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Advances in Decision Sciences
Volume 2011 (2011), Article ID 761961, 15 pages
http://dx.doi.org/10.1155/2011/761961
Research Article

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

Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

Received 27 December 2010; Accepted 13 June 2011

Academic Editor: Henry Schellhorn

Copyright © 2011 G. Darzanou and K. Skouri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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) 𝑀1𝜇<𝑀2<𝑇, (ii) 𝑀1<𝑀2𝜇<𝑇, (iii) 𝜇𝑀1<𝑀2<𝑇, (iv) 𝜇𝑀1<𝑇<𝑀2, (v) 𝜇<𝑇𝑀1<𝑀2, (vi) 𝑀1𝜇<𝑇𝑀2.

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),𝐶𝑝the 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>𝐶𝑝 see Teng et al. [27]),𝑝the unit selling price,𝐼𝑒the interest rate earned, 𝐼𝑐the 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)+𝐶𝑝𝛽(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)=𝑓(𝑡),𝑡<𝜇,𝑓(𝜇),𝑡𝜇,(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), 0𝑡𝑇 satisfies the following differential equations:𝑑𝐼(𝑡)𝑑𝑡+𝜃𝐼(𝑡)=𝐷(𝑡),0𝑡𝑡1(3.1) with boundary condition 𝐼(𝑡1)=0 and𝑑𝐼(𝑡)𝑑𝑡=𝐷(𝑡)𝛽(𝑇𝑡),𝑡1𝑡𝑇(3.2) with boundary condition 𝐼(𝑡1)=0.

From the two possible relations between parameters 𝑡1 and 𝜇, (i) 𝑡1𝜇 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 as𝐶𝑡1=𝐶1𝑡1if𝑡1𝐶𝜇,2𝑡1if𝑡1>𝜇,(3.3) where𝐶1𝑡1=𝑐1𝑡10𝑒𝜃𝑡𝑡1𝑡𝑓(𝑥)𝑒𝜃𝑥𝑑𝑥𝑑𝑡+𝑐2𝜇𝑡1(𝜇𝑡)𝑓(𝑡)𝛽(𝑇𝑡)𝑑𝑡+𝑓(𝜇)𝑇𝜇𝑡𝜇+𝛽(𝑇𝑥)𝑑𝑥𝑑𝑡𝑇𝜇𝜇𝑡1𝑓(𝑥)𝛽(𝑇𝑥)𝑑𝑥𝑑𝑡+𝑐3𝑡10𝑓(𝑡)𝑒𝜃𝑡𝑑𝑡𝑡10𝑓(𝑡)𝑑𝑡+𝑐4𝜇𝑡1(1𝛽(𝑇𝑡))𝑓(𝑡)𝑑𝑡+𝑓(𝜇)𝑇𝜇,𝐶(1𝛽(𝑇𝑡))𝑑𝑡(3.4)2𝑡1=𝑐1𝜇0𝑒𝜃𝑡𝜇𝑡𝑓(𝑥)𝑒𝜃𝑥𝑑𝑥+𝑓(𝜇)𝑡1𝜇𝑒𝜃𝑥𝑑𝑥𝑑𝑡+𝑓(𝜇)𝑡1𝜇𝑒𝜃𝑡𝑡1𝑡𝑒𝜃𝑥𝑑𝑥𝑑𝑡+𝑐3𝜇0𝑓(𝑡)𝑒𝜃𝑡𝑑𝑡+𝑓(𝜇)𝑡1𝜇𝑒𝜃𝑡𝑑𝑡𝜇0𝑡𝑓(𝑡)𝑑𝑡𝑓(𝜇)1𝜇+𝑐2𝑓(𝜇)𝑇𝑡1(𝑇𝑥)𝛽(𝑇𝑥)𝑑𝑥+𝑐4𝑓(𝜇)𝑇𝑡1.(1𝛽(𝑇𝑡))𝑑𝑡(3.5)

4. Model I—The Inventory Model When 𝑀1𝜇<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 (𝑡1𝑀1𝜇<𝑇). The purchasing cost is 𝐶𝐴1,1𝑡1=𝐶𝑝(1𝑟)𝑡10𝑓(𝑥)𝑒𝜃𝑥𝑑𝑥+𝑓(𝜇)𝑇𝜇𝛽(𝑇𝑥)𝑑𝑥+𝜇𝑡1.𝑓(𝑥)𝛽(𝑇𝑥)𝑑𝑥(4.1) The interest earned during the period of positive inventory level is 𝐼𝑇1,1𝑡1=𝑝𝐼𝑒𝑡10𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼𝑒𝑀1𝑡1𝑡10𝑓(𝑥)𝑑𝑥.(4.2) Since 𝑡1𝜇, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.1), and (4.2) 𝑇𝐶11𝑡1=𝐶1𝑡1+𝐶𝐴1,1𝑡1𝐼𝑇1,1𝑡1.(4.3)Subcase 1.2 (𝑀1<𝑡1𝜇<𝑇). The purchasing cost is 𝐶𝐴1,1 (relation (4.1)).
The interest payable for the inventory not being sold after the due date 𝑀1 is 𝑃𝑇2,1𝑡1=𝐶𝑝(1𝑟)𝐼𝑐𝑡1𝑀1𝑒𝜃𝑡𝑡1𝑡𝑓(𝑥)𝑒𝜃𝑥𝑑𝑥𝑑𝑡.(4.4) The interest earned is 𝐼𝑇2,1𝑡1=𝑝𝐼𝑒𝑡10𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡.(4.5) Since again 𝑡1𝜇, the total cost over [0,𝑇] is calculated using the relations (3.4), (4.1), (4.4), and (4.6) and is 𝑇𝐶12𝑡1=𝐶1𝑡1+𝐶𝐴1,1𝑡1+𝑃𝑇2,1𝑡1𝐼𝑇2,1𝑡1.(4.6)
Subcase 1.3 (𝑀1𝜇𝑡1𝑇). The purchasing cost is 𝐶𝐴2,1𝑡1=𝐶𝑝(1𝑟)𝜇0𝑓(𝑥)𝑒𝜃𝑥𝑑𝑥+𝑓(𝜇)𝑡1𝜇𝑒𝜃𝑥𝑑𝑥+𝑓(𝜇)𝑇𝑡1.𝛽(𝑇𝑥)𝑑𝑥(4.7) The interest earned, 𝐼𝑇3,1, is: 𝐼𝑇3,1𝑡1=𝑝𝐼𝑒𝜇0𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑡1𝜇𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑡1𝜇𝑡𝜇.𝑓(𝜇)𝑑𝑥𝑑𝑡(4.8) The interest payable for the inventory not being sold after the due date 𝑀1 is 𝑃𝑇3,1𝑡1=𝐶𝑝(1𝑟)𝐼𝑐𝜇𝑀1𝑒𝜃𝑡𝜇𝑡𝑒𝜃𝑥𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑓(𝜇)𝜇𝑀1𝑒𝜃𝑡𝑡1𝜇𝑒𝜃𝑥𝑑𝑥𝑑𝑡+𝑓(𝜇)𝑡1𝜇𝑒𝜃𝑡𝑡1𝑡𝑒𝜃𝑥.𝑑𝑥𝑑𝑡(4.9) Since 𝜇<𝑡1, the total cost over [0,𝑇] is again calculated from (3.5), (4.7)–(4.9) and is 𝑇𝐶13𝑡1=𝐶2𝑡1+𝐶𝐴2,1𝑡1+𝑃𝑇3,1𝑡1𝐼𝑇3,1𝑡1.(4.10)
The results obtained lead to the following total cost function: 𝑇𝐶1𝑡1=𝑇𝐶1,1𝑡1,𝑡1𝑀1,𝑇𝐶1,2𝑡1,𝑀1<𝑡1𝜇,𝑇𝐶1,3𝑡1,𝜇𝑡1.(4.11) So the problem is min𝑡1𝑇𝐶1𝑡1.(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 𝑇𝐶1,1(𝑡1) is 𝑑𝑇𝐶1,1𝑡1𝑑𝑡1=𝑐1+𝑐3𝜃𝜃𝑒𝜃𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐41𝛽𝑇𝑡1𝑝𝐼𝑒𝑀1𝑡1+𝐶𝑝𝑒(1𝑟)𝜃𝑡1𝛽𝑇𝑡1𝑓𝑡1=0.(4.13) Since 𝑑𝑇𝐶1,1(0)/𝑑𝑡1<0 and 𝑑𝑇𝐶1,1(𝑇)/𝑑𝑡1>0, (4.13) has at least one root. So if 𝑡1,1 is the root of (4.13), this corresponds to minimum since 𝑑𝑇𝐶21,1(𝑡1)𝑑𝑡21|||||𝑡1=𝑡1,1𝑡=𝑓1,1𝑐1+𝑐3𝜃𝑒𝜃𝑡1,1+𝑐2𝛽𝑇𝑡1,1+𝑐2𝑇𝑡1,1𝛽𝑇𝑡1,1𝑐4𝛽𝑇𝑡1,1+𝑝𝐼𝑒+𝐶𝑝(1𝑟)𝜃𝑒𝜃𝑡1,1+𝛽𝑇𝑡1,1>0.(4.14) Consequently, 𝑡1,1 is the unique unconstrained minimum of 𝑇𝐶1,1(𝑡1).
The first-order condition for a minimum of 𝑇𝐶1,2(𝑡1) is 𝑑𝑇𝐶1,2𝑡1𝑑𝑡1=𝑐1+𝑐3𝜃𝜃𝑒𝜃𝑡1𝑓𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑓𝑡1𝑐41𝛽𝑇𝑡1𝑓𝑡1+𝐶𝑝𝑒(1𝑟)𝜃𝑡1𝛽𝑇𝑡1+𝐶𝑝(1𝑟)𝐼𝑐𝜃𝑒𝜃(𝑡1𝑀1)𝑓𝑡11𝑝𝐼𝑒𝑡10𝑓(𝑥)𝑑𝑥=0.(4.15) Let us set (𝑥)=𝑐2𝑥𝛽(𝑥)+𝑐4(1𝛽(𝑥))𝑝𝐼𝑒𝑥. If 𝑡1,2 is the root of (4.15) (this may or may not exist), 𝑓(𝑥) is an increasing function and further if (𝑥)>0, then 𝑑2𝑇𝐶1,2𝑡1𝑑𝑡21||||𝑡1=𝑡1,2=𝑝𝐼𝑒𝑓𝑡1,2𝑡1,20𝑡𝑓(𝑥)𝑑𝑥+𝑓1,2𝑐1+𝑐3𝜃𝑒𝜃𝑡1,2+𝑐2𝛽𝑇𝑡1,2+𝑇𝑡1,2𝛽𝑇𝑡1,2𝑐4𝛽𝑇𝑡1,2+𝐶𝑝(1𝑟)𝐼𝑐𝑒𝜃(𝑡1,2𝑀1)+𝐶𝑝(1𝑟)𝜃𝑒𝜃𝑡1,2+𝛽𝑇𝑡1,2𝑝𝐼𝑒>0,(4.16) and this 𝑡1,2 corresponds to unconstrained minimum of 𝑇𝐶1,2(𝑡1).
The first-order condition for a minimum of 𝑇𝐶1,3(𝑡1) is 𝑑𝑇𝐶1,3𝑡1𝑑𝑡1𝑐=𝑓(𝜇)1+𝑐3𝜃𝜃𝑒𝜃𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐41𝛽𝑇𝑡1𝑝𝐼𝑒𝑡1+𝐶𝜇𝑝(1𝑟)𝐼𝑐𝜃𝑒𝜃(𝑡1𝑀1)1+𝐶𝑝𝑒(1𝑟)𝜃𝑡1𝛽𝑇𝑡1𝑝𝐼𝑒𝜇0𝑓(𝑥)𝑑𝑥=0.(4.17) If 𝑡1,3 is the root of (4.17) (this may or may not exist) and (𝑥)>0, then 𝑑2𝑇𝐶1,3𝑡1𝑑𝑡21=𝑐1+𝑐3𝜃𝑒𝜃𝑡1+𝑐2𝛽𝑇𝑡1+𝑇𝑡1𝛽𝑇𝑡1𝑐4𝛽𝑇𝑡1𝑝𝐼𝑒+𝐶𝑝(1𝑟)𝐼𝑐𝑒𝜃(𝑡1𝑀1)+𝐶𝑝(1𝑟)𝜃𝑒𝜃𝑡1+𝛽𝑇𝑡1𝑓(𝜇)>0,(4.18) this 𝑡1,3 corresponds to unconstrained minimum of 𝑇𝐶1,3(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 𝑇𝐶1,1(𝑡1), say 𝑡1,1,𝑀1, as follows.
Substep 1.1. Compute 𝑡1,1,𝑀1 from (4.13); if 𝑡1,1,𝑀1<𝑀1, then set 𝑡1,1,𝑀1=𝑡1,1,𝑀1 and compute 𝑇𝐶1,1(𝑡1,1,𝑀1) else go to Substep 1.2.Substep 1.2. Find the min{𝑇𝐶1,1(0),𝑇𝐶1,1(𝑀1)} and accordingly set 𝑡1,1,𝑀1.

Step 2. Find the global minimum of 𝑇𝐶1,2(𝑡1), say 𝑡1,2,𝑀1, as follows.
Substep 2.1. Compute 𝑡1,2,𝑀1 from (4.15); if 𝑀1<𝑡1,2,𝑀1<𝜇, then set 𝑡1,2,𝑀1=𝑡1,2,𝑀1 and compute 𝑇𝐶2(𝑡1,2,𝑀1) else go to Substep 2.2.Substep 2.2. Find the min{𝑇𝐶1,2(𝑀1),𝑇𝐶1,2(𝜇)} and accordingly set 𝑡1,2,𝑀1.

Step 3. Find the global minimum of 𝑇𝐶1,3(𝑡1), say 𝑡1,3,𝑀1, as follows.
Substep 3.1. Compute 𝑡1,3,𝑀1 from (4.17); if 𝜇<𝑡1,3,𝑀1, then set 𝑡1,3,𝑀1=𝑡1,3,𝑀1and compute 𝑇𝐶1,3(𝑡1,3,𝑀1) else go to Substep 3.2.Substep 3.2. Find the min{𝑇𝐶1,3(𝜇),𝑇𝐶1,3(𝑇)} and accordingly set 𝑡1,3,𝑀1.

Step 4. Find the min{𝑇𝐶1,1(𝑡1,1,𝑀1),𝑇𝐶1,2(𝑡1,2,𝑀1),𝑇𝐶1,3(𝑡1,3,𝑀1)} and accordingly select the optimal value for 𝑡1 say 𝑡1,𝑀1 with optimal cost 𝐶1(𝑡1,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 (𝑡1𝜇<𝑀2<𝑇). The purchasing cost is. 𝐶𝐴1,2𝑡1=𝐶𝐴1,1𝑡1.1𝑟(4.19) The interest earned during the period of positive inventory level is. 𝐼𝑇1,2𝑡1=𝑝𝐼𝑒𝑡10𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑝𝐼𝑒𝑀2𝑡1𝑡10𝑓(𝑥)𝑑𝑥.(4.20) Since 𝑡1𝜇, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.19), and (4.20) 𝑇𝐶2,1𝑡1=𝐶1𝑡1+𝐶𝐴1,2𝑡1𝐼𝑇1,2𝑡1.(4.21)Subcase 2.2 (𝜇<𝑡1𝑀2<𝑇). The purchasing cost is 𝐶𝐴2,2=𝐶𝐴2,1𝑡1.1𝑟(4.22) The interest earned is 𝐼𝑇2,2𝑡1=𝑝𝐼𝑒𝜇0𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑀2𝜇𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑡1𝜇𝑡𝜇+𝑓(𝜇)𝑑𝑥𝑑𝑡𝑀2𝑡1𝑡1𝜇.𝑓(𝜇)𝑑𝑥𝑑𝑡(4.23) Since again 𝜇𝑡1, the total cost over [0,𝑇] is calculated using the relations (3.5), (4.22), and (4.23) and is 𝑇𝐶2,2𝑡1=𝐶2𝑡1+𝐶𝐴2,2𝑡1𝐼𝑇2,2𝑡1.(4.24)Subcase 2.3 (𝜇𝑀2𝑡1𝑇). The purchasing cost is 𝐶𝐴2,2(𝑡1).
The interest earned, 𝐼𝑇3,2, is 𝐼𝑇3,2𝑡1=𝑝𝐼𝑒𝜇0𝑡0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑡1𝜇𝜇0𝑓(𝑥)𝑑𝑥𝑑𝑡+𝑡1𝜇𝑡𝜇.𝑓(𝜇)𝑑𝑥𝑑𝑡(4.25) The interest payable for the inventory not being sold after the due date 𝑀2 is 𝑃𝑇3,2𝑡1=𝐶𝑝𝐼𝑐𝑓(𝜇)𝑡1𝑀2𝑒𝜃𝑡𝑡1𝑡𝑒𝜃𝑥𝑑𝑥𝑑𝑡.(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 𝑇𝐶2,3𝑡1=𝐶2𝑡1+𝐶𝐴2,2𝑡1+𝑃𝑇3,2𝑡1𝐼𝑇3,2𝑡1.(4.27)
The results obtained lead to the following total cost function: 𝑇𝐶2𝑡1=𝑇𝐶2,1𝑡1,𝑡1𝜇,𝑇𝐶2,2𝑡1,𝜇<𝑡1𝑀2,𝑇𝐶2,3𝑡1,𝑀2𝑡1.(4.28) So the problem is min𝑡1𝑇𝐶2𝑡1.(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 𝑇𝐶2,1(𝑡1) is 𝑑𝑇𝐶2,1𝑡1𝑑𝑡1=𝑐1+𝑐3𝜃𝜃𝑒𝜃𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐41𝛽𝑇𝑡1𝑝𝐼𝑒𝑀2𝑡1+𝐶𝑝𝑒𝜃𝑡1𝛽𝑇𝑡1𝑓𝑡1=0.(4.30) Since 𝑑𝑇𝐶2,1(0)/𝑑𝑡1<0 and 𝑑𝑇𝐶2,1(𝑇)/𝑑𝑡1>0, (4.30) has at least one root. So if 𝑡1,1 is the root of (4.30), this corresponds to minimum as 𝑑𝑇𝐶22,1𝑡1𝑑𝑡21|||||𝑡1=𝑡1,1𝑡=𝑓1,1𝑐1+𝑐3𝜃𝑒𝜃𝑡1,1+𝑐2𝛽𝑇𝑡1,1+𝑐2𝑇𝑡1,1𝛽𝑇𝑡1,1𝑐4𝛽𝑇𝑡1,1+𝑝𝐼𝑒+𝐶𝑝𝜃𝑒𝜃𝑡1,1+𝛽𝑇𝑡1,1>0.(4.31) So 𝑡1,1 is the unconstrained minimum of 𝑇𝐶2,1(𝑡1).
The first-order condition for a minimum of 𝑇𝐶2,2(𝑡1) is 𝑑𝑇𝐶2,2𝑡1𝑑𝑡1𝑐=𝑓(𝜇)1+𝑐3𝜃𝜃𝑒𝜃𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐41𝛽𝑇𝑡1𝑝𝐼𝑒𝑀2𝑡1+𝐶𝑝𝑒𝜃𝑡1𝛽𝑇𝑡1=0.(4.32) If 𝑡1,2 is the root of (4.32) (this may or may not exist), this corresponds to unconstrained minimum of 𝑇𝐶2,2(𝑡1) as 𝑑𝑇𝐶22,2𝑡1𝑑𝑡21|||||𝑡1=𝑡1,2𝑐=𝑓(𝜇)1+𝑐3𝜃𝑒𝜃𝑡1,2+𝑐2𝛽𝑇𝑡1,2+𝑐2𝑇𝑡1,2𝛽𝑇𝑡1,2𝑐4𝛽𝑇𝑡1,2+𝑝𝐼𝑒+𝐶𝑝𝜃𝑒𝜃𝑡1,2+𝛽𝑇𝑡1,2>0.(4.33) The first-order condition for a minimum of 𝑇𝐶2,3(𝑡1) is 𝑑𝑇𝐶2,3𝑡1𝑑𝑡1𝑐=𝑓(𝜇)1+𝑐3𝜃𝜃𝑒𝜃𝑡11𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐41𝛽𝑇𝑡1𝑝𝐼𝑒𝑡1+𝐶𝜇𝑝𝐼𝑐𝜃𝑒𝜃(𝑡1𝑀2)1+𝐶𝑝𝑒𝜃𝑡1𝛽𝑇𝑡1𝑝𝐼𝑒𝜇0𝑓(𝑥)𝑑𝑥=0.(4.34) If 𝑡1,3 is a root of (4.34) (this may or may not exist) and 𝑐1+𝑐3𝜃+𝐶𝑝𝐼𝑐𝑝𝐼𝑒 this corresponds to unconstrained minimum of 𝑇𝐶2,3(𝑡1) as 𝑑𝑇𝐶22,3𝑡1𝑑𝑡21=𝑐1+𝑐3𝜃+𝐶𝑝𝐼𝑐𝑒𝜃𝑡1+𝑐2𝛽𝑇𝑡1+𝑐2𝑇𝑡1𝛽𝑇𝑡1𝑐4𝛽𝑇𝑡1𝑝𝐼𝑒+𝐶𝑝𝜃𝑒𝜃𝑡1+𝛽𝑇𝑡1𝑓(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 𝑇𝐶2,1(𝑡1), say 𝑡1,1,𝑀2, as follows.
Substep 1.1. Compute 𝑡1,1,𝑀2 from (4.30); if 𝑡1,1,𝑀2<𝜇, then set 𝑡1,1,𝑀2=𝑡1,1,𝑀2 and compute 𝑇𝐶2,1(𝑡1,1,𝑀2) else go to Substep 1.2.Substep 1.2. Find the min{𝑇𝐶2,1(0),𝑇𝐶2,1(𝜇)} and accordingly set 𝑡1,1,𝑀2.

Step 2. Find the global minimum of 𝑇𝐶2,2(𝑡1), say 𝑡1,2,𝑀2, as follows.
Substep 2.1. Compute 𝑡1,2,𝑀2 from (4.32); if 𝜇<𝑡1,2,𝑀2<𝑀2, then set 𝑡1,2,𝑀2=𝑡1,2,𝑀2 and compute 𝑇𝐶2,2(𝑡1,2,𝑀2) else go to Substep 2.2.Substep 2.2. Find the min{𝑇𝐶2,2(𝜇),𝑇𝐶2,2(𝑀2)} and accordingly set 𝑡1,2,𝑀2.

Step 3. Find the global minimum of 𝑇𝐶2,3(𝑡1), say 𝑡1,3,𝑀2, as follows.
Substep 3.1. Compute 𝑡1,3,𝑀2 from (4.34); if 𝑀2<𝑡1,3,𝑀2<𝑇, then set 𝑡1,3,𝑀2=𝑡1,3,𝑀2  and compute 𝑇𝐶2,3(𝑡1,3,𝑀2) else go to Substep 3.2.Substep 3.2. Find the min{𝑇𝐶2,3(𝑀2),𝑇𝐶2,3(𝑇)} and accordingly set 𝑡1,3,𝑀2.

Step 4. Find the min{𝑇𝐶2,1(𝑡1,1,𝑀2),𝑇𝐶2,2(𝑡1,2,𝑀2),𝑇𝐶2,3(𝑡1,3,𝑀2)} and accordingly select the optimal value for 𝑡1 say 𝑡1,𝑀2 with optimal cost 𝑇𝐶2(𝑡1,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(𝑡1,𝑀1),𝑇𝐶2(𝑡1,𝑀2)} and accordingly select the optimal value 𝑡1.

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)=3𝑒4.5𝑡 and 𝛽(𝑥)=𝑒0.2𝑥, 𝑀1=0.13 years, 𝑀2=0.43 years, 𝑝=15, 𝐶𝑝=10, 𝐼𝑒=0.12, 𝐼𝑐=0.15.

5.1. The Payment Is Made at 𝑀1

From (4.13), 𝑡1,1,𝑀1=0.399, which is not feasible as 𝑡1,1,𝑀1>𝑀1. Since 𝑇𝐶1,1(0)=55.469 and 𝑇𝐶1,1(𝑀1)=51.9633, it follows that 𝑡1,1,𝑀1=𝑀1. From (4.15), 𝑡1,2,𝑀1=0.426, which is not valid again as 𝑡1,2,𝑀1>𝜇. Since 𝑇𝐶1,1(𝑀1)=𝑇𝐶1,2(𝑀1)=51.9633 and 𝑇𝐶1,2(𝜇)=46.6669, the optimal value for 𝑡11,2,𝑀=𝜇. From (4.17) 𝑡1,3,𝑀1=0.423; this value for 𝑡1 is valid as 𝜇<𝑡1,3,𝑀1<𝑇 so 𝑡1,3,𝑀1=𝑡1,3,𝑀1 and 𝑇𝐶1,3(𝑡1,3,𝑀1)=44.8287.

Finally 𝑇𝐶1,3(𝑡1,3,𝑀1)=min{𝑇𝐶1,1(𝑀1),𝑇𝐶1,2(𝜇),𝑇𝐶1,3(𝑡1,3,𝑀1)}=44.8287 and consequently 𝑡1,𝑀1=0.423.

5.2. The Payment Is Made at 𝑀2

From (4.30), 𝑡1,1,𝑀2=0.424 which is not feasible as 𝑡1,1,𝑀2>𝜇. Since 𝑇𝐶2,1(0)=55.6719𝜅𝛼𝜄  𝑇𝐶2,1(𝜇)=46.2334, it follows that 𝑡1,1,𝑀2=𝜇. From (4.32), 𝑡1,2,𝑀2=0.424 which is valid again as 𝜇<𝑡1,2<𝑀2 so 𝑡1,2,𝑀2=0.424 and 𝑇𝐶2,2(𝑡1,2,𝑀2)=44.3497. From (4.34), 𝑡1,3,𝑀2=0.451; this value for 𝑡1 is also valid as 𝜇<𝑀2<𝑡1,3<𝑇 so 𝑡1,3,𝑀1=𝑡1,3,𝑀1 and 𝑇𝐶2,3(𝑡1,3,𝑀1)=44.3039.

Finally 𝑇𝐶2,3(𝑡1,3,𝑀2)=min{𝑇𝐶2,1(𝜇),𝑇𝐶2,2(𝑡1,3,𝑀2),𝑇𝐶2,3(𝑡1,3,𝑀2)}=44.3039, and consequently 𝑡1,𝑀2=0.451.

So, as 𝑇𝐶2(𝑡1,𝑀2)=min{𝑇𝐶1(𝑡1,𝑀1),𝑇𝐶2(𝑡1,𝑀2)}, the optimal 𝑡1 is 𝑡1=𝑡1,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].

tab1
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, 𝐼𝑝=0, and 𝐼𝑐=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, [2830].

Acknowledgment

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

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