An Inventory System for Deteriorating Products with Ramp-Type Demand Rate under Two-Level Trade Credit Financing

Abstract

An inventory system for deteriorating products, with ramp-type demand rate, under two-level trade credit policy is considered. 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 highlight the obtained results, and sensitivity analysis of the optimal solution with respect to major parameters of the system is carried out.

1. Introduction

In the conventional economic order quantity (EOQ) model, it is assumed that the supplier is paid for the items immediately after the items are received. In practice, the supplier may provide to the retailer a permissible delay in payments. During this credit period, the retailer can accumulate the revenue and earn interest on that revenue. However, beyond this period the supplier charges interest on the unpaid balance. Hence, a permissible delay indirectly reduces the cost of holding stock. On the other hand, trade credit offered by the supplier encourages the retailer to buy more. Thus it is also a powerful promotional tool that attracts new customers, who consider it as an alternative incentive policy to quantity discounts. Hence, trade credit can play a major role in inventory control for both the supplier as well as the retailer (see Jaggi et al. [1]). Three types of trade credit have been appeared, mainly, in the literature: (i)a fixed trade credit period, (Goyal [2] Aggarwal and Jaggi [3], Jamal et al. [4], Chang and Dye [5], Teng [6], Jaber [7], Jaggi et al. [1], Ouyang and Cheng [8], Chung and Huang [9]);(ii)a two-level trade credit known as 𝑟/𝑀₁/𝑀₂. More precisely, 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. (Huang [10], Liao [11], Teng and Chang [12]); (iii)a trade credit period linked to the ordering quantity (Chang et al. [13], Chung and Liao [14], Ouyang et al. [15]).

For a comprehensive review for inventory lot-size models under trade credits, the reader is referred to Chang et al. [16].

In the literature referring to models with permissible 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]). In addition, the demand rate of a seasonable product increases at the beginning of the season up to a certain moment (say, 𝜇) and then remains constant for the rest of the planning horizon, 𝑇. The term “ramp-type” is used to represent such demand pattern. Hill [17] proposed an inventory model with variable branch being any power function of time. Research on this field continues with Mandal and Pal [18], Wu and Ouyang [19], and Wu [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 fall 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 [18] and Wu and Ouyang [19] and discussed both cases. Panda et al. [22] developed an inventory model for deteriorating items (with three-parameter Weibull distributed deterioration rate) with generalized exponential ramp-type demand rate and complete backlogging. Skouri et al. [23] extend the work of Deng et al. [21] by introducing a general ramp-type demand rate and Weibull deterioration rate. Panda et al. [24] presented a production-inventory model with generalized quadratic ramp-type demand rate and constant deterioration rate when shortages are not allowed. Skouri and Konstantaras [25] extended their previous work [23] studying an order level inventory model for deteriorating items based on time-dependent three branches ramp-type demand rate. Lin [26] studied an inventory model with general ramp-type demand rate, constant deterioration rate, complete backlogging, and several replenishment cycles during the finite time and used the hide-and-seek simulated annealing (SA) approach to determine the optimal replenishment policy.

This paper is an extension of the inventory system of Skouri et al. [23] assuming constant deterioration rate, when the two-level trade credit scheme, 𝑟/𝑀₁/𝑀₂, which was described above, is considered. The study of this system requires the examination of the ordering relations between the time parameters 𝑀₁,𝑀₂,𝜇,𝑇, which, actually, lead to the following different models:

(i) 𝑀₁≤𝜇<𝑀₂<𝑇, (ii) 𝑀₁<𝑀₂≤𝜇<𝑇, (iii) 𝜇≤𝑀₁<𝑀₂<𝑇, (iv) 𝜇≤𝑀₁<𝑇<𝑀₂, (v) 𝜇<𝑇≤𝑀₁<𝑀₂, (vi) 𝑀₁≤𝜇<𝑇≤𝑀₂.

Note that from the definition of demand rate 𝜇<𝑇 and from credit scheme 𝑀₁<𝑀₂.

This study can be used: (1) for the determination of the optimal replenishment policy under a specific trade credit settings (corresponding to one of the six models mentioned above) and (2) for supplier’ selection, since it is obvious that the ordering of the parameters 𝜇, 𝑀₁, 𝑀₂, 𝑇 leads to different trade credit offers. Although the analysis of all models is available upon request, in order to reduce the length of the paper, only the first model will be presented.

The paper is organized as follows: the notation and assumptions used are given in Section 2. In Section 3, the quantities and functions, which are common to each of the possible models are derived. The mathematical formulation of the first model and the determination of the optimal policy are provided in Section 4. In Section 5, numerical examples highlighting the results obtained are given, and sensitivity analysis with respect to major parameters of the system is carried out. The paper closes with concluding remarks in Section 6.

2. Notation and Assumptions

The following notation is used through the paper.

2.1. Notation

𝑇is the constant scheduling period (cycle),𝑡₁the time when the inventory level falls to zero, 𝑆the maximum inventory level at each scheduling period (cycle),𝐶_𝑝the unit purchase cost,𝑐₁the inventory holding cost per unit per unit time,𝑐₂the shortage cost per unit per unit time,𝑐₃the cost incurred from the deterioration of one unit,𝑐₄the per unit opportunity cost due to the lost sales (𝑐₄>𝐶_𝑝 see Teng et al. [27]),𝑝the unit selling price,𝐼_𝑒the interest rate earned, 𝐼_𝑐the interest rate charged, 𝑟cash discount rate, 0<𝑟<1,𝑀₁the period of cash discount in years, 𝑀₂the period of permissible delay in payments in years, 𝑀₁<𝑀₂,𝜇the parameter of the ramp-type demand function (time point), and 𝐼(𝑡)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)+𝐶₂𝑇𝛽^(𝑥)+𝐶_𝑝𝛽^(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 [10]).(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 by𝐷(𝑡)=𝑓𝑓((𝑡𝜇)),,𝑡𝑡<≥𝜇𝜇,,(2.1) where 𝑓(𝑡) is a positive, differentiable function of 𝑡∈(0,𝑇].

3. Deriving the Common Quantities for the Inventory Models

In this section, common quantities entering to all models will be derived. Note that these quantities are affected only by the ordering relations between 𝑡₁ and 𝜇. The inventory level 𝐼(𝑡), 0≤𝑡≤𝑇 satisfies the following differential equations:𝑑𝐼(𝑡)𝑑𝑡+𝜃𝐼(𝑡)=−𝐷(𝑡),0≤𝑡≤𝑡¹(3.1) with boundary condition 𝐼(𝑡₁)=0 and𝑑𝐼(𝑡)𝑑𝑡=−𝐷(𝑡)𝛽(𝑇−𝑡),𝑡¹≤𝑡≤𝑇(3.2) with boundary condition 𝐼(𝑡₁)=0.

From the two possible relations between parameters 𝑡₁ and 𝜇, (i) 𝑡₁≤𝜇 and (ii) 𝑡₁>𝜇, and following identical steps as in Skouri et al. [23], the sum of holding, deterioration, shortages, and lost sales cost is obtained as𝐶𝑡₁=𝐶¹𝑡₁if𝑡₁𝐶≤𝜇,₂𝑡₁if𝑡₁>𝜇,(3.3) where𝐶₁𝑡₁=𝑐₁^𝑡1₀𝑒^−𝜃𝑡^𝑡1_𝑡+𝑐₂𝑓(𝑥)𝑒^𝜃𝑥𝑑𝑥𝑑𝑡^𝜇𝑡₁(𝜇−𝑡)𝑓(𝑡)𝛽(𝑇−𝑡)𝑑𝑡+𝑓(𝜇)^𝑇𝜇^𝑡𝜇+𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡_𝑇𝜇^𝜇𝑡₁+𝑐₃𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥𝑑𝑡^𝑡1₀+𝑐₄𝑓(𝑡)𝑒^𝜃𝑡𝑑𝑡−^𝑡1₀𝑓(𝑡)𝑑𝑡^𝜇𝑡₁𝐶(1−𝛽(𝑇−𝑡))𝑓(𝑡)𝑑𝑡+𝑓(𝜇)^𝑇𝜇(1−𝛽(𝑇−𝑡))𝑑𝑡,(3.4)₂𝑡₁=𝑐₁+^𝜇0𝑓𝑒(𝜇^−𝜃𝑡)^𝜇𝑡_𝑡1𝜇𝑓(𝑥)𝑒^𝜃𝑥𝑑𝑥+𝑓(𝜇)^𝑡1_𝜇𝑒^𝜃𝑥𝑑𝑥𝑑𝑡𝑒^−𝜃𝑡^𝑡1_𝑡+𝑐₃𝑒^𝜃𝑥𝑑𝑥𝑑𝑡^𝜇0+𝑐₂𝑓(𝑡)𝑒^𝜃𝑡𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^𝜃𝑡𝑑𝑡−^𝜇0𝑓(𝑡)𝑑𝑡−𝑓(𝜇)𝑡₁−𝜇𝑓(𝜇)^𝑇𝑡₁(𝑇−𝑥)𝛽(𝑇−𝑥)𝑑𝑥+𝑐₄𝑓(𝜇)^𝑇𝑡₁(1−𝛽(𝑇−𝑡))𝑑𝑡.(3.5)

4. Model I—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_𝐴1,1𝑡₁=𝐶_𝑝(1−𝑟)^𝑡1₀𝑓(𝑥)𝑒^𝜃𝑥𝑑𝑥+𝑓(𝜇)^𝑇𝜇𝛽(𝑇−𝑥)𝑑𝑥+^𝜇𝑡₁𝑓(𝑥)𝛽(𝑇−𝑥)𝑑𝑥.(4.1) The interest earned during the period of positive inventory level is 𝐼_𝑇1,1𝑡₁=𝑝𝐼_𝑒^𝑡1₀^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼_𝑒𝑀₁−𝑡₁^𝑡1₀𝑓(𝑥)𝑑𝑥.(4.2) Since 𝑡₁≤𝜇, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.1), and (4.2) 𝑇𝐶₁₁𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,1𝑡₁−𝐼_𝑇1,1𝑡₁.(4.3)Subcase 1.2 (𝑀₁<𝑡₁≤𝜇<𝑇). The purchasing cost is 𝐶_𝐴1,1 (relation (4.1)).
The interest payable for the inventory not being sold after the due date 𝑀₁ is 𝑃_𝑇2,1𝑡₁=𝐶_𝑝(1−𝑟)𝐼_𝑐^𝑡1_𝑀1𝑒^−𝜃𝑡^𝑡1_𝑡𝑓(𝑥)𝑒^𝜃𝑥𝑑𝑥𝑑𝑡.(4.4) The interest earned is 𝐼_𝑇2,1𝑡₁=𝑝𝐼_𝑒^𝑡1₀^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡.(4.5) Since again 𝑡₁≤𝜇_, the total cost over [0,𝑇] is calculated using the relations (3.4), (4.1), (4.4), and (4.6) and is 𝑇𝐶₁₂𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,1𝑡₁+𝑃_𝑇2,1𝑡₁−𝐼_𝑇2,1𝑡₁.(4.6)Subcase 1.3 (𝑀₁≤𝜇≤𝑡₁≤𝑇). The purchasing cost is 𝐶_𝐴2,1𝑡₁=𝐶_𝑝(1−𝑟)^𝜇0𝑓(𝑥)𝑒^𝜃𝑥𝑑𝑥+𝑓(𝜇)^𝑡1_𝜇𝑒^𝜃𝑥𝑑𝑥+𝑓(𝜇)^𝑇𝑡₁𝛽(𝑇−𝑥)𝑑𝑥.(4.7) The interest earned, 𝐼_𝑇3,1, is: 𝐼_𝑇3,1𝑡₁=𝑝𝐼_𝑒^𝜇0^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑡1_𝜇^𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑡1_𝜇^𝑡𝜇𝑓(𝜇)𝑑𝑥𝑑𝑡.(4.8) The interest payable for the inventory not being sold after the due date 𝑀₁ is 𝑃_𝑇3,1𝑡₁=𝐶_𝑝(1−𝑟)𝐼_𝑐^𝜇𝑀₁𝑒^−𝜃𝑡^𝜇𝑡+𝑓(𝜇)_𝜇𝑀1𝑒^𝜃𝑥𝑓(𝑥)𝑑𝑥𝑑𝑡𝑒^−𝜃𝑡^𝑡1_𝜇𝑒^𝜃𝑥𝑑𝑥𝑑𝑡+𝑓(𝜇)^𝑡1_𝜇𝑒^−𝜃𝑡^𝑡1_𝑡𝑒^𝜃𝑥𝑑𝑥𝑑𝑡.(4.9) Since 𝜇<𝑡₁, the total cost over [0,𝑇] is again calculated from (3.5), (4.7)–(4.9) and is 𝑇𝐶₁₃𝑡₁=𝐶₂𝑡₁+𝐶_𝐴2,1𝑡₁+𝑃_𝑇3,1𝑡₁−𝐼_𝑇3,1𝑡₁.(4.10)
The results obtained lead to the following total cost function: 𝑇𝐶₁𝑡₁=⎧⎪⎨⎪⎩𝑇𝐶^1,1𝑇𝐶_1,2𝑡₁,𝑡₁≤𝑀₁,𝑇𝐶_1,3𝑡₁,𝑀₁<𝑡₁≤𝜇,𝑡₁,𝜇≤𝑡₁.(4.11) So the problem is min_𝑡1𝑇𝐶₁𝑡₁.(4.12) Its solution requires, separately, studying each of 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) is 𝑑𝑇𝐶_1,1𝑡₁𝑑𝑡₁=𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1−𝑐₄−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁1−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒𝑀₁−𝑡₁+𝐶_𝑝(1−𝑟)𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁𝑓𝑡₁=0.(4.13) Since 𝑑𝑇𝐶_1,1(0)/𝑑𝑡₁<0 and 𝑑𝑇𝐶_1,1(𝑇)/𝑑𝑡₁>0, (4.13) has at least one root. So if 𝑡_1,1 is the root of (4.13), this corresponds to minimum since 𝑑𝑇𝐶^21,1(𝑡₁)𝑑𝑡²¹|||||_{𝑡1=𝑡1,1}=𝑓𝑡_1,1−𝑐₄𝑐₁+𝑐₃𝜃𝑒^𝜃𝑡1,1+𝑐₂𝛽𝑇−𝑡_1,1+𝑐₂𝑇−𝑡_1,1𝛽^𝑇−𝑡_1,1𝛽′𝑇−𝑡_1,1+𝑝𝐼_𝑒+𝐶_𝑝(1−𝑟)𝜃𝑒^𝜃𝑡1,1+𝛽′𝑇−𝑡_1,1>0.(4.14) 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_1,2𝑡₁𝑑𝑡₁=𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1+𝐶_𝑝−1𝑓𝑡₁−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁𝑓𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁𝑓𝑡₁(1−𝑟)𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁+𝐶^𝑝(1−𝑟)𝐼_𝑐𝜃𝑒^{𝜃(𝑡1−𝑀1)}−𝑝𝐼_𝑒−1𝑓𝑡₁^𝑡1₀𝑓(𝑥)𝑑𝑥=0.(4.15) Let us set ℎ(𝑥)=𝑐₂𝑥𝛽(𝑥)+𝑐₄(1−𝛽(𝑥))−𝑝𝐼_𝑒𝑥. If 𝑡_1,2 is the root of (4.15) (this may or may not exist), 𝑓(𝑥) is an increasing function and further if ℎ^(𝑥)>0, then 𝑑²𝑇𝐶_1,2𝑡₁𝑑𝑡²¹||||_{𝑡1=𝑡1,2}=+𝑝𝐼𝑓_𝑒𝑓𝑡^_1,2𝑡_1,2^𝑡1,2₀𝑓(𝑥)𝑑𝑥−𝑐₁𝑐₄+𝑐₃𝜃𝑒^𝜃𝑡1,2+𝑐₂𝛽𝑇−𝑡_1,2+𝑇−𝑡_1,2𝛽^𝑇−𝑡_1,2𝛽^𝑇−𝑡_1,2+𝐶_𝑝(1−𝑟)𝐼_𝑐𝑒^{𝜃(𝑡1,2−𝑀1)}+𝐶_𝑝(1−𝑟)𝜃𝑒^𝜃𝑡1,2+𝛽^𝑇−𝑡_1,2−𝑝𝐼_𝑒>0,(4.16) 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_1,3𝑡₁𝑑𝑡₁=𝑓(𝜇)𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1−𝑝𝐼_𝑒−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁𝑡₁−𝜇+𝐶^𝑝(1−𝑟)𝐼_𝑐𝜃𝑒^{𝜃(𝑡1−𝑀1)}−𝑝𝐼_𝑒−1+𝐶_𝑝(1−𝑟)𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁^𝜇0𝑓(𝑥)𝑑𝑥=0.(4.17) If 𝑡_1,3 is the root of (4.17) (this may or may not exist) and ℎ^(𝑥)>0, then 𝑑²𝑇𝐶_1,3𝑡₁𝑑𝑡²¹=−𝑐𝑝₁𝐼+_𝑒𝑐₃𝜃𝑒^𝜃𝑡1+𝑐₂𝛽𝑇−𝑡₁+𝑇−𝑡₁𝛽^𝑇−𝑡₁−𝑐₄𝛽^𝑇−𝑡₁+𝐶_𝑝(1−𝑟)𝐼_𝑐𝑒^{𝜃(𝑡1−𝑀1)}+𝐶_𝑝(1−𝑟)𝜃𝑒^𝜃𝑡1+𝛽^𝑇−𝑡₁𝑓(𝜇)>0,(4.18) this 𝑡_1,3 corresponds to unconstrained minimum of 𝑇𝐶_1,3(𝑡₁).

Remark 4.1. 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 (4.13); 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,𝑀1 from (4.15); if 𝑀₁<𝑡_1,2,𝑀1<𝜇, then set 𝑡^∗1,2,𝑀₁=𝑡_1,2,𝑀1 and compute 𝑇𝐶₂(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 (4.17); if 𝜇<𝑡_1,3,𝑀1, then set 𝑡^∗1,3,𝑀₁=𝑡_1,3,𝑀1and 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 𝐶₁(𝑡_1,𝑀1).

Case 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_𝐴1,2𝑡₁=𝐶^𝐴1,1𝑡₁1−𝑟.(4.19) The interest earned during the period of positive inventory level is. 𝐼_𝑇1,2𝑡₁=𝑝𝐼_𝑒^𝑡1₀^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼_𝑒𝑀₂−𝑡₁^𝑡1₀𝑓(𝑥)𝑑𝑥.(4.20) Since 𝑡₁≤𝜇, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.19), and (4.20) 𝑇𝐶_2,1𝑡₁=𝐶₁𝑡₁+𝐶_𝐴1,2𝑡₁−𝐼_𝑇1,2𝑡₁.(4.21)Subcase 2.2 (𝜇<𝑡₁≤𝑀₂<𝑇). The purchasing cost is 𝐶_𝐴2,2=𝐶^𝐴2,1𝑡₁1−𝑟.(4.22) The interest earned is 𝐼_𝑇2,2𝑡₁=𝑝𝐼_𝑒^𝜇0^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑀2_𝜇^𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑡1_𝜇^𝑡𝜇+𝑓(𝜇)𝑑𝑥𝑑𝑡_𝑀2𝑡1^𝑡1_𝜇𝑓(𝜇)𝑑𝑥𝑑𝑡.(4.23) Since again 𝜇≤𝑡₁, the total cost over [0,𝑇] is calculated using the relations (3.5), (4.22), and (4.23) and is 𝑇𝐶_2,2𝑡₁=𝐶₂𝑡₁+𝐶_𝐴2,2𝑡₁−𝐼_𝑇2,2𝑡₁.(4.24)Subcase 2.3 (𝜇≤𝑀₂≤𝑡₁≤𝑇). The purchasing cost is 𝐶_𝐴2,2(𝑡₁).
The interest earned, 𝐼_𝑇3,2, is 𝐼_𝑇3,2𝑡₁=𝑝𝐼_𝑒^𝜇0^𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑡1_𝜇^𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+^𝑡1_𝜇^𝑡𝜇𝑓(𝜇)𝑑𝑥𝑑𝑡.(4.25) The interest payable for the inventory not being sold after the due date 𝑀₂ is 𝑃_𝑇3,2𝑡₁=𝐶_𝑝𝐼_𝑐𝑓(𝜇)^𝑡1_𝑀2𝑒^−𝜃𝑡^𝑡1_𝑡𝑒^𝜃𝑥𝑑𝑥𝑑𝑡.(4.26) Since 𝜇<𝑡₁, the total cost over [0,𝑇] is again calculated from (3.5), (4.22), (4.25), and (4.26) and is 𝑇𝐶_2,3𝑡₁=𝐶₂𝑡₁+𝐶_𝐴2,2𝑡₁+𝑃_𝑇3,2𝑡₁−𝐼_𝑇3,2𝑡₁.(4.27)
The results obtained lead to the following total cost function: 𝑇𝐶₂𝑡₁=⎧⎪⎨⎪⎩𝑇𝐶^2,1𝑇𝐶_2,2𝑡₁,𝑡₁≤𝜇,𝑇𝐶_2,3𝑡₁,𝜇<𝑡₁≤𝑀₂,𝑡₁,𝑀₂≤𝑡₁.(4.28) So the problem is min_𝑡1𝑇𝐶₂𝑡₁.(4.29) Its solution, as in the previous case, requires, separately, studying each of 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_2,1𝑡₁𝑑𝑡₁=𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1−𝑝𝐼_𝑒−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁𝑀₂−𝑡₁+𝐶_𝑝𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁𝑓𝑡₁=0.(4.30) Since 𝑑𝑇𝐶_2,1(0)/𝑑𝑡₁<0 and 𝑑𝑇𝐶_2,1(𝑇)/𝑑𝑡₁>0, (4.30) has at least one root. So if 𝑡_1,1 is the root of (4.30), this corresponds to minimum as 𝑑𝑇𝐶^22,1𝑡₁𝑑𝑡²¹|||||_{𝑡1=𝑡1,1}=𝑓𝑡_1,1−𝑐₄𝑐₁+𝑐₃𝜃𝑒^𝜃𝑡1,1+𝑐₂𝛽𝑇−𝑡_1,1+𝑐₂𝑇−𝑡_1,1𝛽^𝑇−𝑡_1,1𝛽^𝑇−𝑡_1,1+𝑝𝐼_𝑒+𝐶_𝑝𝜃𝑒^𝜃𝑡1,1+𝛽^𝑇−𝑡_1,1>0.(4.31) 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_2,2𝑡₁𝑑𝑡₁=𝑓(𝜇)𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1−𝑝𝐼_𝑒−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁𝑀₂−𝑡₁+𝐶_𝑝𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁=0.(4.32) If 𝑡_1,2 is the root of (4.32) (this may or may not exist), this corresponds to unconstrained minimum of 𝑇𝐶_2,2(𝑡₁) as 𝑑𝑇𝐶^22,2𝑡₁𝑑𝑡²¹|||||_{𝑡1=𝑡1,2}=𝑓(𝜇)−𝑐𝑐₁₄+𝑐₃𝜃𝑒^𝜃𝑡1,2+𝑐₂𝛽𝑇−𝑡_1,2+𝑐₂𝑇−𝑡_1,2𝛽^𝑇−𝑡_1,2𝛽^𝑇−𝑡_1,2+𝑝𝐼_𝑒+𝐶_𝑝𝜃𝑒^𝜃𝑡1,2+𝛽^𝑇−𝑡_1,2>0.(4.33) The first-order condition for a minimum of 𝑇𝐶_2,3(𝑡₁) is 𝑑𝑇𝐶_2,3𝑡₁𝑑𝑡₁=𝑓(𝜇)𝑐¹+𝑐₃𝜃𝜃𝑒^𝜃𝑡1+𝐶−1−𝑐₂𝑇−𝑡₁𝛽𝑇−𝑡₁−𝑐₄1−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒𝑡₁−𝜇_𝑝𝐼_𝑐𝜃𝑒^{𝜃(𝑡1−𝑀2)}−1+𝐶_𝑝𝑒^𝜃𝑡1−𝛽𝑇−𝑡₁−𝑝𝐼_𝑒^𝜇0𝑓(𝑥)𝑑𝑥=0.(4.34) If 𝑡_1,3 is a root of (4.34) (this may or may not exist) and 𝑐₁+𝑐₃𝜃+𝐶_𝑝𝐼_𝑐≥𝑝𝐼_𝑒 this corresponds to unconstrained minimum of 𝑇𝐶_2,3(𝑡₁) as 𝑑𝑇𝐶^22,3𝑡₁𝑑𝑡²¹=−𝑐𝑐₁₄+𝑐₃𝜃+𝐶_𝑝𝐼_𝑐𝑒^𝜃𝑡1+𝑐₂𝛽𝑇−𝑡₁+𝑐₂𝑇−𝑡₁𝛽^𝑇−𝑡₁𝛽^𝑇−𝑡₁−𝑝𝐼_𝑒+𝐶_𝑝𝜃𝑒^𝜃𝑡1+𝛽^𝑇−𝑡₁𝑓(𝜇).(4.35)

Remark 4.2. The function 𝑇𝐶₂(𝑡₁) is not differentiable in 𝑀₂.
The procedure for the determination of the optimal replenishment policy when payment is made at time 𝑀₂ is as follows.

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 (4.30); 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 (4.32); 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 (4.34); 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 some important model’s parameters, is carried out.

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, 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, 𝐼_𝑐=0.15.

5.1. The Payment Is Made at 𝑀₁

From (4.13), 𝑡_1,1,𝑀1=0.399, which is not feasible as 𝑡_1,1,𝑀1>𝑀₁. Since 𝑇𝐶_1,1(0)=55.469 and 𝑇𝐶_1,1(𝑀₁)=51.9633, it follows that 𝑡^∗1,1,𝑀₁=𝑀₁. From (4.15), 𝑡_1,2,𝑀1=0.426, which is not valid again as 𝑡_1,2,𝑀1>𝜇. Since 𝑇𝐶_1,1(𝑀₁)=𝑇𝐶_1,2(𝑀₁)=51.9633 and 𝑇𝐶_1,2(𝜇)=46.6669, the optimal value for 𝑡^∗_11,2,𝑀=𝜇. From (4.17) 𝑡_1,3,𝑀1=0.423; 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)=44.8287.

Finally 𝑇𝐶_1,3(𝑡^∗1,3,𝑀₁)=min{𝑇𝐶_1,1(𝑀₁),𝑇𝐶_1,2(𝜇),𝑇𝐶_1,3(𝑡^∗1,3,𝑀₁)}=44.8287 and consequently 𝑡^∗1,𝑀₁=0.423.

5.2. The Payment Is Made at 𝑀₂

From (4.30), 𝑡_1,1,𝑀2=0.424 which is not feasible as 𝑡_1,1,𝑀2>𝜇. Since 𝑇𝐶_2,1(0)=55.6719 𝜅𝛼𝜄  𝑇𝐶_2,1(𝜇)=46.2334, it follows that 𝑡^∗1,1,𝑀₂=𝜇. From (4.32), 𝑡_1,2,𝑀2=0.424 which is valid again as 𝜇<𝑡_1,2<𝑀₂ so 𝑡^∗1,2,𝑀₂=0.424 and 𝑇𝐶_2,2(𝑡^∗1,2,𝑀₂)=44.3497. From (4.34), 𝑡_1,3,𝑀2=0.451; this value for 𝑡₁ is also valid as 𝜇<𝑀₂<𝑡_1,3<𝑇 so 𝑡^∗1,3,𝑀₁=𝑡_1,3,𝑀1 and 𝑇𝐶_2,3(𝑡^∗1,3,𝑀₁)=44.3039.

Finally 𝑇𝐶_2,3(𝑡^∗1,3,𝑀₂)=min{𝑇𝐶_2,1(𝜇),𝑇𝐶_2,2(𝑡^∗1,3,𝑀₂),𝑇𝐶_2,3(𝑡^∗1,3,𝑀₂)}=44.3039, and consequently 𝑡^∗1,𝑀₂=0.451.

So, as 𝑇𝐶₂(𝑡_1,𝑀2)=min{𝑇𝐶₁(𝑡_1,𝑀1),𝑇𝐶₂(𝑡_1,𝑀2)}, the optimal 𝑡₁ is 𝑡^∗1=𝑡^∗1,𝑀₂=0.451, which leads to a payment at 𝑀₂.

Using the data of the previous example, a sensitivity analysis is carried out to explore the effect of change on some, of the basic, model’s parameters (𝜇,𝑀₁,𝑀₂,𝑇,𝑟) to the optimal policy (i.e., 𝑡₁ time of payment and optimal total cost). The results are presented in Table 1 and some interesting findings are summarized as follows.(1)The changes of parameters 𝑀₁, 𝑀₂ and 𝑟 have no impact on the optimal 𝑡₁, the time of payment and the optimal cost.(2)The error on the parameters’ estimation of 𝜇 has no impact on the time of payment, small impact on the optimal 𝑡₁, but high impact on the total optimal cost. This last observation is in line with the relative findings in Deng et al. [21].

Table 1 

Sensitivity analysis: the effect of changing the parameter (i) keeping all other parameters unchanged.

6. Conclusions

In this paper, the following interrelated factors, which have appeared in the literature of inventory control, are incorporated: (i) the product’s life cycle, which implies that its demand can be described as a ramp-type function of time, (ii) the effect of deterioration, (iii) the 𝑟/𝑀₁/𝑀₂ credit scheme, which can be offered by supplier to the retailer for stimulating the demand, and (iv) the diminished, with the waiting time, backlogging rate, which is described as a decreasing function of time. As a result, this paper is a modification of the inventory system presented by Skouri et al. [23] when the 𝑟/𝑀₁/𝑀₂ credit scheme is considered. The study of this system requires the examination of the ordering relations between the time parameters 𝑀₁, 𝑀₂, 𝜇, 𝑇, which, actually, lead to the six different models. This inventory system, setting 𝑓(𝑡)=𝐷₀𝑡, 𝛽(𝑥)=1, 𝑀₁=𝑀₂=0, 𝐼_𝑝=0, and 𝐼_𝑐=0, can give as special cases the ones presented by Mandal and Pal [18], Wu and Ouyang [19], and Deng et al. [21]. This model could be extended assuming several replenishment cycles during the planning horizon. For this extension, the application of some popular heuristic optimization algorithm (like Particle Swarm Optimization or Differential Evolution) may be useful, [28–30].

Acknowledgment

The authors thank the research committee of the University of Ioannina for the financial support.