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

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.


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  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.

Notation and Assumptions
The following notation is used through the paper.

Notation
T is the constant scheduling period cycle ,

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 C 2 β x C 2 Tβ x C p β 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 M 1 ; otherwise, the full payment is paid within M 2 , 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 by where f t is a positive, differentiable function of t ∈ 0, T .

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 t 1 and μ. where 3.5

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 M 1 , there are two payment policies for the buyer.Either the payment is paid at time M 1 to receive the cash discount Case 1 or the payment is paid at time M 2 so as not to receive the cash discount Case 2 .Then, these two cases will be discussed.
Case 1 payment is made at time M 1 .In this case, the following subcases should be considered.
The interest earned during the period of positive inventory level is Since t 1 ≤ μ, the total cost in the time interval 0, T is calculated using 3.4 , 4.1 , and 4.2 The interest payable for the inventory not being sold after the due date M 1 is The interest earned is Since again t 1 ≤ μ , the total cost over 0, T is calculated using the relations 3.4 , 4.1 , 4.4 , and 4.6 and is The interest earned, I T 3,1 , is: The interest payable for the inventory not being sold after the due date M 1 is

4.9
Since μ < t 1 , the total cost over 0, T is again calculated from 3.5 , 4.7 -4.9 and is

4.10
The results obtained lead to the following total cost function:

4.11
So the problem is min 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 TC 1 t 1 is continuous at the points M 1 and μ.
The first-order condition for a minimum of TC 1,1 t 1 is

4.13
Since dT C 1,1 0 /dt 1 < 0 and dT C 1,1 T /dt 1 > 0, 4.13 has at least one root.So if t 1,1 is the root of 4.13 , this corresponds to minimum since The first-order condition for a minimum of TC 1,2 t 1 is

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Let us set h x c 2 xβ x c 4 1 − β x − pI e x.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 4.16 and this t 1,2 corresponds to unconstrained minimum of TC 1,2 t 1 .
The first-order condition for a minimum of TC 1,3 t 1 is 4.17 If t 1,3 is the root of 4.17 this may or may not exist and h x > 0, then this t 1,3 corresponds to unconstrained minimum of TC 1,3 t 1 .
Remark 4.1.The function Then, the following procedure summarizes the previous results for the determination of the optimal replenishment policy, when payment is made at time M 1 .
Case 2 payment is made at time M 2 .When the payment is made at time M 2 the following cases should be considered.
The interest earned during the period of positive inventory level is.

4.20
Since t 1 ≤ μ, the total cost in the time interval 0, T is calculated using 3.4 , 4.19 , and 4.20 The interest earned is

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Since again μ ≤ t 1 , the total cost over 0, T is calculated using the relations 3.5 , 4.22 , and 4.23 and is The interest earned, I T 3,2 , is The interest payable for the inventory not being sold after the due date M 2 is e θx dx dt.

4.26
Since μ < t 1 , the total cost over 0, T is again calculated from 3.5 , 4.22 , 4.25 , and 4.26 and is

4.27
The results obtained lead to the following total cost function:

4.28
So the problem is min 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 TC 2 t 1 is continuous at the points M 2 and μ.
The first-order condition for the minimum for TC 2,1 t 1 is

4.30
Since dT C 2,1 0 /dt 1 < 0 and dT C 2,1 T /dt 1 > 0, 4.30 has at least one root.So if t 1,1 is the root of 4.30 , this corresponds to minimum as The first-order condition for a minimum of TC 2,2 t 1 is

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 t 1 as

4.33
The first-order condition for a minimum of TC 2,3 t 1 is

4.34
If t 1,3 is a root of 4.34 this may or may not exist and c 1 c 3 θ C p I c ≥ pI e this corresponds to unconstrained minimum of TC 2,3 t 1 as

4.35
Remark 4.2.The function The procedure for the determination of the optimal replenishment policy when payment is made at time M 2 is as follows.
Finally to find the overall optimum t 1 for the problem under consideration, the results obtained for the two presented cases i.e., payment is made at M 1 and payment is made at M 2 are combined, that is, find min{TC 1 t 1,M 1 , TC 2 t 1,M 2 } and accordingly select the optimal value t * 1 .

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 c   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, M 1 , M 2 , T, r to the optimal policy i.e., t 1 time of payment and optimal total cost .The results are presented in Table 1 and some interesting findings are summarized as follows.

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r/M 1 /M 2 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/M 1 /M 2 credit scheme is considered.The study of this system requires the examination of the ordering relations between the time parameters M 1 , M 2 , μ, T, which, actually, lead to the six different models.This inventory system, setting f t D 0 t, β x 1, M 1 M 2 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 .

t 1
the time when the inventory level falls to zero, S the maximum inventory level at each scheduling period cycle , C p the unit purchase cost, c 1 the inventory holding cost per unit per unit time, c 2 the shortage cost per unit per unit time, c 3 the cost incurred from the deterioration of one unit, c 4 the per unit opportunity cost due to the lost sales c 4 > C p see Teng et al. 27 , p the unit selling price, I e the interest rate earned, I c the interest rate charged, r cash discount rate, 0 < r < 1, M 1 the period of cash discount in years, M 2 the period of permissible delay in payments in years, M 1 < M 2 , μ the parameter of the ramp-type demand function time point , and I t the inventory level at time t.

Table 1 :
Sensitivity analysis: the effect of changing the parameter i keeping all other parameters unchanged.TC 2 t 1,M 2 min{TC 1 t 1,M 1 , TC 2 t 1,M 2 }, the optimal t 1 is t *