#### 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 . 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 βramp-typeβ 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 supplierβ 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

is the constant scheduling period (cycle),the time when the inventory level falls to zero, the maximum inventory level at each scheduling period (cycle),the unit purchase cost,the inventory holding cost per unit per unit time,the shortage cost per unit per unit time,the cost incurred from the deterioration of one unit,the per unit opportunity cost due to the lost sales ( see Teng et al. [27]),the unit selling price,the interest rate earned, the interest rate charged, cash discount rate, ,the period of cash discount in years, the period of permissible delay in payments in years, ,the parameter of the ramp-type demand function (time point), and the inventory level at time .##### 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 ,ββ 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 IβThe 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)β(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 ββand 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 modelβs parameters, is carried out.

The input parameters are ββ¬ per unit per unit time, ββ¬ per unit per unit time, ββ¬ per unit, ββ¬ per unit per unit time, , years, , years, and , years, years, , , , .

##### 5.1. The Payment Is Made at

From (4.13), , which is not feasible as . Since and , it follows that . From (4.15), , which is not valid again as . Since and , the optimal value for . From (4.17) ; this value for is valid as so and .

Finally and consequently .

##### 5.2. The Payment Is Made at

From (4.30), which is not feasible as . Since βββ, it follows that . From (4.32), which is valid again as so and . From (4.34), ; this value for is also valid as so and .

Finally , and consequently .

So, as , the optimal is , which leads to a payment at .

Using the data of the previous example, a sensitivity analysis is carried out to explore the effect of change on some, of the basic, modelβs parameters to the optimal policy (i.e., 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 , and have no impact on the optimal , the time of payment and the optimal cost.(2)The error on the parametersβ estimation of has no impact on the time of payment, small impact on the optimal , but high impact on the total optimal cost. This last observation is in line with the relative findings in Deng et al. [21].

#### 6. Conclusions

In this paper, the following interrelated factors, which have appeared in the literature of inventory control, are incorporated: (i) the productβs life cycle, which implies that its demand can be described as a ramp-type function of time, (ii) the effect of deterioration, (iii) the 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 credit scheme is considered. The study of this system requires the examination of the ordering relations between the time parameters , , , , which, actually, lead to the six different models. This inventory system, setting , , , , and , 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, [28β30].

#### Acknowledgment

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