A Global Optimizing Policy for Decaying Items with Ramp-Type Demand Rate under Two-Level Trade Credit Financing Taking Account of Preservation Technology

Abstract

An inventory system for deteriorating items, with ramp-type demand rate, under two-level trade credit policy taking account of preservation technology is considered. The objective of this study is to develop a deteriorating inventory policy when the supplier provides to the retailer a permissible delay in payments, and during this credit period, the retailer accumulates the revenue and earns interest on that revenue; also the retailer invests on the preservation technology to reduce the rate of product deterioration. Shortages are allowed and partially backlogged. Sufficient conditions of the existence and uniqueness of the optimal replenishment policy are provided, and an algorithm, for its determination, is proposed. Numerical examples draw attention to the obtained results, and the sensitivity analysis of the optimal solution with respect to leading parameters of the system is carried out.

1. Introduction

In broad spectrum, deterioration is defined as the damage, spoilage, dryness, vaporization, and so forth, that result in the decrease of usefulness of the original one. In the past few decades, inventory problems for deteriorating items have been widely studied. The first attempt to derive optimal policies for deteriorating items was made by Ghare and Schrader [1], who derived a revised form of the economic order quantity (EOQ) model assuming exponential decay. This model was extended to consider the Weibull distribution deterioration by Covert and Philip [2]. Raafat [3] presented a complete survey of the available inventory literature for deteriorating inventory models. Goyal and Giri [4] also provided a detailed review of deteriorating inventory literatures. Teng et al. [5] developed an inventory model for deteriorating items with time varying demand and partial backlogging. Recently, C. Singh and S. R. Singh [6] presented an inventory model considering the Weibull distribution deterioration.

Investing in preservation technology (PT) for reducing deterioration rate has received little attention in the past years. The consideration of PT is important due to rapid social changes and the fact that P can reduce the deterioration rate significantly. Moreover, sales, inventories, and order quantities are very sensitive to the rate of deterioration, especially for fast deteriorating products. The higher rate of deterioration would result in a higher total annual relevant cost and a lower demand rate [7, 8]. Ouyang et al. [9] found that if the retailer can reduce effectively the deteriorating rate of item by improving the storage facility, the total annual relevant inventory cost will be reduced. Many enterprises invest in equipments to reduce the deterioration rate and extend the product expiration date. For example, refrigeration equipments are used to reduce the deterioration rate of fruits, flowers, and sea foods in the supermarket. Murr and Morris [10] showed that a lower temperature will increase the storage life and decrease decay. Wee et al. [11] presented a model using preservation technology.

In global market, supplier uses trade credit as a promotion tool to increase his sale and attract new retailers. Therefore, in practice, the supplier will allow a certain fixed period (credit period) for settling the amount that the supplier owes to retailer for the items supplied. Before the end of the trade credit period, the the retailer can sell the goods, accumulate revenue, and earn interest. However, beyond this period the supplier charges interest on the unpaid balance. Hence, trade credit can play a major role in inventory control for both the supplier and the retailer (see Jaggi et al. [12]). Huang and Chung [13] extended Goyal’s [14] model to discuss the replenishment and payment policies to minimize the annual total average cost under cash discount and payment delay from the retailer’s point of view. Some relevant models related to this research area are found in the works of Huang [15], Yang [7], Darzanou and Skouri [16], Singh et al. [17], and so forth.

In the literature referring to models with allowable delay in payments, the demand is, mostly, treated either as constant or as continuous differentiable function of time. However, in the case of a new brand of consumer good coming to the market, its demand rate increases in its growth stage (i.e., [0,𝜇]) and then remains stable in its maturity stage (i.e., [𝜇,𝑇]). The term “ramp type” is used to represent such demand pattern. Hill [18] proposed an inventory model with variable branch, being any power function of time. Research on this field continues with Mandal and Pal [19] and Wu and Ouyang [20]. In the above cited papers, the optimal replenishment policy requires to determine the decision time (say, 𝑡₁) at which the inventory level falls to zero. Consequently, the following two cases should be examined: (1) the inventory level falls to zero before the demand reaches constant (i.e., 𝑡₁<𝜇) and (2) the inventory level falls to zero after the demand reaches constant (i.e., 𝑡₁>𝜇). Almost all of the researchers examined only the first case. Deng et al. [21] first reconsidered the inventory models proposed by Mandal and Pal [19] and Wu and Ouyang [20] and discussed both cases. Skouri et al. [22] extend the work of Deng et al. [21] by introducing a general ramp-type demand rate and Weibull deterioration rate.

The present paper is an extension of the inventory system of Darzanou and Skouri [16] in the sense of preservation technology for deteriorating items when the two-level trade credit scheme, 𝑟/𝑀₁/𝑀₂, (in which the supplier provides 𝑟 discount off the price if the payment is made within period 𝑀₁; otherwise, the full payment is due within period 𝑀₂) is considered. The study of this organism requires the inspection of the ordering relations between the time parameters 𝑀₁,𝑀₂,𝜇,𝑇, which, actually, lead to the various models. To reduce the length of the paper, one of these models (𝑀₁≤𝜇<𝑀₂<𝑇) will be presented at this time. The optimal solution of the proposed model not only exists but also is unique. To illustrate the theory of the proposed model the numerical example is provided, and sensitivity analysis with respect to parameters of the system is carried out. The original deterioration rate is assumed to be 𝜃. A reduced deterioration rate of 𝑚(𝜉) is assumed when the retailer’s investment cost of preservation equipments or technology is 𝜉.

2. Notation and Assumptions

The following notation is used through the paper.

2.1. Notation

 𝑇 is the constant scheduling period (cycle), 𝑡₁ is the time when the inventory level falls to zero, 𝐶_𝑝 is the unit purchase cost, 𝑐₁ is the inventory holding cost per unit per unit time, 𝑐₂ is the shortage cost per unit per unit time, 𝑐₃ is the cost incurred from the deterioration of one unit, 𝑐₄ is the per unit opportunity cost due to the lost sales (𝑐₄>𝐶_𝑝 see Teng et al. [5]), 𝑆 is the maximum inventory level at the scheduling period (cycle), 𝜉 is the preservation technology (PT) cost for reducing deterioration rate in order to preserve the products, 𝜉≥0, 𝑚(𝜉) is reduced deterioration rate, a function of 𝜉,  𝑝 is the unit selling price, 𝐼_𝑒 is the interest rate earned, 𝐼_𝑐 is the interest rate charged, 𝑟 is cash discount rate, 0<𝑟<1, 𝑀₁ is the period of cash discount in years, 𝑀₂ is the period of permissible delay in payments in years, 𝑀₁<𝑀₂, 𝜇 is the parameter of the ramp-type demand function (time point), 𝐼(𝑡) is the inventory level at time 𝑡.

2.2. Assumptions

The inventory model is developed under the following 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 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 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0, where 𝛽^′(𝑥) is the derivate of 𝛽(𝑥). The case with 𝛽(𝑥)=1 corresponds to complete backlogging model.(3)The supplier offers cash discount if payment is paid within 𝑀₁; otherwise, the full payment is paid within 𝑀₂ (see Huang [15]).(4)The on-hand inventory deteriorates at a constant rate 𝜃(0<𝜃<1) per time unit. The deteriorated items are withdrawn immediately from the warehouse, and there is no provision for repair or replacement.(5)The demand rate 𝐷(𝑡) is a ramp-type function of time given by 𝐷(𝑡)=𝑓(𝑡)𝑓𝜇,𝑡<𝜇,,𝑡≥𝜇, (1) where 𝑓(𝑡) is a positive, differentiable function of 𝑡∈(0,𝑇].

3. Deriving the Common Quantities for the Inventory Models

The inventory level 𝐼(𝑡), 0≤𝑡≤𝑇 satisfies the following differential equations: 𝐼^′(𝑡)+(𝜃−𝑚(𝜉))𝐼(𝑡)=−𝐷(𝑡),0≤𝑡≤𝑡₁ (2) with boundary condition 𝐼(𝑡₁) = 0, and 𝐼^′(𝑡)=−𝐷(𝑡)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝑇, (3) with boundary condition 𝐼(𝑡₁)=0.

From the two possible relations between parameters 𝑡₁ and 𝜇, (i) 𝑡₁≤𝜇 and (ii) 𝑡₁>𝜇, the sum of holding, deterioration, shortages, and lost sales cost is obtained as 𝐶𝑡₁=𝐶₁𝑡₁if𝑡₁𝐶≤𝜇,₂𝑡₁if𝑡₁>𝜇, (4) where, 𝐶₁(𝑡₁) and 𝐶₂(𝑡₁) are calculated as follows.

Case 1 ((𝑡₁≤𝜇) (see Figure 1)). In this section, the inventory model starting with no shortages is studied. 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.
The backlogged demand is satisfied at the next replenishment. The inventory level, 𝐼(𝑡), 0≤𝑡≤𝑇 satisfies the following differential equations.
In this case, (2) becomes 𝐼^′(𝑡)+(𝜃−𝑚(𝜉))𝐼(𝑡)=−𝑓(𝑡),0≤𝑡≤𝑡₁,𝐼𝑡₁=0. (5) Equation (3) leads to the following two equations: 𝐼^′(𝑡)=−𝑓(𝑡)𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝜇,𝐼𝑡₁𝐼=0,_′(𝑡)=−𝑓𝜇𝛽(𝑇−𝑡),𝜇≤𝑡≤𝑇,𝐼𝜇₋=𝐼𝜇₊. (6)(7) The solutions of (5)–(7) are, respectively, 𝐼(𝑡)=𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥,0≤𝑡≤𝑡₁,𝐼(𝑡)=−^𝑡_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥,𝑡₁𝐼(𝑡)𝜇≤𝑡≤𝜇,=−𝑓^𝑡_𝜇𝛽()−𝑇−𝑥𝑑𝑥_𝜇𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥,𝜇≤𝑡≤𝑇. (8)(9)(10) The total amount of deteriorated items during [0,𝑡₁] is 𝐷=^𝑡1₀𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥−^𝑡1₀𝑓(𝑡)𝑑𝑡. (11) The cumulative inventory carried in the interval [0,𝑡₁] is found from (8) and is 𝐼₁=^𝑡1₀𝐼(𝑡)𝑑𝑡=^𝑡1₀𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡. (12) Due to (9) and (10), the time-weighted backorders due to shortages during the interval [𝑡₁,𝑇] are 𝐼₂=^𝑇_𝑡1[−𝐼(𝑡)]𝑑𝑡=^𝜇_𝑡1[−𝐼(𝑡)]𝑑𝑡+^𝑇_𝜇[(𝑡=−𝐼)]𝑑𝑡_𝜇𝑡1𝑓(𝑡)𝛽()𝜇𝜇−𝑡𝑇−𝑡𝑑𝑡+𝑓^𝑇_𝜇^𝑡_𝜇𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+^𝑇_𝜇^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡. (13) The amount of lost sales during [𝑡₁,𝑇] is 𝐿=^𝜇_𝑡11−𝛽(𝑇−𝑡)𝑓(𝑡)𝑑𝑡+𝑓𝜇^𝑇_𝜇1−𝛽(𝑇−𝑡)𝑑𝑡. (14) The cost 𝐶₁(𝑡₁) in the time interval [0,𝑇] is the sum of holding, shortage, deterioration, and opportunity costs and is given by 𝐶₁𝑡₁=𝜉+𝑐₁^𝑡1₀𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡+𝑐₃^𝑡1₀𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥−^𝑡1₀𝑓(𝑡)𝑑𝑡+𝑐₂^𝜇_𝑡1𝑓(𝑡)𝛽()𝜇𝜇−𝑡𝑇−𝑡𝑑𝑡+𝑓^𝑇_𝜇^𝑡_𝜇𝛽()+𝑇−𝑥𝑑𝑥𝑑𝑡_{𝑇𝜇𝜇𝑡1}𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡+𝑐₄^𝜇_𝑡1()𝑓(𝑡)𝜇1−𝛽𝑇−𝑡𝑑𝑡+𝑓^𝑇_𝜇1−𝛽(𝑇−𝑡)𝑑𝑡. (15)

Figure 1 

Inventory level for the model starting with no shortage over the cycle (case 𝑡1<𝜇).

Case 2 ((𝑡₁>𝜇) (see Figure 2)). In this case, (2) reduces to the following two equations: 𝐼^′(𝑡)+((𝜉𝐼(𝑡)(𝑡)𝐼𝜇𝜃−𝑚))=−𝑓,0≤𝑡≤𝜇,₋=𝐼𝜇⁺,𝐼^′(𝑡)+(𝜃−𝑚(𝜉))𝐼(𝑡)=−𝑓𝜇,𝜇≤𝑡≤𝑡₁,𝐼𝑡₁=0. (16) Equation (3) becomes 𝐼^′(𝑡)=−𝑓𝜇𝛽(𝑇−𝑡),𝑡₁≤𝑡≤𝑇,𝐼𝑡₁=0. (17) Their solutions are, respectively, 𝐼(𝑡)=𝑒^{−(𝜃−𝑚(𝜉))𝑡}×^𝜇_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥+𝑓𝜇^𝑡1_𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥},𝐼(𝑡)𝜇𝑒𝑑𝑥0≤𝑡≤𝜇,=𝑓^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥,𝜇≤𝑡≤𝑡₁,𝐼(𝑡)=−𝑓𝜇^𝑡1_𝑡𝛽(𝑇−𝑥)𝑑𝑥,𝑡₁≤𝑡≤𝑇. (18)(19)(20) The total amount of deteriorated items during [0,𝑡₁] is 𝐷=∫^𝜇₀𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥+𝑓^𝑡1_𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥−^𝜇₀𝑓(𝑡)𝑑𝑡−^𝑡1_𝜇𝑓𝜇𝑑𝑡. (21) The total inventory carried in the interval [0,𝑡₁] is found from (18) and (19) and is 𝐼₁=^𝑡1₀𝐼(𝑡)𝑑𝑡=^𝜇₀𝐼(𝑡)𝑑𝑡+^𝑡1_𝜇𝐼(𝑡)=𝑑𝑡_𝜇0𝑒^{−(𝜃−𝑚(𝜉))𝑡}×^𝜇_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥+𝑓𝜇^𝑡1_𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥𝑑𝑡+𝑓^𝑡1_𝜇𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡. (22) The time-weighted backorders due to shortages during the interval [𝑡₁,𝑇] are 𝐼₂=^𝑇_𝑡1[−𝐼(𝑡)]𝑑𝑡=𝑓𝜇^𝑇_𝑡1^𝑡_𝑡1𝛽()𝜇𝑇−𝑥𝑑𝑥𝑑𝑡=𝑓^𝑇_𝑡1(𝑇−𝑥)𝛽(𝑇−𝑥)𝑑𝑥. (23) The amount of lost sales during [𝑡₁,𝑇] is 𝐿=𝑓𝜇^𝑇_𝑡11−𝛽(𝑇−𝑡)𝑑𝑡. (24) The cost 𝐶₂(𝑡₁) in the time interval [0,𝑇] is the sum of holding, shortage, deterioration and opportunity costs and is given by 𝐶₂𝑡₁=𝜉+𝑐₁^𝜇₀𝑒^{−(𝜃−𝑚(𝜉))𝑡}×^𝜇_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥+𝑓^𝑡1_𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥𝑑𝑡+𝑓^𝑡1_𝜇𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡+𝑐₃^𝜇₀𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇×𝑑𝑥+𝑓_𝑡1𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥−^𝜇₀𝑓(𝑡)𝑑𝑡−^𝑡1_𝜇𝑓𝜇𝑑𝑡+𝑐₂𝑓𝜇^𝑇_𝑡1(𝑇−𝑥)𝛽(𝑇−𝑥)𝑑𝑥+𝑐₄𝑓𝜇^𝑇_𝑡11−𝛽(𝑇−𝑡)𝑑𝑡. (25)

Figure 2 

Inventory level for the model starting with no shortage over the cycle (case 𝑡1>𝜇).

4. The Inventory Model When 𝑀₁≤𝜇<𝑀₂<𝑇

In order to obtain the total cost for this model, the purchasing cost, interest charges for the items kept in stock, and the interest earned should be taken into account. Since the supplier offers cash discount if payment is paid within 𝑀₁, there are two payment policies for the buyer. Either the payment is paid at time 𝑀₁ to receive the cash discount (Case 1) or the payment is paid at time 𝑀₂ so as not to receive the cash discount (Case 2). Then, these two cases will be discussed.

Case 1 (payment is made at time 𝑀₁). In this case, the following subcases should be considered.

Subcase 1.1 ( 𝑡₁≤𝑀₁≤𝜇<𝑇). The purchasing cost is 𝐶_𝐴1,1𝑡₁=𝐶_𝑝()×1−𝑟_𝑡10𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇×𝑑𝑥+𝑓_𝑇𝜇𝛽(𝑇−𝑥)𝑑𝑥+^𝜇_𝑡1𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥. (26) The interest earned, 𝐼_𝑇1,1, during the period of positive inventory level is 𝐼_𝑇1,1𝑡₁=𝑝𝐼_𝑒^𝑡1₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼_𝑒𝑀₁−𝑡₁^𝑡1₀𝑓(𝑥)𝑑𝑥. (27) Since 𝑡₁≤𝜇, the total cost in the time interval [0,𝑇] is calculated using (15), (26), and (27): 𝑇𝐶_1,1𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,1𝑡₁−𝐼_𝑇1,1𝑡₁. (28)

Subcase 1.2 ( 𝑀₁<𝑡₁≤𝜇<𝑇). The purchasing cost is 𝐶_𝐴1,1 (relation (26)).

The interest payable for the inventory not being sold after the due date 𝑀₁ is 𝑃_𝑇2,1𝑡₁=𝐶_𝑝(1−𝑟)𝐼_𝑐^𝑡1_𝑀1𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡. (29) The interest earned, 𝐼_𝑇2,1, is 𝐼_𝑇2,1𝑡₁=𝑝𝐼_𝑒^𝑀1₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡. (30) Since again 𝑡₁≤𝜇, the total cost over [0,𝑇] is calculated using the relations (15), (26), (29), and (30) and is 𝑇𝐶_1,2𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,1𝑡₁+𝑃_𝑇2,1𝑡₁−𝐼_𝑇2,1𝑡₁. (31)

Subcase 1.3 ( 𝑀₁≤𝜇≤𝑡₁≤𝑇). The purchasing cost is 𝐶_𝐴2,1𝑡₁=𝐶_𝑝()×1−𝑟_𝜇0𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇×𝑑𝑥+𝑓_𝑡1𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥+𝑓𝜇^𝑇_𝑡1𝛽(𝑇−𝑥)𝑑𝑥. (32) The interest earned, 𝐼_𝑇3,1, is 𝐼_𝑇3,1𝑡₁=𝑝𝐼_𝑒^𝑀1₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡. (33) The interest payable for the inventory not being sold after the due date 𝑀₁ is 𝑃_𝑇3,1𝑡₁=𝐶_𝑝(1−𝑟)𝐼_𝑐×^𝜇_𝑀1𝑒^{−(𝜃−𝑚(𝜉))𝑡𝜇}_𝑡𝑓(𝑥)𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥𝑑𝑡+𝑓^𝜇_𝑀1𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝜇𝑒^{(𝜃−𝑚(𝜉))𝑥}𝜇𝑑𝑥𝑑𝑡+𝑓^𝑡1_𝜇𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡. (34) Since 𝜇<𝑡₁, the total cost over (0,𝑇) is again calculated from (25), (32)–(34) and is 𝑇𝐶_1,3𝑡₁=𝐶₂𝑡₁+𝐶_𝐴3,1𝑡₁+𝑃_𝑇3,1𝑡₁−𝐼_𝑇3,1𝑡₁. (35) The results obtained lead to the following total cost function: 𝑇𝐶₁𝑡₁=𝑇𝐶_1,1𝑡₁,𝑡₁≤𝑀₁≤𝜇<𝑇,𝑇𝐶_1,2𝑡₁,𝑀₁<𝑡₁≤𝜇<𝑇,𝑇𝐶_1,3𝑡₁,𝑀₁≤𝜇≤𝑡₁≤𝑇. (36) So the problem is min𝑇𝐶_1𝑡1𝑡₁. (37) Its solution requires, separately, studying each of the three branches and then combining the results to obtain the optimal policy. It is easy to check that 𝑇𝐶₁(𝑡₁) is continuous at the points 𝑀₁ and 𝜇.

The first-order condition for a minimum of 𝑇𝐶_1,1(𝑡₁) is𝑑𝑇𝐶_1,1𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒𝑀₁−𝑡₁+𝐶_𝑝()×𝑒1−𝑟_{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁𝑓𝑡₁=0. (38) Since 𝑑𝑇𝐶_1,1(0)/𝑑𝑡₁<0 and 𝑑𝑇𝐶_1,1(𝑇)/𝑑𝑡₁>0, (38) has at least one root. So if 𝑡_1,1 is the root of (38) and if 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0, then this corresponds to minimum as 𝑑²𝑇𝐶_1,1𝑡₁𝑑𝑡²_{1𝑡1=𝑡1,1}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,1}+𝑐₂𝛽𝑇−𝑡_1,1+𝑐₂𝑇−𝑡_1,1×𝛽^′𝑇−𝑡_1,1−𝑐₄𝛽^′𝑇−𝑡_1,1+𝑝𝐼_𝑒+𝐶_𝑝()×(𝜉𝑒1−𝑟(𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1,1}+𝛽^′𝑇−𝑡₁𝑓𝑡_1,1>0. (39) Consequently, 𝑡_1,1 is the unique unconstrained minimum of 𝑇𝐶_1,1(𝑡₁).

The first-order condition for a minimum of 𝑇𝐶_1,2(𝑡₁) is 𝑑𝑇𝐶_1,2𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁+𝐶_𝑝(1−𝑟)𝑒^{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁+𝐶_𝑝(1−𝑟)𝐼_𝑐((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))(𝑡1−𝑀1)}−1𝑓𝑡₁=0. (40) If 𝑡_1,2 is the root of (40) (this may or may not exist) and further if 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0, then 𝑑²𝑇𝐶_1,2𝑡₁𝑑𝑡²_{1𝑡1=𝑡1,2}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,2}+𝐶_𝑝(1−𝑟)𝐼_𝑐𝑒^{(𝜃−𝑚(𝜉))(𝑡1,2−𝑀1)}+𝑐₂𝛽𝑇−𝑡_1,2+𝑐₂𝑇−𝑡_1,2×𝛽^′𝑇−𝑡_1,2−𝑐₄𝛽^′𝑇−𝑡_1,2+𝐶_𝑝()×(𝜉𝑒1−𝑟(𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1,2}+𝛽^′𝑇−𝑡_1,2𝑓𝑡_1,2>0, (41) and this 𝑡_1,2 corresponds to unconstrained minimum of 𝑇𝐶_1,2(𝑡₁).

The first-order condition for a minimum of 𝑇𝐶_1,3(𝑡₁) is 𝑑𝑇𝐶_1,3𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁+𝐶_𝑝(1−𝑟)𝑒^{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁+𝐶_𝑝(1−𝑟)𝐼_𝑐((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))(𝑡1−𝑀1)}−1𝑓𝜇=0. (42) If 𝑡_1,3 is the root of (42) (this may or may not exist) and 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0, then 𝑑²𝑇𝐶_1,3𝑡₁𝑑𝑡²_{1𝑡1=𝑡1,3}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,3}+𝐶_𝑝(1−𝑟)𝐼_𝑐𝑒^{(𝜃−𝑚(𝜉))(𝑡1,3−𝑀1)}+𝐶_𝑝(1−𝑟)(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,3}+𝛽^′𝑇−𝑡_1,3+𝑐₂𝛽𝑇−𝑡_1,3+𝑐₂𝑇−𝑡_1,3𝛽^′𝑇−𝑡_1,3−𝑐₄𝛽^′𝑇−𝑡_1,3𝑓𝜇>0. (43) This 𝑡_1,3 corresponds to unconstrained minimum of 𝑇𝐶_1,3(𝑡₁).

Remark 1. The function 𝑇𝐶₁(𝑡₁) is not differentiable in 𝑀₁.

Then, the following procedure summarizes the previous results for the determination of the optimal replenishment policy, when payment is made at time 𝑀₁.

Step 1. Find the global minimum of 𝑇𝐶_1,1(𝑡₁), say 𝑡^∗_1,1,𝑀1, as follows.

Substep 1.1. Compute 𝑡_1,1,𝑀1 from (38); if 𝑡_1,1,𝑀1<𝑀₁, then set 𝑡^∗_1,1,𝑀1=𝑡_1,1,𝑀1 and compute 𝑇𝐶_1,1(𝑡^∗_1,1,𝑀1); else go to Substep 1.2.

Substep 1.2. Find the min{𝑇𝐶_1,1(0),𝑇𝐶_1,1(𝑀₁)} and accordingly set 𝑡^∗_1,1,𝑀1.

Step 2. Find the global minimum of 𝑇𝐶_1,2(𝑡₁), say 𝑡^∗_1,2,𝑀1, as follows.

Substep 2.1. Compute 𝑡_1,2,𝑀1from (40); if 𝑀₁<𝑡_1,2,𝑀1<𝜇, then set 𝑡^∗_1,2,𝑀1=𝑡_1,2,𝑀1 and compute 𝑇𝐶_1,2(𝑡^∗_1,2,𝑀1); else go to Substep 2.2.

Substep 2.2. Find the min{𝑇𝐶_1,2(𝑀₁),𝑇𝐶_1,2(𝜇)} and accordingly set 𝑡^∗_1,2,𝑀1.

Step 3. Find the global minimum of 𝑇𝐶_1,3(𝑡₁), say 𝑡^∗_1,3,𝑀1, as follows.

Substep 3.1. Compute 𝑡_1,3,𝑀1 from (42); if 𝜇<𝑡_1,3,𝑀1, then set 𝑡^∗_1,3,𝑀1=𝑡_1,3,𝑀1 and compute 𝑇𝐶_1,3(𝑡^∗_1,3,𝑀1); else go to Substep 3.2.

Substep 3.2. Find the min{𝑇𝐶_1,3(𝜇),𝑇𝐶_1,3(𝑇)} and accordingly set 𝑡^∗_1,3,𝑀1.

Step 4. Find the min{𝑇𝐶_1,1(𝑡^∗_1,1,𝑀1),𝑇𝐶_1,2(𝑡^∗_1,2,𝑀1),𝑇𝐶_1,3(𝑡^∗_1,3,𝑀1)} and accordingly select the optimal value for 𝑡₁, say 𝑡^∗_1,𝑀1, with optimal cost 𝑇𝐶₁(𝑡^∗_1,𝑀1).

Case 2 2 (payment is made at time 𝑀₂). When the payment is made at time 𝑀₂ the following cases should be considered.

Subcase 2.1 ( 𝑡₁≤𝜇<𝑀₂<𝑇). The purchasing cost is 𝐶_𝐴1,2𝑡₁=𝐶_𝐴1,1𝑡₁().1−𝑟 (44) The interest earned during the period of positive inventory level is 𝐼_𝑇1,2𝑡₁=𝑝𝐼_𝑒^𝑡1₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼_𝑒𝑀₂−𝑡₁^𝑡1₀𝑓(𝑥)𝑑𝑥. (45) Since 𝑡₁≤𝜇, the total cost in the time interval [0,𝑇] is calculated using (15), (44), and (45): 𝑇𝐶_2,1𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,2𝑡₁−𝐼_𝑇1,2𝑡₁. (46)

Subcase 2.2 ( 𝜇<𝑡₁≤𝑀₂<𝑇). The purchasing cost is 𝐶_𝐴2,2𝑡₁=𝐶_𝐴2,1𝑡₁().1−𝑟 (47) The interest earned, 𝐼_𝑇2,2(𝑡₁), is 𝐼_𝑇2,2𝑡₁=𝑝𝐼_𝑒^𝜇₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑀2_𝜇^𝜇₀𝑓(𝑥)+𝑑𝑥𝑑𝑡_{𝑡1𝜇𝑡𝜇}𝑓𝜇𝑑𝑥𝑑𝑡+^𝑀2_𝑡1^𝑡1_𝜇𝑓𝜇𝑑𝑥𝑑𝑡. (48) Since again 𝜇≤𝑡₁, the total cost over [0,𝑇] is calculated using the relations (25), (47), and (48) and is 𝑇𝐶_2,2𝑡₁=𝐶₂𝑡₁+𝐶_𝐴2,2𝑡₁−𝐼_𝑇2,2𝑡₁. (49)

Subcase 2.3 ( 𝜇≤𝑀₂≤𝑡₁≤𝑇). The purchasing cost is 𝐶_𝐴2,2(𝑡₁).

The interest earned, 𝐼_𝑇3,2, is 𝐼_𝑇3,2𝑡₁=𝑝𝐼_𝑒^𝜇₀^𝑡₀𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑀2_𝜇^𝜇₀𝑓(𝑥)𝑑𝑥𝑑𝑡. (50) The interest payable for the inventory not being sold after the due date 𝑀₂ is 𝑃_𝑇3,2𝑡₁=𝐶_𝑝𝐼_𝑐𝑓𝜇^𝑡1_𝑀2𝑒^{−(𝜃−𝑚(𝜉))𝑡𝑡1}_𝑡𝑒^{(𝜃−𝑚(𝜉))𝑥}𝑑𝑥𝑑𝑡. (51) Since 𝜇<𝑡₁, the total cost over [0,𝑇] is again calculated from (25), (47), (50), and (51) and is 𝑇𝐶_2,3𝑡₁=𝐶₂𝑡₁+𝐶_𝐴2,2𝑡₁+𝑃_𝑇3,2𝑡₁−𝐼_𝑇3,2𝑡₁. (52) The results obtained lead to the following total cost function: 𝑇𝐶₂𝑡₁=𝑇𝐶_2,1𝑡₁,𝑡₁≤𝜇<𝑀₂<𝑇,𝑇𝐶_2,2𝑡₁,𝜇<𝑡₁≤𝑀₂<𝑇,𝑇𝐶_2,3𝑡₁,𝜇≤𝑀₂≤𝑡₁≤𝑇. (53) So the problem is min𝑇𝐶_2𝑡1𝑡₁. (54) Its solution, as in the previous case, requires, separately, studying each of the three branches and then combining the results to obtain the optimal policy. It is easy to check that 𝑇𝐶₂(𝑡₁) is continuous at the points 𝑀₂ and 𝜇.

The first-order condition for the minimum for 𝑇𝐶_2,1(𝑡₁) is 𝑑𝑇𝐶_2,1𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒𝑀₂−𝑡₁+𝐶_𝑝𝑒^{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁𝑓𝑡₁=0. (55) Since 𝑑𝑇𝐶_2,1(0)/𝑑𝑡₁<0 and 𝑑𝑇𝐶_2,1(𝑇)/𝑑𝑡₁>0, (55) has at least one root. So, if 𝑡_1,1 is the root of (55), this corresponds to minimum if 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0 as 𝑑²𝑇𝐶_2,1𝑡₁𝑑𝑡²_{1𝑡1=𝑡1,1}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,1}+𝑐₂𝛽𝑇−𝑡_1,1+𝑐₂𝑇−𝑡_1,1𝛽^′𝑇−𝑡_1,1−𝑐₄𝛽^′𝑇−𝑡_1,1+𝑝𝐼_𝑒+𝐶_𝑝(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,1}+𝛽^′𝑇−𝑡₁𝑓𝑡_1,1>0. (56) So, 𝑡_1,1 is the unconstrained minimum of 𝑇𝐶_2,1(𝑡₁).

The first-order condition for a minimum of 𝑇𝐶_2,2(𝑡₁) is 𝑑𝑇𝐶_2,2𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁+𝐶_𝑝𝑒^{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒𝑀₂−𝑡₁𝑓𝜇=0. (57) If 𝑡_1,2 is the root of (57) (this may or may not exist), this corresponds to unconstrained minimum of 𝑇𝐶_2,2(𝑡₁) if 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0 as 𝑑²𝑇𝐶_2,2(𝑡₁)𝑑𝑡²_{1𝑡1=𝑡1,2}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,2}+𝑐₂𝛽𝑇−𝑡_1,2+𝑐₂𝑇−𝑡_1,2×𝛽^′𝑇−𝑡_1,2−𝑐₄𝛽^′𝑇−𝑡_1,2+𝑝𝐼_𝑒+𝐶_𝑝(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,2}+𝛽^′𝑇−𝑡_1,2𝑓𝜇>0. (58) The first-order condition for a minimum of 𝑇𝐶_2,3(𝑡₁) is 𝑑𝑇𝐶_2,3𝑡₁𝑑𝑡₁=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))𝑡1}−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁+𝐶_𝑝𝑒^{(𝜃−𝑚(𝜉))𝑡1}−𝛽𝑇−𝑡₁+𝐶_𝑝𝐼_𝑐((𝜉𝑒𝜃−𝑚))^{(𝜃−𝑚(𝜉))(𝑡1−𝑀2)}−1𝑓𝜇=0. (59) If 𝑡_1,3 is a root of (59) (this may or may not exist) and this corresponds to unconstrained minimum of 𝑇𝐶_2,3(𝑡₁) if 𝑐₂𝛽(𝑥)+𝑐₂𝑥𝛽^′(𝑥)−𝑐₄𝛽^′(𝑥)≥0 as 𝑑²𝑇𝐶_2,3𝑡₁𝑑𝑡²_{1𝑡1=𝑡1,3}=𝑐₁+𝑐₃(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,3}+𝑐₂𝛽𝑇−𝑡_1,3+𝑐₂𝑇−𝑡_1,3×𝛽^′𝑇−𝑡_1,3−𝑐₄𝛽^′𝑇−𝑡_1,3+𝐶_𝑝(𝜃−𝑚(𝜉))𝑒^{(𝜃−𝑚(𝜉))𝑡1,3}+𝛽^′𝑇−𝑡_1,3+𝐶_𝑝𝐼_𝑐𝑒^{(𝜃−𝑚(𝜉))(𝑡1,3−𝑀2)}𝑓𝜇>0. (60)

Remark 2. The function 𝑇𝐶₂(𝑡₁) is not differentiable in 𝑀₂.

Then, the following procedure summarizes the previous results for the determination of the optimal replenishment policy, when payment is made at time 𝑀₂.

Step 1. Find the global minimum of 𝑇𝐶_2,1(𝑡₁), say 𝑡^∗_1,1,𝑀2, as follows.

Substep 1.1. Compute 𝑡_1,1,𝑀2 from (55); if 𝑡_1,1,𝑀2<𝜇, then set 𝑡^∗_1,1,𝑀2=𝑡_1,1,𝑀2 and compute 𝑇𝐶_2,1(𝑡^∗_1,1,𝑀2); else go to Substep 1.2.

Substep 1.2. Find the min{𝑇𝐶_2,1(0),𝑇𝐶_2,1(𝜇)} and accordingly set 𝑡^∗_1,1,𝑀2.

Step 2. Find the global minimum of 𝑇𝐶_2,2(𝑡₁), say 𝑡^∗_1,2,𝑀2, as follows.

Substep 2.1. Compute 𝑡_1,2,𝑀2 from (57); if 𝜇<𝑡_1,2,𝑀2<𝑀₂, then set 𝑡^∗_1,2,𝑀2=𝑡_1,2,𝑀2 and compute 𝑇𝐶_2,2(𝑡^∗_1,2,𝑀2); else go to Substep 2.2.

Substep 2.2. Find the min{𝑇𝐶_2,2(𝜇),𝑇𝐶_2,2(𝑀₂)} and accordingly set 𝑡^∗_1,2,𝑀2.

Step 3. Find the global minimum of 𝑇𝐶_2,3(𝑡₁), say 𝑡^∗_1,3,𝑀2, as follows.

Substep 3.1. Compute 𝑡_1,3,𝑀2 from (59); if 𝑀₂<𝑡_1,3,𝑀2<𝑇, then set 𝑡^∗_1,3,𝑀2=𝑡_1,3,𝑀2 and compute 𝑇𝐶_2,3(𝑡^∗_1,3,𝑀2); else go to Substep 3.2.

Substep 3.2. Find the min{𝑇𝐶_2,3(𝑀₂),𝑇𝐶_2,3(𝑇)} and accordingly set 𝑡^∗_1,3,𝑀2.

Step 4. Find the min{𝑇𝐶_2,1(𝑡^∗_1,1,𝑀2),𝑇𝐶_2,2(𝑡^∗_1,2,𝑀2),𝑇𝐶_2,3(𝑡^∗_1,3,𝑀2)} and accordingly select the optimal value for 𝑡₁ say 𝑡^∗_1,𝑀2, with optimal cost 𝑇𝐶₂(𝑡^∗_1,𝑀2).

Finally to find the overall optimum 𝑡₁ for the problem under consideration, the results obtained for the two presented cases (i.e., payment is made at 𝑀₁ and payment is made at 𝑀₂) are combined; that is, find min{𝑇𝐶₁(𝑡^∗_1,𝑀1),𝑇𝐶₂(𝑡^∗_1,𝑀2)} and accordingly select the optimal value 𝑡^∗₁.

5. Numerical Examples and Sensitivity Analysis

In this section, a numerical example is provided to illustrate the results obtained in previous sections. In addition, a sensitivity analysis, with respect to system parameters, is carried out. Here, 𝑚(𝜉) is a function of the preservation technology cost 𝜉 such that 𝑚(𝜉)=𝜃(1−𝑒^−𝑎𝜉), 𝑎≥0, and 𝑎 is the simulation coefficient representing the percentage increase in 𝑚(𝜉) per euro increase in 𝜉, which means 𝑚(𝜉) is an increasing function bounded above by 𝜃.

The input parameters are 𝑐₁=3€ per unit per unit time, 𝑐₂=15€ per unit per unit time, 𝑐₃=5€ per unit, 𝑐₄=20€ per unit per unit time, 𝑟=0.005, 𝜇=0.3 years, 𝜃=0.001, 𝑎=0.001, 𝜉=20€, 𝑇=0.5 years, 𝑓(𝑡)=3𝑒^4.5𝑡 and 𝛽(𝑥)=𝑒^−0.2𝑥, 𝑀₁=0.13 years, 𝑀₂=0.43 years, 𝑝=15, 𝐶_𝑝=10€, 𝐼_𝑒=0.12, and 𝐼_𝑐=0.15.

5.1. The Payment Is Made at 𝑀₁

From (38), 𝑡_1,1,𝑀1=0.398924, which is not feasible as 𝑡_1,1,𝑀1>𝑀₁. Since 𝑇𝐶_1,1(0)=75.469 and 𝑇𝐶_1,1(𝑀₁)=71.9633, it follows that 𝑡^∗_1,1,𝑀1=𝑀₁. From (40), 𝑡_1,2,𝑀1=0.40288, which is not valid again as 𝑡_1,2,𝑀1>𝜇. Since 𝑇𝐶_1,1(𝑀₁)=𝑇𝐶_1,2(𝑀₁)=71.9633 and 𝑇𝐶_1,2(𝜇)=67.0128, the optimal value for 𝑡^∗_1,2,𝑀1=𝜇. From (42), 𝑡_1,3,𝑀1=0.412557; this value for 𝑡₁ is valid as 𝜇<𝑡_1,3,𝑀1<𝑇, so 𝑡^∗_1,3,𝑀1=𝑡_1,3,𝑀1 and 𝑇𝐶_1,3(𝑡^∗_1,3,𝑀1)=65.5819.

Finally, 𝑇𝐶₁(𝑡^∗_1,𝑀1)=min{𝑇𝐶_1,1(𝑀₁),𝑇𝐶_1,2(𝜇),𝑇𝐶_1,3(𝑡^∗_1,3,𝑀1)}=65.5819, and consequently 𝑡^∗_1,𝑀1=0.412557.

5.2. The Payment Is Made at 𝑀₂

From (55), 𝑡_1,1,𝑀2=0.424279 which is not feasible as 𝑡_1,1,𝑀2>𝜇. Since 𝑇𝐶_2,1(0)=75.6719, 𝑇𝐶_2,1(𝜇)=66.2333, it follows that 𝑡^∗_1,1,𝑀2=𝜇. From (57), 𝑡_1,2,𝑀2=0.424279 which is valid again as 𝜇<𝑡_1,2<𝑀₂ so 𝑡^∗_1,2,𝑀2=0.424279 and 𝑇𝐶_2,2(𝑡^∗_1,2,𝑀2)=64.3494. From (59), 𝑡_1,3,𝑀2=0.424197; this value for 𝑡₁ is not valid as 𝑡_1,3<𝑀₂. Since 𝑇𝐶_2,3(𝑀₂)=64.5295, 𝑇𝐶_2,3(𝑇)=65.2301 so 𝑡^∗_1,3,𝑀2=𝑀₂ and 𝑇𝐶_2,3(𝑡^∗_1,3,𝑀2)=64.5295.

Finally, 𝑇𝐶₂(𝑡^∗_1,𝑀2)=min{𝑇𝐶_2,1(𝜇),𝑇𝐶_2,2(𝑡^∗_1,2,𝑀2),𝑇𝐶_2,3(𝑀₂)}=64.3494, and consequently 𝑡^∗_1,𝑀2=0.424279.

So, as 𝑇𝐶(𝑡^∗₁)=min{𝑇𝐶₁(𝑡^∗_1,𝑀1),𝑇𝐶₂(𝑡^∗_1,𝑀2)}=64.3494, the optimal 𝑡₁ is 𝑡^∗₁=𝑡^∗_1,𝑀2=0.424279, which leads to a payment at 𝑀₂.

6. Sensitivity Analysis

Using the data of the previous example, a sensitivity analysis is carried out to explore the effect of change on the basic model’s parameters to the optimal policy (i.e., 𝑡₁ time of payment and optimal total cost). The results are presented in Tables 1 and 2, and some observations are summarized as follows.(1)It is observed that as the holding cost 𝑐₁ decreases or increases the optimal time 𝑡^∗₁ increases or decreases, and the optimal total cost 𝑇𝐶(𝑡^∗₁) decreases or increases, respectively.(2)It is seen that as the shortage cost 𝑐₂ decreases or increases both the optimal time 𝑡^∗₁ and the optimal total cost 𝑇𝐶(𝑡^∗₁) decrease or increase, respectively.(3)It is examined that as the time (at which demand becomes constant) 𝜇 decreases or increases there is no effect on the optimal time 𝑡^∗₁ but the optimal total cost 𝑇𝐶(𝑡^∗₁) decreases or increases, respectively.(4)It is observed that as the deterioration cost 𝑐₃ decreases or increases the optimal time 𝑡^∗₁ slightly increases or decreases, and the optimal total cost 𝑇𝐶(𝑡^∗₁) also slightly decreases or increases, respectively.(5)It is seen that as the lost sales cost 𝑐₄ decrease or increase both the optimal time 𝑡^∗₁ and the optimal total cost 𝑇𝐶(𝑡^∗₁) slightly decrease or increase, respectively.(6)It is seen that as the decay rate 𝜃 decreases or increases the optimal time 𝑡^∗₁ increases or decreases and the optimal total cost 𝑇𝐶(𝑡^∗₁) slightly decreases or increases, respectively.(7)It is observed that as the preservation cost 𝜉 decreases or increases the optimal time 𝑡^∗₁ and the optimal total cost 𝑇𝐶(𝑡^∗₁) decrease or increase, respectively.(8)It is noticed that as the rate of interest earned 𝐼_𝑒 decreases or increases the optimal time 𝑡^∗₁ decreases or increases, and the optimal total cost 𝑇𝐶(𝑡^∗₁) slightly increases or decreases, respectively.(9)It is noticed that as the selling price 𝑝 decreases or increases the optimal time 𝑡^∗₁ decreases or increases, but the optimal total cost 𝑇𝐶(𝑡^∗₁) increases or decreases, respectively.(10)It is seen that as the purchasing cost 𝐶_𝑝 decreases or increases the optimal time 𝑡^∗₁ increases or decreases, and the optimal total cost 𝑇𝐶(𝑡^∗₁) decreases or increases, respectively.(11)It is observed that as the preservation parameter 𝑎 decreases or increases the optimal time 𝑡^∗₁ decreases or increases, but the optimal total cost 𝑇𝐶(𝑡^∗₁) increases or decreases, respectively.

Table 1 

The effect of changing the parameter (i) while keeping all other parameters unchanged.

Table 2 

The effect of simulation coefficient (a) while keeping all other parameters unchanged.

7. Concluding Remarks

In this paper, we have developed an inventory model with ramp type demand rate, partially backlogged shortages, preservation technology, and two-level trade credit. The model is discussed under the two cases (1) when the payment is made at time 𝑀₁ to get the price discount and (2) when the payment is made at time 𝑀₂ with no price discount. The effect of preservation technology is also taken into consideration throughout the model. The presented model is illustrated through numerical experiments with sensitivity. From the sensitivity table, it is observed that the changes in the parameters preservation technology cost (𝜉), time at which demand becomes constant (𝜇), and purchasing cost (𝐶_𝑝) have greater effect on the system than the other parameters of the system. The convexity of the cost function is also exposed analytically.

This model could be inclusive taking into consideration numerous replenishment cycles throughout the planning horizon.

Acknowledgment

S. Sharma wishes to thank the Council of Scientific and Industrial Research (New Delhi, India) for providing financial help in the form of JRF vide letter no. 08/017(0017)/2011-EMR-I.