑡₁<𝑇. (3.41) Following the previous procedure for the optimal replenishment policy, the optimal value of 𝑡₁, say 𝑡^∗₁, is given by the very simple and known, in classical order level inventory system (Naddor [31]), equation:

𝑡^∗₁=𝑐₂𝑇𝑐₁+𝑐₂. (3.42)

4. The Mathematical Formulation of the Model Starting with Shortages

In this section the inventory model starting with shortages is studied. The cycle now starts with shortages, which occur during the period (0,𝑡₁), and are partially backlogged. At time 𝑡₁ a replenishment brings the inventory level up to 𝑆. Demand and deterioration of the items deplete the inventory level during the period (𝑡₁,𝑇) until this falls to zero at 𝑡=𝑇. Again the three cases 𝑡₁<𝜇<𝛾<𝑇, 𝜇<𝑡₁<𝛾<𝑇, and 𝜇<𝛾<𝑡₁<𝑇 must be examined.

Case 4 (𝑡₁<𝜇<𝛾<𝑇). The inventory level, 𝐼(𝑡), 0≤𝑡≤𝑇 satisfies the following differential equations: 𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝑡)𝛽(𝑡₁−𝑡),0≤𝑡≤𝑡₁,𝐼(0)=0,𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝑡),𝑡₁≤𝑡≤𝜇,𝐼(𝜇₋)=𝐼(𝜇₊),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝜇),𝜇≤𝑡≤𝛾,𝐼(𝛾₋)=𝐼(𝛾₊),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑔(𝑡),𝛾≤𝑡≤𝑇,𝐼(𝑇)=0. (4.1) The solutions of (4.1), are, respectively, 𝐼(𝑡)=−^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥,0≤𝑡≤𝑡₁,𝐼(𝑡)=𝑒^{−𝛼𝑡𝑏}[^𝜇_𝑡𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥],𝑡₁≤𝑡≤𝜇,𝐼(𝑡)=𝑒^{−𝛼𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥],𝜇≤𝑡≤𝛾,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏𝑇}_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥,𝛾≤𝑡≤𝑇. (4.2)(4.3)(4.4)(4.5) The total amount of deteriorated units during [𝑡₁,𝑇] is 𝐷=𝑒^{−𝑎𝑡𝑏1}[^𝜇_𝑡1𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏−𝑔(𝑥)𝑑𝑥]_𝜇𝑡1𝑓(𝑥)𝑑𝑥−(𝛾−𝜇)𝑓(𝜇)−^𝑇_𝛾𝑔(𝑥)𝑑𝑥. (4.6) The total inventory carried during the interval [𝑡₁,𝑇] is found using (4.3), (4.4), and (4.5) and is 𝐼₁=^𝜇_𝑡1𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥]𝑑𝑡_𝛾𝜇𝑒^{−𝛼𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥]𝑑𝑡+^𝑇_𝛾𝑒^{−𝑎𝑡𝑏}[^𝑇_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥]𝑑𝑡. (4.7) Due to (4.2) the time-weighted backorders during the time interval [0,𝑡₁] are 𝐼₂=^𝑡1₀^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥𝑑𝑡. (4.8) The amount of lost sales during [0,𝑡₁] is 𝐿=^𝑡1₀[1−𝛽(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡. (4.9) The inventory cost during the time interval [0,𝑇] is the sum of holding, shortage, deterioration, and opportunity costs and is given by 𝑇𝐶₁(𝑡₁)=𝑐₁𝐼₁+𝑐₂𝐼₂+𝑐₃𝐷+𝑐₄𝐿=𝑐₁(^𝜇_𝑡1𝑒^{−𝑎𝑡𝑏}[^𝜇_𝑡𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥]𝑑𝑡)+𝑐₁(^𝛾_𝜇𝑒^{−𝛼𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥]𝑑𝑡_𝑇𝛾𝑒^{−𝑎𝑡𝑏}[^𝑇_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥]𝑑𝑡)+𝑐₂^𝑡1₀^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥𝑑𝑡+𝑐₄^𝑡1₀[1−𝛽(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑐₃[𝑒^{−𝑎𝑡𝑏1}[^𝜇_𝑡1𝑓(𝑥)𝑒^𝑎𝑥𝑏𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏−𝑔(𝑥)𝑑𝑥]_𝜇𝑡1𝑓(𝑥)𝑑𝑥−(𝛾−𝜇)𝑓(𝜇)−^𝑇_𝛾𝑔(𝑥)𝑑𝑥]. (4.10)

Case 5 (𝜇<𝑡₁<𝛾<𝑇). The inventory level, 𝐼(𝑡), 0≤𝑡≤𝑇 satisfies the following differential equations: 𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝑡)𝛽(𝑡₁−𝑡),0≤𝑡≤𝜇,𝐼(0)=0,𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝜇)𝛽(𝑡₁−𝑡),𝜇≤𝑡≤𝑡₁,𝐼(𝜇⁻)=𝐼(𝜇⁺),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑓(𝜇),𝑡₁≤𝑡≤𝛾,𝐼(𝛾₋)=𝐼(𝛾₊),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑔(𝑡),𝛾≤𝑡≤𝑇,𝐼(𝑇)=0. (4.11) The solutions of (4.11), are, respectively, 𝐼(𝑡)=−^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥,0≤𝑡≤𝜇,𝐼(𝑡)=−^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥−𝑓(𝜇)^𝑡_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥,𝜇≤𝑡≤𝑡₁,𝐼(𝑡)=𝑒^{−𝛼𝑡𝑏}[𝑓(𝜇)^𝛾_𝑡𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥],𝑡₁≤𝑡≤𝛾,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏𝑇}_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥,𝛾≤𝑡≤𝑇. (4.12) The total cost of this case is obtained with a similar way of the previous cases and is, 𝑇𝐶₂(𝑡₁)=𝑐₁[^𝛾_𝑡1𝑒^{−𝛼𝑡𝑏}(𝑓(𝜇)^𝛾_𝑡𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥)𝑑𝑡_𝑇𝛾(𝑒^{−𝑎𝑡𝑏𝑇}_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥)𝑑𝑡]+𝑐₃[𝑒^{−𝑎𝑡𝑏1}(^𝛾_𝑡1𝑓(𝜇)𝑒^𝑎𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥)−𝑓(𝜇)(𝛾−𝑡₁)−^𝑇_𝛾𝑔(𝑥)𝑑𝑥]+𝑐₂[^𝜇₀[^𝑡₀𝑓(𝑥)𝛽(𝑡₁+−𝑥)𝑑𝑥]𝑑𝑡_𝑡1𝜇[^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥+𝑓(𝜇)^𝑡_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥]𝑑𝑡]+𝑐₄[^𝜇₀[1−𝛽(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇[1−𝛽(𝑡₁−𝑡)]𝑑𝑡]. (4.13)

Case 6 (𝜇<𝛾<𝑡_𝟏<𝑇). The inventory level, 𝐼(𝑡), 0≤𝑡≤𝑇 for this case satisfies the following differential equations: 𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝑡)𝛽(𝑡₁−𝑡),0≤𝑡≤𝜇,𝐼(0)=0,𝑑𝐼(𝑡)𝑑𝑡=−𝑓(𝜇)𝛽(𝑡₁−𝑡),𝜇≤𝑡≤𝛾,𝐼(𝜇⁻)=𝐼(𝜇⁺),𝑑𝐼(𝑡)𝑑𝑡=−𝑔(𝑡)𝛽(𝑡₁−𝑡),𝛾≤𝑡≤𝑡₁,𝐼(𝛾₋)=𝐼(𝛾₊),𝑑𝐼(𝑡)𝑑𝑡+𝑎𝑏𝑡^𝑏−1𝐼(𝑡)=−𝑔(𝑡),𝑡₁≤𝑡≤𝑇,𝐼(𝑇)=0. (4.14)(4.15)(4.16)(4.17) The solutions of (4.14), (4.15), (4.16), and (4.17), are, respectively, 𝐼(𝑡)=−^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥,0≤𝑡≤𝜇,𝐼(𝑡)=−^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥−𝑓(𝜇)^𝑡_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥,𝜇≤𝑡≤𝛾,𝐼(𝑡)=−^𝑡_𝛾𝑔(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥−^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥−𝑓(𝜇)^𝛾_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥,𝛾≤𝑡≤𝑡₁,𝐼(𝑡)=𝑒^{−𝑎𝑡𝑏𝑇}_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥,𝑡₁≤𝑡≤𝑇. (4.18) The total cost of this case is obtained with a similar way of the previous cases and is, 𝑇𝐶₃(𝑡₁)=𝑐₁[^𝑇_𝑡1𝑒^{−𝛼𝑡𝑏𝑇}_𝑡𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥𝑑𝑡]+𝑐₃[𝑒^{−𝑎𝑡𝑏1𝑇}_𝑡1𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥−^𝑇_𝑡1𝑔(𝑥)𝑑𝑥]+𝑐₂[^𝜇₀[^𝑡₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥]𝑑𝑡+^𝛾_𝜇[^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥+𝑓(𝜇)^𝑡_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥]𝑑𝑡]+𝑐₂[^𝑡1_𝛾(^𝑡_𝛾𝑔(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥+^𝜇₀𝑓(𝑥)𝛽(𝑡₁−𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝛽(𝑡₁−𝑥)𝑑𝑥)𝑑𝑡]+𝑐₄[^𝜇₀[1−𝛽(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[1−𝛽(𝑡₁−𝑡)]𝑑𝑡+^𝑡1_𝛾[1−𝛽(𝑡₁−𝑡)]𝑔(𝑡)𝑑𝑡]. (4.19) Finally the total cost function of the system over [0,𝑇] takes the following form: 𝑇𝐶(𝑡₁)=𝑇𝐶₁(𝑡₁),if𝑡₁≤𝜇,𝑇𝐶₂(𝑡₁),if𝜇<𝑡₁<𝛾,𝑇𝐶₃(𝑡₁),if𝛾≤𝑡₁. (4.20) It is easy to check that this function is continuous at 𝜇 and 𝛾. The problem now is the minimization of this, three branches, function 𝑇𝐶(𝑡₁). This requires, separately, studying each of these branches and then combining the results to state the algorithm giving the optimal policy.

4.1. The Optimal Replenishment Policy

In this subsection we derive the optimal replenishment policy, that is, we calculate the value, say 𝑡^∗₁, which minimizes the total cost function. Taking the first-order derivative of 𝑇𝐶₁(𝑡₁), say 𝐾₁(𝑡₁), and equating it to zero gives:

𝐾₁(𝑡₁)=−(𝑐₁+𝑐₃𝛼𝑏𝑡₁^𝑏−1)𝑒^{−𝛼𝑡𝑏1}[^𝜇_𝑡1𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥]_𝑡10[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡=0. (4.21) If 𝑡^∗₁ is a root of (4.21), for this root the second-order condition for minimum is

[𝑐₁+𝑐₃𝛼𝑏(𝑡^∗₁)^𝑏−1−𝑐₃(𝑏−1)(𝑡^∗₁)⁻¹]𝛼𝑏(𝑡^∗₁)^𝑏−1𝑒^{−𝛼𝑡1∗𝑏}×[^𝜇_𝑡∗1𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥]+[𝑐₁+𝑐₃𝛼𝑏(𝑡^∗₁)^𝑏−1]𝑓(𝑡^∗₁)+[𝑐₂−𝑐₄𝛽^′(0)]𝑓(𝑡^∗₁)+^𝑡∗1₀[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁−𝑡)]𝑓(𝑡)𝑑𝑡>0. (4.22) So, if (4.22) holds and 𝑡^∗₁≤𝜇, then the value of order level, 𝑆, is

𝑆^∗=𝐼(𝑡^∗₁)=𝑒^{−𝛼𝑡∗1𝑏}[^𝜇_𝑡∗1𝑒^𝛼𝑥𝑏𝑓(𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥], (4.23) the ordering quantity is

𝑄^∗=^𝑡∗1₀𝑓(𝑥)𝛽(𝑡^∗₁−𝑥)𝑑𝑥+𝑆^∗, (4.24) and the total cost is𝑇𝐶₁(𝑡^∗₁).

Equating the first-order derivative of 𝑇𝐶₂(𝑡₁), say 𝐾₂(𝑡₁), to zero gives

𝐾₂(𝑡₁)=−(𝑐₁+𝑐₃𝛼𝑏𝑡₁^𝑏−1)𝑒^{−𝛼𝑡𝑏1}(𝑓(𝜇)^𝛾_𝑡1𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥)_𝜇0[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁−𝑡)]𝑑𝑡=0. (4.25) If 𝑡^∗₁ is a root of (4.25), for this root the second-order condition for minimum is

[𝑐₁+𝑐₃𝛼𝑏𝑡^∗₁^𝑏−1]𝛼𝑏𝑡^∗₁^𝑏−1𝑒^{−𝛼(𝑡∗1)𝑏}(𝑓(𝜇)^𝛾_𝑡∗1𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥)−𝑐₃𝛼𝑏(𝑏−1)𝑡^∗₁^(𝑏−2)𝑒^{−𝛼(𝑡∗1)𝑏}(𝑓(𝜇)^𝛾_𝑡∗1𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥)+[𝑐₁+𝑐₃𝛼𝑏(𝑡^∗₁)^𝑏−1]𝑓(𝜇)+𝑓(𝜇)(𝑐₂−𝑐₄𝛽^′(0))+^𝜇₀[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝑡∗1_𝜇[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁−𝑡)]𝑑𝑡>0. (4.26) So, if (4.26) holds and 𝜇<𝑡^∗₁<𝛾, then the value of S is

𝑆^∗=𝐼(𝑡^∗₁)=𝑒^{−𝛼𝑡∗1𝑏}[𝑓(𝜇)^𝛾_𝑡∗1𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥]. (4.27) the ordering quantity is

𝑄^∗=^𝜇₀𝑓(𝑥)𝛽(𝑡^∗₁−𝑥)𝑑𝑥+𝑓(𝜇)^𝑡∗1_𝜇𝛽(𝑡^∗₁−𝑥)𝑑𝑥+𝑆^∗, (4.28) and the total cost is 𝑇𝐶₂(𝑡^∗₁).

Equating the first-order derivative of 𝑇𝐶₃(𝑡₁), say 𝐾₃(𝑡₁), to zero gives

𝐾₃(𝑡₁)=−(𝑐₁+𝑐₃𝛼𝑏𝑡₁^𝑏−1)𝑒^{−𝛼𝑡𝑏1𝑇}_𝑡1𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥_𝜇0[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁+−𝑡)]𝑑𝑡_𝑡1𝛾[𝑐₂𝛽(𝑡₁−𝑡)+𝑐₂(𝑡₁−𝑡)𝛽^′(𝑡₁−𝑡)−𝑐₄𝛽^′(𝑡₁−𝑡)]𝑔(𝑡)𝑑𝑡=0. (4.29) If 𝑡^∗₁ is a root of (4.29), for this root the second-order condition for minimum is

[𝑐₁+𝑐₃𝛼𝑏𝑡^∗₁^𝑏−1]𝛼𝑏𝑡^∗₁^𝑏−1𝑒^{−𝛼(𝑡∗1)𝑏𝑇}_𝑡∗1𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥−𝑐₃𝛼𝑏(𝑏−1)𝑡^∗₁^(𝑏−2)𝑒^{−𝛼(𝑡∗1)𝑏𝑇}_𝑡∗1𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥+[𝑐₁+𝑐₃𝛼𝑏(𝑡^∗₁)^𝑏−1]𝑔(𝑡^∗₁)+𝑔(𝑡^∗₁)(𝑐₂−𝑐₄𝛽^′(0))+^𝜇₀[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁+−𝑡)]𝑑𝑡_𝑡∗1𝛾[2𝑐₂𝛽^′(𝑡^∗₁−𝑡)+𝑐₂(𝑡^∗₁−𝑡)𝛽^′′(𝑡^∗₁−𝑡)−𝑐₄𝛽^′′(𝑡^∗₁−𝑡)]𝑔(𝑡)𝑑𝑡>0. (4.30) So, if (4.30) holds and 𝑡^∗₁>𝛾, then the value of S is

𝑆^∗=𝐼(𝑡^∗₁)=𝑒^{−𝑎(𝑡∗1)𝑏𝑇}_𝑡∗1𝑒^𝑎𝑥𝑏𝑔(𝑥)𝑑𝑥, (4.31) the ordering quantity is

𝑄^∗=^𝜇₀𝑓(𝑥)𝛽(𝑡^∗₁−𝑥)𝑑𝑥+𝑓(𝜇)^𝛾_𝜇𝛽(𝑡^∗₁−𝑥)𝑑𝑥+^𝑡∗1_𝛾𝑔(𝑥)𝛽(𝑡^∗₁−𝑥)𝑑𝑥+𝑆^∗, (4.32) and the total cost is 𝑇𝐶₃(𝑡^∗₁).

Remark 4.1. Due to (4.21), (4.25), and (4.29) the function 𝑇𝐶(𝑡₁) is differentiable at the point 𝜇 and 𝛾.

In the previous analysis there is no guarantee that 𝑡^∗₁ exists and corresponds to the minimum. Its uniqueness is also another issue. The proposition, which follows, provides sufficient conditions for existence, uniqueness, and validity of 𝑡^∗₁.

Let us set

Δ₁=𝐾₁(𝜇)=−(𝑐₁+𝑐₃𝛼𝑏𝜇^𝑏−1)𝑒^{−𝛼𝜇𝑏}[𝑓(𝜇)^𝛾_𝜇𝑒^𝛼𝑥𝑏𝑑𝑥+^𝑇_𝛾𝑒^𝛼𝑥𝑏+𝑔(𝑥)𝑑𝑥]_𝜇0[𝑐₂𝛽(𝜇−𝑡)+𝑐₂(𝜇−𝑡)𝛽^′(𝜇−𝑡)−𝑐₄𝛽^′Δ(𝜇−𝑡)]𝑓(𝑡)𝑑𝑡,₂=𝐾₂(𝛾)=−(𝑐₁+𝑐₃𝛼𝑏𝛾^𝑏−1)𝑒^{−𝛼𝛾𝑏}(^𝑇_𝛾𝑒^𝛼𝑥𝑏𝑔(𝑥)𝑑𝑥)+^𝜇₀[𝑐₂𝛽(𝛾−𝑡)+𝑐₂(𝛾−𝑡)𝛽^′(𝛾−𝑡)−𝑐₄𝛽^′(𝛾−𝑡)]𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝛾_𝜇[𝑐₂𝛽(𝛾−𝑡)+𝑐₂(𝛾−𝑡)𝛽^′(𝛾−𝑡)−𝑐₄𝛽^′(𝛾−𝑡)]𝑑𝑡 (4.33)

and ℎ(𝑡)=𝑐₂𝛽(𝑡)+𝑐₂𝑡𝛽^′(𝑡)−𝑐₄𝛽^′(𝑡).

The following proposition can be easily proved observing that 𝐾₁(0)<0 and 𝐾₃(𝑇)>0.

Proposition 4.2. If 𝑏<1,ℎ^′(𝑡)>0,𝑡∈[0,𝑇]and(1)Δ₁>0, then (4.21) has one root, say t^∗₁, which is the unique optimal value of the problem.(2)Δ₁<0, and Δ₂>0 then (4.25) has one root, say t^∗₁, which is the unique optimal value of the problem.(3)Δ₁<0, and Δ₂<0 then (4.29) has one root, say t^∗₁, which is the unique optimal value of the problem.

We note that the above proposition ensures the existence, uniqueness, and validity of 𝑡₁. If the conditions of this proposition do not hold, then the following procedure can be used to calculate the optimal replenishment policy.

Step 1. (1) Find the global minimizing point, 𝑡^∗₁, for 𝑇𝐶₁(𝑡₁). This will be one of the following points: (a)root of (4.21), (an interior point of [0,𝜇]) which satisfies (4.22)(b)𝑡^∗₁=0,(c)𝑡^∗₁=𝜇.Then calculate 𝑇𝐶₁(𝑡^∗₁). 
(2) Find the global minimizing, 𝑡^∗₁, for 𝑇𝐶₂(𝑡₁). This will be one of the following points:(a)a root of (4.25), (an interior point of [𝜇,𝑇]) which satisfies (4.26),(b)𝑡^∗₁=𝜇,(c)𝑡^∗₁=𝛾.Then calculate 𝑇𝐶₂(𝑡^∗₁).
(3) Find the global minimizing, 𝑡^∗₁, for 𝑇𝐶₃(𝑡₁). This will be one of the following points:(a)a root of (4.29), (an interior point of [𝜇,𝑇]) which satisfies (4.30),(b)𝑡^∗₁=𝛾,(c)𝑡^∗₁=T.Then calculate 𝑇𝐶₃(𝑡^∗₁).

Step 2. Find the min {𝑇𝐶₁(𝑡^∗₁),𝑇𝐶₂(𝑡^∗₁),𝑇𝐶₃(𝑡^∗₁)} and accordingly select the optimum 𝑡^∗₁.

Remark 4.3. The analysis shows that, in this model, 𝑡^∗₁ is dependent from the demand rate 𝐷(𝑡).

5. Numerical Examples

The examples, which follow, illustrate the results obtained.

Example 5.1. The input parameters are 𝑐₁=$3 per unit per year, 𝑐₂=$15 per unit per year, 𝑐₃=$5 per unit, 𝑐₄=$20 per unit, 𝜇=0.12 year, 𝛾=0.9, 𝑎=0.001, 𝑏=2, 𝑇=1 year, 𝑓(𝑡)=3𝑒^4.5𝑡, 𝑔(𝑡)=3𝑒^{4.5𝜇−0.8(𝑡−𝛾)}, and 𝛽(𝑥)=𝑒^−0.2𝑥.
Model Starting with No Shortages
Using (3.37) the optimal value of 𝑡₁ is 𝜇<𝑡^∗₁=0.860<𝛾, and consequently the optimal ordering quantity is 𝑄^∗=4.98 (from (3.39)) and the minimum cost is 𝑇𝐶(𝑡^∗₁)=6.631 (from (3.23)).Model Starting with Shortages
Using (4.25) the optimal value of 𝑡₁ is 𝜇<𝑡^∗₁=0.163<𝛾, and consequently the optimal ordering quantity is 𝑄^∗=4.98 (from (4.28)), and the minimum cost is 𝑇𝐶(𝑡^∗₁)=6.329 (from (4.13)).

Example 5.2. This example is identical to Example 5.1, except that 𝛾=0.7, 𝑓(𝑡)=20𝑒^3.5𝑡, and 𝑔(𝑡)=20𝑒^3.5𝜇−50(𝑡−𝛾).
Model Starting with No Shortages
Again 𝑡^∗₁=0.860 but now 𝛾<𝑡^∗₁, the optimal ordering quantity is 𝑄^∗=27.49 (from (3.40)), and the minimum cost is 𝑇𝐶(𝑡^∗₁)=35.855 (from (3.34)).Model Starting with Shortages
Using (4.25) the optimal value of 𝑡₁ is 𝜇<𝑡^∗₁=0.148<𝛾, and consequently the optimal ordering quantity is 𝑄^∗=27.48 (from (4.28)) and the minimum cost is 𝑇𝐶(𝑡^∗₁)=33.008 (from (4.13)).

In Tables 1 and 2 some sensitivity analysis for the models starting without and with shortages, respectively, is performed, for the above examples, by changing the cost parameter values −75%,−50%,+50% and, +100% taking one at a time and keeping the remaining unchanged. From these two examples and the sensitivity analysis is evident the following.

Table 1 

Sensitivity analysis for model starting without shortages.

Table 2 

Sensitivity analysis for model starting with shortages.

(1) For the model starting without shortages, although the time when shortages occur (𝑡^∗₁) is identical for the two examples, the ordering quantities and costs are significant different, obviously because of the demand rate.

(2) For the two models the changes in the total optimal costs indicate that the models are highly sensitive to the error on the estimation of the parameter value 𝑐₁, moderately sensitive to the error on 𝑐₂, while low sensitivity is to the error on the estimation of the parameters 𝑐₃ and 𝑐₄. In presented examples, the costs related to storage inventory (𝑐₁ and 𝑐₃) are less than costs related to unsatisfied demand (𝑐₂ and 𝑐₄) maybe this explains the low sensitivity to the error on the estimation of the parameter 𝑐₄. While the small deterioration rate maybe implies low sensitivity with respect to 𝑐₃.

(3) The errors in cost parameters are attenuated when translated into changes in the optimal ordering quantity.

6. Concluding Remarks

In this paper, an order level inventory model for deteriorating items has been studied. The basic assumption of the model is based on time dependent three branches ramp type demand rate. The demand of seasonable and fashionable products can be described well with this function, as the nature of demand of these products is increasing at the beginning of the season, steady in the mid of the season, and decreasing at the end of the season. To the best of our knowledge this demand pattern studied for the first time, at least, using so general functions for the nonsteady periods. In addition a time dependent backlogging and deterioration rate are assumed. The inventory model is studied under two different replenishment policies: (a) starting with no shortages and (b) starting with shortages. An algorithm to obtain the optimal policy is proposed.

Moreover from this model follows as special cases the following ones.

(1)If 𝑓(𝑡)=𝐷₀𝑡,𝑡∈[0,𝜇],𝑔(𝑥)=𝑓(𝜇),𝜇=𝛾 and 𝛽(𝑥)=1/(1+𝛿𝑥), this model reduces to that of Wu [12]. (2)If in addition to (1), 𝛽(𝑥)=1 (case of complete backlogging), then it further reduces to that of Wu et al. [10]. (3)If 𝑏=1 (i.e., constant deterioration rate), 𝑓(𝑡)=𝐷₀𝑡,𝑔(𝑥)=𝑓(𝜇),𝜇=𝛾, and 𝛽(𝑥)=1, then the models reduce to those of Wu and Ouyang [11], Mandal and Pal [9], and Deng et al. [15]. (4)If 𝑔(𝑥)=𝑓(𝜇),𝜇=𝛾, this model gives the one model proposed by Skouri et al. [16].

Acknowledgment

The authors would like to thank the referees for their valuable comments and suggestions that improved the paper.