Advances in Decision Sciences

Advances in Decision Sciences / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 761961 | https://doi.org/10.1155/2011/761961

G. Darzanou, K. Skouri, "An Inventory System for Deteriorating Products with Ramp-Type Demand Rate under Two-Level Trade Credit Financing", Advances in Decision Sciences, vol. 2011, Article ID 761961, 15 pages, 2011. https://doi.org/10.1155/2011/761961

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

Academic Editor: Henry Schellhorn
Received27 Dec 2010
Accepted13 Jun 2011
Published12 Sep 2011

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 π‘Ÿ/𝑀1/𝑀2. More precisely, the supplier provides π‘Ÿ 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,πœ‡]) and then remains stable in its maturity stage (i.e., [πœ‡,𝑇]). In addition, the demand rate of a seasonable product increases at the beginning of the season up to a certain moment (say, πœ‡) and then remains constant for the rest of the planning horizon, 𝑇. The term β€œramp-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, 𝑑1) 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., 𝑑1<πœ‡) and (2) the inventory level falls to zero after the demand reaches constant (i.e., 𝑑1>πœ‡). 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, π‘Ÿ/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,πœ‡,𝑇, which, actually, lead to the following different models:

(i) 𝑀1β‰€πœ‡<𝑀2<𝑇, (ii) 𝑀1<𝑀2β‰€πœ‡<𝑇, (iii) πœ‡β‰€π‘€1<𝑀2<𝑇, (iv) πœ‡β‰€π‘€1<𝑇<𝑀2, (v) πœ‡<𝑇≀𝑀1<𝑀2, (vi) 𝑀1β‰€πœ‡<𝑇≀𝑀2.

Note that from the definition of demand rate πœ‡<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 πœ‡, 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),𝑑1the time when the inventory level falls to zero, 𝑆the maximum inventory level at each scheduling period (cycle),𝐢𝑝the 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>𝐢𝑝 see Teng et al. [27]),𝑝the unit selling price,𝐼𝑒the interest rate earned, 𝐼𝑐the interest rate charged, π‘Ÿcash discount rate, 0<π‘Ÿ<1,𝑀1the period of cash discount in years, 𝑀2the period of permissible delay in payments in years, 𝑀1<𝑀2,πœ‡the parameter of the ramp-type demand function (time point), and 𝐼(𝑑)the inventory level at time 𝑑.
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(π‘₯) which is a nonincreasing function of π‘₯ with 0<𝛽(π‘₯)≀1,  𝛽(0)=1 and π‘₯ is the waiting time up to the next replenishment. Moreover, it is assumed that 𝛽(π‘₯) satisfies the relation 𝐢2𝛽(π‘₯)+𝐢2π‘‡π›½ξ…ž(π‘₯)+πΆπ‘π›½ξ…ž(π‘₯)β‰₯0, where π›½ξ…ž(π‘₯) is the derivate of 𝛽(π‘₯). The case with 𝛽(π‘₯)=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 πœƒ (0<πœƒ<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(𝑑) is a ramp-type function of time given by𝐷(𝑑)=𝑓(𝑑),𝑑<πœ‡,𝑓(πœ‡),𝑑β‰₯πœ‡,(2.1) where 𝑓(𝑑) is a positive, differentiable function of π‘‘βˆˆ(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 𝑑1 and πœ‡. The inventory level 𝐼(𝑑), 0≀𝑑≀𝑇 satisfies the following differential equations:𝑑𝐼(𝑑)𝑑𝑑+πœƒπΌ(𝑑)=βˆ’π·(𝑑),0≀𝑑≀𝑑1(3.1) with boundary condition 𝐼(𝑑1)=0 and𝑑𝐼(𝑑)𝑑𝑑=βˆ’π·(𝑑)𝛽(π‘‡βˆ’π‘‘),𝑑1≀𝑑≀𝑇(3.2) with boundary condition 𝐼(𝑑1)=0.

From the two possible relations between parameters 𝑑1 and πœ‡, (i) 𝑑1β‰€πœ‡ and (ii) 𝑑1>πœ‡, and following identical steps as in Skouri et al. [23], the sum of holding, deterioration, shortages, and lost sales cost is obtained as𝐢𝑑1ξ€Έ=𝐢1𝑑1ξ€Έif𝑑1πΆβ‰€πœ‡,2𝑑1ξ€Έif𝑑1>πœ‡,(3.3) where𝐢1𝑑1ξ€Έ=𝑐1ξ‚»ξ€œπ‘‘10π‘’βˆ’πœƒπ‘‘ξ‚Έξ€œπ‘‘1𝑑𝑓(π‘₯)π‘’πœƒπ‘₯𝑑π‘₯𝑑𝑑+𝑐2ξ‚»ξ€œπœ‡π‘‘1ξ€œ(πœ‡βˆ’π‘‘)𝑓(𝑑)𝛽(π‘‡βˆ’π‘‘)𝑑𝑑+𝑓(πœ‡)π‘‡πœ‡ξ‚Έξ€œπ‘‘πœ‡ξ‚Ή+ξ€œπ›½(π‘‡βˆ’π‘₯)𝑑π‘₯π‘‘π‘‘π‘‡πœ‡ξ‚Έξ€œπœ‡π‘‘1𝑓(π‘₯)𝛽(π‘‡βˆ’π‘₯)𝑑π‘₯𝑑𝑑+𝑐3ξ‚»ξ€œπ‘‘10𝑓(𝑑)π‘’πœƒπ‘‘ξ€œπ‘‘π‘‘βˆ’π‘‘10𝑓(𝑑)𝑑𝑑+𝑐4ξ‚»ξ€œπœ‡π‘‘1ξ€œ(1βˆ’π›½(π‘‡βˆ’π‘‘))𝑓(𝑑)𝑑𝑑+𝑓(πœ‡)π‘‡πœ‡ξ‚Ό,𝐢(1βˆ’π›½(π‘‡βˆ’π‘‘))𝑑𝑑(3.4)2𝑑1ξ€Έ=𝑐1ξ‚»ξ€œπœ‡0π‘’βˆ’πœƒπ‘‘ξ‚Έξ€œπœ‡π‘‘π‘“(π‘₯)π‘’πœƒπ‘₯ξ€œπ‘‘π‘₯+𝑓(πœ‡)𝑑1πœ‡π‘’πœƒπ‘₯ξ‚Ήξ€œπ‘‘π‘₯𝑑𝑑+𝑓(πœ‡)𝑑1πœ‡π‘’βˆ’πœƒπ‘‘ξ‚Έξ€œπ‘‘1π‘‘π‘’πœƒπ‘₯𝑑π‘₯𝑑𝑑+𝑐3ξ‚»ξ€œπœ‡0𝑓(𝑑)π‘’πœƒπ‘‘ξ€œπ‘‘π‘‘+𝑓(πœ‡)𝑑1πœ‡π‘’πœƒπ‘‘ξ€œπ‘‘π‘‘βˆ’πœ‡0𝑑𝑓(𝑑)π‘‘π‘‘βˆ’π‘“(πœ‡)1ξ€Έξ‚Όβˆ’πœ‡+𝑐2ξ‚»ξ€œπ‘“(πœ‡)𝑇𝑑1ξ‚Ό(π‘‡βˆ’π‘₯)𝛽(π‘‡βˆ’π‘₯)𝑑π‘₯+𝑐4ξ‚»ξ€œπ‘“(πœ‡)𝑇𝑑1ξ‚Ό.(1βˆ’π›½(π‘‡βˆ’π‘‘))𝑑𝑑(3.5)

4. Model Iβ€”The Inventory Model When 𝑀1β‰€πœ‡<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 (𝑑1≀𝑀1β‰€πœ‡<𝑇). The purchasing cost is 𝐢𝐴1,1𝑑1ξ€Έ=πΆπ‘ξ‚Έξ€œ(1βˆ’π‘Ÿ)𝑑10𝑓(π‘₯)π‘’πœƒπ‘₯ξ€œπ‘‘π‘₯+𝑓(πœ‡)π‘‡πœ‡ξ€œπ›½(π‘‡βˆ’π‘₯)𝑑π‘₯+πœ‡π‘‘1ξ‚Ή.𝑓(π‘₯)𝛽(π‘‡βˆ’π‘₯)𝑑π‘₯(4.1) The interest earned during the period of positive inventory level is 𝐼𝑇1,1𝑑1ξ€Έ=π‘πΌπ‘’ξ€œπ‘‘10ξ€œπ‘‘0𝑓(π‘₯)𝑑π‘₯𝑑𝑑+𝑝𝐼𝑒𝑀1βˆ’π‘‘1ξ€Έξ€œπ‘‘10𝑓(π‘₯)𝑑π‘₯.(4.2) Since 𝑑1β‰€πœ‡, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.1), and (4.2) 𝑇𝐢11𝑑1ξ€Έ=𝐢1𝑑1ξ€Έ+𝐢𝐴1,1𝑑1ξ€Έβˆ’πΌπ‘‡1,1𝑑1ξ€Έ.(4.3)Subcase 1.2 (𝑀1<𝑑1β‰€πœ‡<𝑇). The purchasing cost is 𝐢𝐴1,1 (relation (4.1)).
The interest payable for the inventory not being sold after the due date 𝑀1 is 𝑃𝑇2,1𝑑1ξ€Έ=𝐢𝑝(1βˆ’π‘Ÿ)πΌπ‘ξ€œπ‘‘1𝑀1π‘’βˆ’πœƒπ‘‘ξ‚Έξ€œπ‘‘1𝑑𝑓(π‘₯)π‘’πœƒπ‘₯𝑑π‘₯𝑑𝑑.(4.4) The interest earned is 𝐼𝑇2,1𝑑1ξ€Έ=π‘πΌπ‘’ξ€œπ‘‘10ξ€œπ‘‘0𝑓(π‘₯)𝑑π‘₯𝑑𝑑.(4.5) Since again 𝑑1β‰€πœ‡, the total cost over [0,𝑇] is calculated using the relations (3.4), (4.1), (4.4), and (4.6) and is 𝑇𝐢12𝑑1ξ€Έ=𝐢1𝑑1ξ€Έ+𝐢𝐴1,1𝑑1ξ€Έ+𝑃𝑇2,1𝑑1ξ€Έβˆ’πΌπ‘‡2,1𝑑1ξ€Έ.(4.6)
Subcase 1.3 (𝑀1β‰€πœ‡β‰€π‘‘1≀𝑇). The purchasing cost is 𝐢𝐴2,1𝑑1ξ€Έ=πΆπ‘ξ‚Έξ€œ(1βˆ’π‘Ÿ)πœ‡0𝑓(π‘₯)π‘’πœƒπ‘₯ξ€œπ‘‘π‘₯+𝑓(πœ‡)𝑑1πœ‡π‘’πœƒπ‘₯ξ€œπ‘‘π‘₯+𝑓(πœ‡)𝑇𝑑1ξ‚Ή.𝛽(π‘‡βˆ’π‘₯)𝑑π‘₯(4.7) The interest earned, 𝐼𝑇3,1, is: 𝐼𝑇3,1𝑑1ξ€Έ=π‘πΌπ‘’ξ‚΅ξ€œπœ‡0ξ€œπ‘‘0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑑1πœ‡ξ€œπœ‡0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑑1πœ‡ξ€œπ‘‘πœ‡ξ‚Ά.𝑓(πœ‡)𝑑π‘₯𝑑𝑑(4.8) The interest payable for the inventory not being sold after the due date 𝑀1 is 𝑃𝑇3,1𝑑1ξ€Έ=𝐢𝑝(1βˆ’π‘Ÿ)πΌπ‘ξ‚΅ξ€œπœ‡π‘€1π‘’βˆ’πœƒπ‘‘ξ€œπœ‡π‘‘π‘’πœƒπ‘₯ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑓(πœ‡)πœ‡π‘€1π‘’βˆ’πœƒπ‘‘ξ€œπ‘‘1πœ‡π‘’πœƒπ‘₯ξ€œπ‘‘π‘₯𝑑𝑑+𝑓(πœ‡)𝑑1πœ‡π‘’βˆ’πœƒπ‘‘ξ€œπ‘‘1π‘‘π‘’πœƒπ‘₯ξ‚Ά.𝑑π‘₯𝑑𝑑(4.9) Since πœ‡<𝑑1, the total cost over [0,𝑇] is again calculated from (3.5), (4.7)–(4.9) and is 𝑇𝐢13𝑑1ξ€Έ=𝐢2𝑑1ξ€Έ+𝐢𝐴2,1𝑑1ξ€Έ+𝑃𝑇3,1𝑑1ξ€Έβˆ’πΌπ‘‡3,1𝑑1ξ€Έ.(4.10)
The results obtained lead to the following total cost function: 𝑇𝐢1𝑑1ξ€Έ=⎧βŽͺ⎨βŽͺβŽ©π‘‡πΆ1,1𝑑1ξ€Έ,𝑑1≀𝑀1,𝑇𝐢1,2𝑑1ξ€Έ,𝑀1<𝑑1β‰€πœ‡,𝑇𝐢1,3𝑑1ξ€Έ,πœ‡β‰€π‘‘1.(4.11) So the problem is min𝑑1𝑇𝐢1𝑑1ξ€Έ.(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 𝑇𝐢1(𝑑1) is continuous at the points 𝑀1 and πœ‡.
The first-order condition for a minimum of 𝑇𝐢1,1(𝑑1) is 𝑑𝑇𝐢1,1𝑑1𝑑𝑑1=𝑐1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έβˆ’1βˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€·π‘€1βˆ’π‘‘1ξ€Έ+𝐢𝑝𝑒(1βˆ’π‘Ÿ)πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1𝑓𝑑1ξ€Έ=0.(4.13) Since 𝑑𝑇𝐢1,1(0)/𝑑𝑑1<0 and 𝑑𝑇𝐢1,1(𝑇)/𝑑𝑑1>0, (4.13) has at least one root. So if 𝑑1,1 is the root of (4.13), this corresponds to minimum since 𝑑𝑇𝐢21,1(𝑑1)𝑑𝑑21|||||𝑑1=𝑑1,1𝑑=𝑓1,1𝑐1+𝑐3πœƒξ€Έπ‘’πœƒπ‘‘1,1+𝑐2π›½ξ€·π‘‡βˆ’π‘‘1,1ξ€Έ+𝑐2ξ€·π‘‡βˆ’π‘‘1,1ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1,1ξ€Έβˆ’π‘4ξ€·π›½β€²π‘‡βˆ’π‘‘1,1ξ€Έ+𝑝𝐼𝑒+𝐢𝑝(1βˆ’π‘Ÿ)πœƒπ‘’πœƒπ‘‘1,1ξ€·+π›½β€²π‘‡βˆ’π‘‘1,1ξ€Έξ€Έξ€Ύ>0.(4.14) Consequently, 𝑑1,1 is the unique unconstrained minimum of 𝑇𝐢1,1(𝑑1).
The first-order condition for a minimum of 𝑇𝐢1,2(𝑑1) is 𝑑𝑇𝐢1,2𝑑1𝑑𝑑1=𝑐1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έπ‘“ξ€·π‘‘βˆ’11ξ€Έβˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1𝑓𝑑1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1𝑓𝑑1ξ€Έ+𝐢𝑝𝑒(1βˆ’π‘Ÿ)πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1+𝐢𝑝(1βˆ’π‘Ÿ)πΌπ‘πœƒξ€·π‘’πœƒ(𝑑1βˆ’π‘€1)ξ€Έπ‘“ξ€·π‘‘βˆ’11ξ€Έβˆ’π‘πΌπ‘’ξ€œπ‘‘10𝑓(π‘₯)𝑑π‘₯=0.(4.15) Let us set β„Ž(π‘₯)=𝑐2π‘₯𝛽(π‘₯)+𝑐4(1βˆ’π›½(π‘₯))βˆ’π‘πΌπ‘’π‘₯. If 𝑑1,2 is the root of (4.15) (this may or may not exist), 𝑓(π‘₯) is an increasing function and further if β„Žξ…ž(π‘₯)>0, then 𝑑2𝑇𝐢1,2𝑑1𝑑𝑑21||||𝑑1=𝑑1,2=π‘πΌπ‘’π‘“ξ…žξ€·π‘‘1,2ξ€Έξ€œπ‘‘1,20𝑑𝑓(π‘₯)𝑑π‘₯+𝑓1,2𝑐1+𝑐3πœƒξ€Έπ‘’πœƒπ‘‘1,2+𝑐2ξ€Ίπ›½ξ€·π‘‡βˆ’π‘‘1,2ξ€Έ+ξ€·π‘‡βˆ’π‘‘1,2ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έξ€»βˆ’π‘4π›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έ+𝐢𝑝(1βˆ’π‘Ÿ)πΌπ‘π‘’πœƒ(𝑑1,2βˆ’π‘€1)+𝐢𝑝(ξ€·1βˆ’π‘Ÿ)πœƒπ‘’πœƒπ‘‘1,2+π›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€Ύ>0,(4.16) and this 𝑑1,2 corresponds to unconstrained minimum of 𝑇𝐢1,2(𝑑1).
The first-order condition for a minimum of 𝑇𝐢1,3(𝑑1) is 𝑑𝑇𝐢1,3𝑑1𝑑𝑑1𝑐=𝑓(πœ‡)1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έβˆ’1βˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€·π‘‘1ξ€Έ+πΆβˆ’πœ‡π‘(1βˆ’π‘Ÿ)πΌπ‘πœƒξ€·π‘’πœƒ(𝑑1βˆ’π‘€1)ξ€Έβˆ’1+𝐢𝑝𝑒(1βˆ’π‘Ÿ)πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1ξ‚Όξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€œπœ‡0𝑓(π‘₯)𝑑π‘₯=0.(4.17) If 𝑑1,3 is the root of (4.17) (this may or may not exist) and β„Žξ…ž(π‘₯)>0, then 𝑑2𝑇𝐢1,3𝑑1𝑑𝑑21=𝑐1+𝑐3πœƒξ€Έπ‘’πœƒπ‘‘1+𝑐2ξ€Ίπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έ+ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1ξ€Έξ€»βˆ’π‘4π›½ξ…žξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘πΌπ‘’+𝐢𝑝(1βˆ’π‘Ÿ)πΌπ‘π‘’πœƒ(𝑑1βˆ’π‘€1)+𝐢𝑝(1βˆ’π‘Ÿ)πœƒπ‘’πœƒπ‘‘1+π›½ξ…žξ€·π‘‡βˆ’π‘‘1𝑓(πœ‡)>0,(4.18) this 𝑑1,3 corresponds to unconstrained minimum of 𝑇𝐢1,3(𝑑1).

Remark 4.1. The function 𝑇𝐢1(𝑑1) 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 𝑇𝐢1,1(𝑑1), say π‘‘βˆ—1,1,𝑀1, as follows.
Substep 1.1. Compute 𝑑1,1,𝑀1 from (4.13); if 𝑑1,1,𝑀1<𝑀1, then set π‘‘βˆ—1,1,𝑀1=𝑑1,1,𝑀1 and compute 𝑇𝐢1,1(π‘‘βˆ—1,1,𝑀1) else go to Substep 1.2.Substep 1.2. Find the min{𝑇𝐢1,1(0),𝑇𝐢1,1(𝑀1)} and accordingly set π‘‘βˆ—1,1,𝑀1.

Step 2. Find the global minimum of 𝑇𝐢1,2(𝑑1), say π‘‘βˆ—1,2,𝑀1, as follows.
Substep 2.1. Compute 𝑑1,2,𝑀1 from (4.15); if 𝑀1<𝑑1,2,𝑀1<πœ‡, then set π‘‘βˆ—1,2,𝑀1=𝑑1,2,𝑀1 and compute 𝑇𝐢2(π‘‘βˆ—1,2,𝑀1) else go to Substep 2.2.Substep 2.2. Find the min{𝑇𝐢1,2(𝑀1),𝑇𝐢1,2(πœ‡)} and accordingly set π‘‘βˆ—1,2,𝑀1.

Step 3. Find the global minimum of 𝑇𝐢1,3(𝑑1), say π‘‘βˆ—1,3,𝑀1, as follows.
Substep 3.1. Compute 𝑑1,3,𝑀1 from (4.17); if πœ‡<𝑑1,3,𝑀1, then set π‘‘βˆ—1,3,𝑀1=𝑑1,3,𝑀1and compute 𝑇𝐢1,3(π‘‘βˆ—1,3,𝑀1) else go to Substep 3.2.Substep 3.2. Find the min{𝑇𝐢1,3(πœ‡),𝑇𝐢1,3(𝑇)} and accordingly set π‘‘βˆ—1,3,𝑀1.

Step 4. Find the min{𝑇𝐢1,1(π‘‘βˆ—1,1,𝑀1),𝑇𝐢1,2(π‘‘βˆ—1,2,𝑀1),𝑇𝐢1,3(π‘‘βˆ—1,3,𝑀1)} and accordingly select the optimal value for 𝑑1 say 𝑑1,𝑀1 with optimal cost 𝐢1(𝑑1,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 (𝑑1β‰€πœ‡<𝑀2<𝑇). The purchasing cost is. 𝐢𝐴1,2𝑑1ξ€Έ=𝐢𝐴1,1𝑑1ξ€Έ.1βˆ’π‘Ÿ(4.19) The interest earned during the period of positive inventory level is. 𝐼𝑇1,2𝑑1ξ€Έ=π‘πΌπ‘’ξ€œπ‘‘10ξ€œπ‘‘0𝑓(π‘₯)𝑑π‘₯𝑑𝑑+𝑝𝐼𝑒𝑀2βˆ’π‘‘1ξ€Έξ€œπ‘‘10𝑓(π‘₯)𝑑π‘₯.(4.20) Since 𝑑1β‰€πœ‡, the total cost in the time interval [0,𝑇] is calculated using (3.4), (4.19), and (4.20) 𝑇𝐢2,1𝑑1ξ€Έ=𝐢1𝑑1ξ€Έ+𝐢𝐴1,2𝑑1ξ€Έβˆ’πΌπ‘‡1,2𝑑1ξ€Έ.(4.21)Subcase 2.2 (πœ‡<𝑑1≀𝑀2<𝑇). The purchasing cost is 𝐢𝐴2,2=𝐢𝐴2,1𝑑1ξ€Έ.1βˆ’π‘Ÿ(4.22) The interest earned is 𝐼𝑇2,2𝑑1ξ€Έ=π‘πΌπ‘’ξ‚Έξ€œπœ‡0ξ€œπ‘‘0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑀2πœ‡ξ€œπœ‡0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑑1πœ‡ξ€œπ‘‘πœ‡+ξ€œπ‘“(πœ‡)𝑑π‘₯𝑑𝑑𝑀2𝑑1ξ€œπ‘‘1πœ‡ξ‚Ή.𝑓(πœ‡)𝑑π‘₯𝑑𝑑(4.23) Since again πœ‡β‰€π‘‘1, the total cost over [0,𝑇] is calculated using the relations (3.5), (4.22), and (4.23) and is 𝑇𝐢2,2𝑑1ξ€Έ=𝐢2𝑑1ξ€Έ+𝐢𝐴2,2𝑑1ξ€Έβˆ’πΌπ‘‡2,2𝑑1ξ€Έ.(4.24)Subcase 2.3 (πœ‡β‰€π‘€2≀𝑑1≀𝑇). The purchasing cost is 𝐢𝐴2,2(𝑑1).
The interest earned, 𝐼𝑇3,2, is 𝐼𝑇3,2𝑑1ξ€Έ=π‘πΌπ‘’ξ‚΅ξ€œπœ‡0ξ€œπ‘‘0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑑1πœ‡ξ€œπœ‡0ξ€œπ‘“(π‘₯)𝑑π‘₯𝑑𝑑+𝑑1πœ‡ξ€œπ‘‘πœ‡ξ‚Ά.𝑓(πœ‡)𝑑π‘₯𝑑𝑑(4.25) The interest payable for the inventory not being sold after the due date 𝑀2 is 𝑃𝑇3,2𝑑1ξ€Έ=πΆπ‘πΌπ‘ξ€œπ‘“(πœ‡)𝑑1𝑀2π‘’βˆ’πœƒπ‘‘ξ€œπ‘‘1π‘‘π‘’πœƒπ‘₯𝑑π‘₯𝑑𝑑.(4.26) Since πœ‡<𝑑1, the total cost over [0,𝑇] is again calculated from (3.5), (4.22), (4.25), and (4.26) and is 𝑇𝐢2,3𝑑1ξ€Έ=𝐢2𝑑1ξ€Έ+𝐢𝐴2,2𝑑1ξ€Έ+𝑃𝑇3,2𝑑1ξ€Έβˆ’πΌπ‘‡3,2𝑑1ξ€Έ.(4.27)
The results obtained lead to the following total cost function: 𝑇𝐢2𝑑1ξ€Έ=⎧βŽͺ⎨βŽͺβŽ©π‘‡πΆ2,1𝑑1ξ€Έ,𝑑1β‰€πœ‡,𝑇𝐢2,2𝑑1ξ€Έ,πœ‡<𝑑1≀𝑀2,𝑇𝐢2,3𝑑1ξ€Έ,𝑀2≀𝑑1.(4.28) So the problem is min𝑑1𝑇𝐢2𝑑1ξ€Έ.(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 𝑇𝐢2(𝑑1) is continuous at the points 𝑀2 and πœ‡.
The first-order condition for the minimum for 𝑇𝐢2,1(𝑑1) is 𝑑𝑇𝐢2,1𝑑1𝑑𝑑1=𝑐1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έβˆ’1βˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€·π‘€2βˆ’π‘‘1ξ€Έ+πΆπ‘ξ€·π‘’πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1𝑓𝑑1ξ€Έ=0.(4.30) Since 𝑑𝑇𝐢2,1(0)/𝑑𝑑1<0 and 𝑑𝑇𝐢2,1(𝑇)/𝑑𝑑1>0, (4.30) has at least one root. So if 𝑑1,1 is the root of (4.30), this corresponds to minimum as 𝑑𝑇𝐢22,1𝑑1𝑑𝑑21|||||𝑑1=𝑑1,1𝑑=𝑓1,1𝑐1+𝑐3πœƒξ€Έπ‘’πœƒπ‘‘1,1+𝑐2π›½ξ€·π‘‡βˆ’π‘‘1,1ξ€Έ+𝑐2ξ€·π‘‡βˆ’π‘‘1,1ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1,1ξ€Έβˆ’π‘4π›½ξ…žξ€·π‘‡βˆ’π‘‘1,1ξ€Έ+𝑝𝐼𝑒+πΆπ‘ξ€·πœƒπ‘’πœƒπ‘‘1,1+π›½ξ…žξ€·π‘‡βˆ’π‘‘1,1ξ€Έξ€Έξ€Ύ>0.(4.31) So 𝑑1,1 is the unconstrained minimum of 𝑇𝐢2,1(𝑑1).
The first-order condition for a minimum of 𝑇𝐢2,2(𝑑1) is 𝑑𝑇𝐢2,2𝑑1𝑑𝑑1𝑐=𝑓(πœ‡)1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έβˆ’1βˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€·π‘€2βˆ’π‘‘1ξ€Έ+πΆπ‘ξ€·π‘’πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1ξ‚Όξ€Έξ€Έ=0.(4.32) If 𝑑1,2 is the root of (4.32) (this may or may not exist), this corresponds to unconstrained minimum of 𝑇𝐢2,2(𝑑1) as 𝑑𝑇𝐢22,2𝑑1𝑑𝑑21|||||𝑑1=𝑑1,2𝑐=𝑓(πœ‡)ξ€½ξ€·1+𝑐3πœƒξ€Έπ‘’πœƒπ‘‘1,2+𝑐2π›½ξ€·π‘‡βˆ’π‘‘1,2ξ€Έ+𝑐2ξ€·π‘‡βˆ’π‘‘1,2ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έβˆ’π‘4π›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έ+𝑝𝐼𝑒+πΆπ‘ξ€·πœƒπ‘’πœƒπ‘‘1,2+π›½ξ…žξ€·π‘‡βˆ’π‘‘1,2ξ€Έξ€Έξ€Ύ>0.(4.33) The first-order condition for a minimum of 𝑇𝐢2,3(𝑑1) is 𝑑𝑇𝐢2,3𝑑1𝑑𝑑1𝑐=𝑓(πœ‡)1+𝑐3πœƒπœƒξ€·π‘’πœƒπ‘‘1ξ€Έβˆ’1βˆ’π‘2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4ξ€·ξ€·1βˆ’π›½π‘‡βˆ’π‘‘1ξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€·π‘‘1ξ€Έ+πΆβˆ’πœ‡π‘πΌπ‘πœƒξ€·π‘’πœƒ(𝑑1βˆ’π‘€2)ξ€Έβˆ’1+πΆπ‘ξ€·π‘’πœƒπ‘‘1ξ€·βˆ’π›½π‘‡βˆ’π‘‘1ξ‚Όξ€Έξ€Έβˆ’π‘πΌπ‘’ξ€œπœ‡0𝑓(π‘₯)𝑑π‘₯=0.(4.34) If 𝑑1,3 is a root of (4.34) (this may or may not exist) and 𝑐1+𝑐3πœƒ+𝐢𝑝𝐼𝑐β‰₯𝑝𝐼𝑒 this corresponds to unconstrained minimum of 𝑇𝐢2,3(𝑑1) as 𝑑𝑇𝐢22,3𝑑1𝑑𝑑21=𝑐1+𝑐3πœƒ+πΆπ‘πΌπ‘ξ€Έπ‘’πœƒπ‘‘1+𝑐2π›½ξ€·π‘‡βˆ’π‘‘1ξ€Έ+𝑐2ξ€·π‘‡βˆ’π‘‘1ξ€Έπ›½ξ…žξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘4π›½ξ…žξ€·π‘‡βˆ’π‘‘1ξ€Έβˆ’π‘πΌπ‘’+πΆπ‘ξ€·πœƒπ‘’πœƒπ‘‘1+π›½ξ…žξ€·π‘‡βˆ’π‘‘1𝑓(πœ‡).(4.35)

Remark 4.2. The function 𝑇𝐢2(𝑑1) 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 𝑇𝐢2,1(𝑑1), say π‘‘βˆ—1,1,𝑀2, as follows.
Substep 1.1. Compute 𝑑1,1,𝑀2 from (4.30); if 𝑑1,1,𝑀2<πœ‡, then set π‘‘βˆ—1,1,𝑀2=𝑑1,1,𝑀2 and compute 𝑇𝐢2,1(π‘‘βˆ—1,1,𝑀2) else go to Substep 1.2.Substep 1.2. Find the min{𝑇𝐢2,1(0),𝑇𝐢2,1(πœ‡)} and accordingly set π‘‘βˆ—1,1,𝑀2.

Step 2. Find the global minimum of 𝑇𝐢2,2(𝑑1), say π‘‘βˆ—1,2,𝑀2, as follows.
Substep 2.1. Compute 𝑑1,2,𝑀2 from (4.32); if πœ‡<𝑑1,2,𝑀2<𝑀2, then set π‘‘βˆ—1,2,𝑀2=𝑑1,2,𝑀2 and compute 𝑇𝐢2,2(π‘‘βˆ—1,2,𝑀2) else go to Substep 2.2.Substep 2.2. Find the min{𝑇𝐢2,2(πœ‡),𝑇𝐢2,2(𝑀2)} and accordingly set π‘‘βˆ—1,2,𝑀2.

Step 3. Find the global minimum of 𝑇𝐢2,3(𝑑1), say π‘‘βˆ—1,3,𝑀2, as follows.
Substep 3.1. Compute 𝑑1,3,𝑀2 from (4.34); if 𝑀2<𝑑1,3,𝑀2<𝑇, then set π‘‘βˆ—1,3,𝑀2=𝑑1,3,𝑀2  and compute 𝑇𝐢2,3(π‘‘βˆ—1,3,𝑀2) else go to Substep 3.2.Substep 3.2. Find the min{𝑇𝐢2,3(𝑀2),𝑇𝐢2,3(𝑇)} and accordingly set π‘‘βˆ—1,3,𝑀2.

Step 4. Find the min{𝑇𝐢2,1(π‘‘βˆ—1,1,𝑀2),𝑇𝐢2,2(π‘‘βˆ—1,2,𝑀2),𝑇𝐢2,3(π‘‘βˆ—1,3,𝑀2)} and accordingly select the optimal value for 𝑑1 say 𝑑1,𝑀2 with optimal cost 𝑇𝐢2(𝑑1,𝑀2).
Finally to find the overall optimum 𝑑1 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{𝑇𝐢1(𝑑1,𝑀1),𝑇𝐢2(𝑑1,𝑀2)} and accordingly select the optimal value π‘‘βˆ—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, π‘Ÿ=0.005, πœ‡=0.3 years, πœƒ=0.001, 𝑇=0.5 years, 𝑓(𝑑)=3𝑒4.5𝑑 and 𝛽(π‘₯)=π‘’βˆ’0.2π‘₯, 𝑀1=0.13 years, 𝑀2=0.43 years, 𝑝=15, 𝐢𝑝=10, 𝐼𝑒=0.12, 𝐼𝑐=0.15.

5.1. The Payment Is Made at 𝑀1

From (4.13), 𝑑1,1,𝑀1=0.399, which is not feasible as 𝑑1,1,𝑀1>𝑀1. Since 𝑇𝐢1,1(0)=55.469 and 𝑇𝐢1,1(𝑀1)=51.9633, it follows that π‘‘βˆ—1,1,𝑀1=𝑀1. From (4.15), 𝑑1,2,𝑀1=0.426, which is not valid again as 𝑑1,2,𝑀1>πœ‡. Since 𝑇𝐢1,1(𝑀1)=𝑇𝐢1,2(𝑀1)=51.9633 and 𝑇𝐢1,2(πœ‡)=46.6669, the optimal value for π‘‘βˆ—11,2,𝑀=πœ‡. From (4.17) 𝑑1,3,𝑀1=0.423; this value for 𝑑1 is valid as πœ‡<𝑑1,3,𝑀1<𝑇 so π‘‘βˆ—1,3,𝑀1=𝑑1,3,𝑀1 and 𝑇𝐢1,3(π‘‘βˆ—1,3,𝑀1)=44.8287.

Finally 𝑇𝐢1,3(π‘‘βˆ—1,3,𝑀1)=min{𝑇𝐢1,1(𝑀1),𝑇𝐢1,2(πœ‡),𝑇𝐢1,3(π‘‘βˆ—1,3,𝑀1)}=44.8287 and consequently π‘‘βˆ—1,𝑀1=0.423.

5.2. The Payment Is Made at 𝑀2

From (4.30), 𝑑1,1,𝑀2=0.424 which is not feasible as 𝑑1,1,𝑀2>πœ‡. Since 𝑇𝐢2,1(0)=55.6719β€‰πœ…π›Όπœ„β€‰β€‰π‘‡πΆ2,1(πœ‡)=46.2334, it follows that π‘‘βˆ—1,1,𝑀2=πœ‡. From (4.32), 𝑑1,2,𝑀2=0.424 which is valid again as πœ‡<𝑑1,2<𝑀2 so π‘‘βˆ—1,2,𝑀2=0.424 and 𝑇𝐢2,2(π‘‘βˆ—1,2,𝑀2)=44.3497. From (4.34), 𝑑1,3,𝑀2=0.451; this value for 𝑑1 is also valid as πœ‡<𝑀2<𝑑1,3<𝑇 so π‘‘βˆ—1,3,𝑀1=𝑑1,3,𝑀1 and 𝑇𝐢2,3(π‘‘βˆ—1,3,𝑀1)=44.3039.

Finally 𝑇𝐢2,3(π‘‘βˆ—1,3,𝑀2)=min{𝑇𝐢2,1(πœ‡),𝑇𝐢2,2(π‘‘βˆ—1,3,𝑀2),𝑇𝐢2,3(π‘‘βˆ—1,3,𝑀2)}=44.3039, and consequently π‘‘βˆ—1,𝑀2=0.451.

So, as 𝑇𝐢2(𝑑1,𝑀2)=min{𝑇𝐢1(𝑑1,𝑀1),𝑇𝐢2(𝑑1,𝑀2)}, the optimal 𝑑1 is π‘‘βˆ—1=π‘‘βˆ—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 (πœ‡,𝑀1,𝑀2,𝑇,π‘Ÿ) to the optimal policy (i.e., 𝑑1 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 π‘Ÿ have no impact on the optimal 𝑑1, the time of payment and the optimal cost.(2)The error on the parameters’ estimation of πœ‡ has no impact on the time of payment, small impact on the optimal 𝑑1, but high impact on the total optimal cost. This last observation is in line with the relative findings in Deng et al. [21].


Parameter (i)Percentage of changes (%) 𝑑 βˆ— 1 Time of paymentTC( 𝑑 βˆ— 1 )

πœ‡ βˆ’500.46 𝑀 2 27.0538
βˆ’250.44 𝑀 2 44.4206
+250.45 𝑀 2 52.8972
+500.45 𝑀 2 58.8673

𝑀 1 βˆ’500.451 𝑀 2 44.3039
βˆ’250.451 𝑀 2 44.3039
+250.451 𝑀 2 44.3039
+500.451 𝑀 2 44.3039

𝑀 2 βˆ’500.434 𝑀 2 44.4075
βˆ’250.443 𝑀 2 44.4206
+250.43 𝑀 2 43.6966
+500.43 𝑀 2 43.5449

π‘Ÿ βˆ’500.451 𝑀 2 44.3039
βˆ’250.451 𝑀 2 44.3039
+250.451 𝑀 2 44.3039
+500.451 𝑀 2 44.3039

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 π‘Ÿ/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 π‘Ÿ/𝑀1/𝑀2 credit scheme is considered. The study of this system requires the examination of the ordering relations between the time parameters 𝑀1, 𝑀2, πœ‡, 𝑇, which, actually, lead to the six different models. This inventory system, setting 𝑓(𝑑)=𝐷0𝑑, 𝛽(π‘₯)=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, [28–30].

Acknowledgment

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

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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.


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