Advances in Decision Sciences

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 . More precisely, the supplier provides discount off the price if the payment is made within period ; otherwise, the full payment is due within period . (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., ) and then remains stable in its maturity stage (i.e., ). In addition, the demand rate of a seasonable product increases at the beginning of the season up to a certain moment (say, ) and then remains constant for the rest of the planning horizon, . 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, ) 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., ) and (2) the inventory level falls to zero after the demand reaches constant (i.e., ). 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, , which was described above, is considered. The study of this system requires the examination of the ordering relations between the time parameters , which, actually, lead to the following different models:

(i) , (ii) , (iii) , (iv) , (v) , (vi) .

Note that from the definition of demand rate and from credit scheme .

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 , , , 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

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 . Replenishment rate is infinite.(2)Shortages are backlogged at a rate which is a nonincreasing function of with ,   and is the waiting time up to the next replenishment. Moreover, it is assumed that satisfies the relation , where is the derivate of . The case with corresponds to complete backlogging model.(3)The supplier offers cash discount if payment is paid within ; otherwise, the full payment is paid within , (see Huang [10]).(4)The on-hand inventory deteriorates at a constant rate () 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 is a ramp-type function of time given by where is a positive, differentiable function of .

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 and . The inventory level , satisfies the following differential equations: with boundary condition and with boundary condition .

From the two possible relations between parameters and , (i) and (ii) , and following identical steps as in Skouri et al. [23], the sum of holding, deterioration, shortages, and lost sales cost is obtained as where

4. Model I—The Inventory Model When

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 , there are two payment policies for the buyer. Either the payment is paid at time to receive the cash discount (Case 1) or the payment is paid at time so as not to receive the cash discount (Case 2). Then, these two cases will be discussed.

Case 1 (payment is made at time ). In this case, the following subcases should be considered.
Subcase 1.1 (). The purchasing cost is The interest earned during the period of positive inventory level is Since , the total cost in the time interval is calculated using (3.4), (4.1), and (4.2) Subcase 1.2 (). The purchasing cost is (relation (4.1)).
The interest payable for the inventory not being sold after the due date is The interest earned is Since again _, the total cost over is calculated using the relations (3.4), (4.1), (4.4), and (4.6) and is Subcase 1.3 (). The purchasing cost is The interest earned, , is: The interest payable for the inventory not being sold after the due date is Since , the total cost over is again calculated from (3.5), (4.7)–(4.9) and is
The results obtained lead to the following total cost function: So the problem is 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 is continuous at the points and .
The first-order condition for a minimum of is Since and , (4.13) has at least one root. So if is the root of (4.13), this corresponds to minimum since Consequently, is the unique unconstrained minimum of .
The first-order condition for a minimum of is Let us set . If is the root of (4.15) (this may or may not exist), is an increasing function and further if , then and this corresponds to unconstrained minimum of .
The first-order condition for a minimum of is If is the root of (4.17) (this may or may not exist) and , then this corresponds to unconstrained minimum of .

Remark 4.1. The function is not differentiable in .
Then, the following procedure summarizes the previous results for the determination of the optimal replenishment policy, when payment is made at time .

Step 1. Find the global minimum of , say , as follows.
Substep 1.1. Compute from (4.13); if , then set and compute else go to Substep 1.2.Substep 1.2. Find the min and accordingly set .

Step 2. Find the global minimum of , say , as follows.
Substep 2.1. Compute from (4.15); if , then set and compute else go to Substep 2.2.Substep 2.2. Find the min and accordingly set .

Step 3. Find the global minimum of , say , as follows.
Substep 3.1. Compute from (4.17); if , then set and compute else go to Substep 3.2.Substep 3.2. Find the min and accordingly set .

Step 4. Find the and accordingly select the optimal value for say with optimal cost .

Case 2 (payment is made at time ). When the payment is made at time the following cases should be considered.
Subcase 2.1 (). The purchasing cost is. The interest earned during the period of positive inventory level is. Since , the total cost in the time interval is calculated using (3.4), (4.19), and (4.20) Subcase 2.2 (). The purchasing cost is The interest earned is Since again , the total cost over is calculated using the relations (3.5), (4.22), and (4.23) and is Subcase 2.3 (). The purchasing cost is .
The interest earned, , is The interest payable for the inventory not being sold after the due date is Since , the total cost over is again calculated from (3.5), (4.22), (4.25), and (4.26) and is
The results obtained lead to the following total cost function: So the problem is 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 is continuous at the points and .
The first-order condition for the minimum for is Since and , (4.30) has at least one root. So if is the root of (4.30), this corresponds to minimum as So is the unconstrained minimum of .
The first-order condition for a minimum of is If is the root of (4.32) (this may or may not exist), this corresponds to unconstrained minimum of as The first-order condition for a minimum of is If is a root of (4.34) (this may or may not exist) and this corresponds to unconstrained minimum of as

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

Step 1. Find the global minimum of , say , as follows.
Substep 1.1. Compute from (4.30); if , then set and compute ) else go to Substep 1.2.Substep 1.2. Find the min and accordingly set .

Step 2. Find the global minimum of , say , as follows.
Substep 2.1. Compute from (4.32); if , then set and compute else go to Substep 2.2.Substep 2.2. Find the min and accordingly set .

Step 3. Find the global minimum of , say , as follows.
Substep 3.1. Compute from (4.34); if , then set   and compute else go to Substep 3.2.Substep 3.2. Find the min and accordingly set .

Step 4. Find the and accordingly select the optimal value for say with optimal cost .
Finally to find the overall optimum for the problem under consideration, the results obtained for the two presented cases (i.e., payment is made at and payment is made at ) are combined, that is, find and accordingly select the optimal value .

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  € per unit per unit time,  € per unit per unit time,  € per unit,  € per unit per unit time, , years, , years, and , years, years, , , , .