# A Global Optimizing Policy for Decaying Items with Ramp-Type Demand Rate under Two-Level Trade Credit Financing Taking Account of Preservation Technology

Authors: S. R. Singh; Swati Sharma
Journal: Advances in Decision Sciences
 (2013)

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## Abstract

An inventory system for deteriorating items, with ramp-type demand rate, under two-level trade credit policy taking account of preservation technology is considered. The objective of this study is to develop a deteriorating inventory policy when the supplier provides to the retailer a permissible delay in payments, and during this credit period, the retailer accumulates the revenue and earns interest on that revenue; also the retailer invests on the preservation technology to reduce the rate of product deterioration. 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 draw attention to the obtained results, and the sensitivity analysis of the optimal solution with respect to leading parameters of the system is carried out.


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## Body

## 1.  Introduction

In broad spectrum, deterioration is defined as the damage, spoilage, dryness, vaporization, and so forth, that result in the decrease of usefulness of the original one. In the past few decades, inventory problems for deteriorating items have been widely studied. The first attempt to derive optimal policies for deteriorating items was made by Ghare and Schrader [1], who derived a revised form of the economic order quantity (EOQ) model assuming exponential decay. This model was extended to consider the Weibull distribution deterioration by Covert and Philip [2]. Raafat [3] presented a complete survey of the available inventory literature for deteriorating inventory models. Goyal and Giri [4] also provided a detailed review of deteriorating inventory literatures. Teng et al. [5] developed an inventory model for deteriorating items with time varying demand and partial backlogging. Recently, C. Singh and S. R. Singh [6] presented an inventory model considering the Weibull distribution deterioration.Investing in preservation technology (PT) for reducing deterioration rate has received little attention in the past years. The consideration of PT is important due to rapid social changes and the fact that P  can reduce the deterioration rate significantly. Moreover, sales, inventories, and order quantities are very sensitive to the rate of deterioration, especially for fast deteriorating products. The higher rate of deterioration would result in a higher total annual relevant cost and a lower demand rate [7, 8]. Ouyang et al. [9] found that if the retailer can reduce effectively the deteriorating rate of item by improving the storage facility, the total annual relevant inventory cost will be reduced. Many enterprises invest in equipments to reduce the deterioration rate and extend the product expiration date. For example, refrigeration equipments are used to reduce the deterioration rate of fruits, flowers, and sea foods in the supermarket. Murr and Morris [10] showed that a lower temperature will increase the storage life and decrease decay. Wee et al. [11] presented a model using preservation technology.In global market, supplier uses trade credit as a promotion tool to increase his sale and attract new retailers. Therefore, in practice, the supplier will allow a certain fixed period (credit period) for settling the amount that the supplier owes to retailer for the items supplied. Before the end of the trade credit period, the the retailer can sell the goods, accumulate revenue, and earn interest. However, beyond this period the supplier charges interest on the unpaid balance. Hence, trade credit can play a major role in inventory control for both the supplier and the retailer (see Jaggi et al. [12]). Huang and Chung [13] extended Goyal’s [14] model to discuss the replenishment and payment policies to minimize the annual total average cost under cash discount and payment delay from the retailer’s point of view. Some relevant models related to this research area are found in the works of Huang [15], Yang [7], Darzanou and Skouri [16], Singh et al. [17], and so forth.In the literature referring to models with allowable 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]). The term “ramp type” is used to represent such demand pattern. Hill [18] proposed an inventory model with variable branch, being any power function of time. Research on this field continues with Mandal and Pal [19] and Wu and Ouyang [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 falls 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 [19] and Wu and Ouyang [20] and discussed both cases. Skouri et al. [22] extend the work of Deng et al. [21] by introducing a general ramp-type demand rate and Weibull deterioration rate.The present paper is an extension of the inventory system of Darzanou and Skouri [16] in the sense of preservation technology for deteriorating items when the two-level trade credit scheme, r/M1/M2, (in which 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) is considered. The study of this organism requires the inspection of the ordering relations between the time parameters M1,M2,μ,T, which, actually, lead to the various models. To reduce the length of the paper, one of these models (M1≤μ<M2<T) will be presented at this time. The optimal solution of the proposed model not only exists but also is unique. To illustrate the theory of the proposed model the numerical example is provided, and sensitivity analysis with respect to parameters of the system is carried out. The original deterioration rate is assumed to be θ. A reduced deterioration rate of m(ξ) is assumed when the retailer’s investment cost of preservation equipments or technology is ξ.


## 2. Notation and Assumptions

The following notation is used through the paper.


### 2.1. Notation

T is the constant scheduling period (cycle),
t
1 is  the time when the inventory level falls to zero,
C
p is the unit purchase cost,
c
1 is the inventory holding cost per unit per unit time,
c
2 is the shortage cost per unit per unit time,
c
3 is the cost incurred from the deterioration of one unit,
c
4 is the per unit opportunity cost due to the lost sales (c4>Cp see Teng et al. [5]),
S is the maximum inventory level at the scheduling period (cycle),
ξ is the preservation technology (PT) cost for reducing deterioration rate in order to preserve the products, ξ≥0,
m
(
ξ
) is reduced deterioration rate, a function of ξ,
p is the unit selling price,
I
e is the interest rate earned,
I
c is the interest rate charged,
r is cash discount rate, 0<r<1,
M
1 is the period of cash discount in years,
M
2 is the period of permissible delay in payments in years, M1<M2,
μ is the parameter of the ramp-type demand function (time point),
I
(
t
) is 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 levelS. Replenishment rate is infinite.(2)
Shortages are backlogged at a rateβ(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)+c2xβ′(x)-c4β′(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 withinM1; otherwise, the full payment is paid within M2 (see Huang [15]).(4)
The on-hand inventory deteriorates at a constant rateθ(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 rateD(t) is a ramp-type function of time given by
(1)D(t)={f(t),t<μ,f(μ),t≥μ,
where f(t) is a positive, differentiable function of t∈(0,T].


## 2.1. Notation

T is the constant scheduling period (cycle),
t
1 is  the time when the inventory level falls to zero,
C
p is the unit purchase cost,
c
1 is the inventory holding cost per unit per unit time,
c
2 is the shortage cost per unit per unit time,
c
3 is the cost incurred from the deterioration of one unit,
c
4 is the per unit opportunity cost due to the lost sales (c4>Cp see Teng et al. [5]),
S is the maximum inventory level at the scheduling period (cycle),
ξ is the preservation technology (PT) cost for reducing deterioration rate in order to preserve the products, ξ≥0,
m
(
ξ
) is reduced deterioration rate, a function of ξ,
p is the unit selling price,
I
e is the interest rate earned,
I
c is the interest rate charged,
r is cash discount rate, 0<r<1,
M
1 is the period of cash discount in years,
M
2 is the period of permissible delay in payments in years, M1<M2,
μ is the parameter of the ramp-type demand function (time point),
I
(
t
) is 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 levelS. Replenishment rate is infinite.(2)
Shortages are backlogged at a rateβ(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)+c2xβ′(x)-c4β′(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 withinM1; otherwise, the full payment is paid within M2 (see Huang [15]).(4)
The on-hand inventory deteriorates at a constant rateθ(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 rateD(t) is a ramp-type function of time given by
(1)D(t)={f(t),t<μ,f(μ),t≥μ,
where f(t) is a positive, differentiable function of t∈(0,T].


## 3. Deriving the Common Quantities for the Inventory Models

The inventory levelI(t), 0≤t≤T satisfies the following differential equations:
(2)I′(t)+(θ-m(ξ))I(t)=-D(t),0≤t≤t1
with boundary condition I(t1) = 0, and
(3)I′(t)=-D(t)β(T-t),t1≤t≤T,
with boundary condition I(t1)=0.From the two possible relations between parameterst1 and μ, (i) t1≤μ and (ii) t1>μ, the sum of holding, deterioration, shortages, and lost sales cost is obtained as
(4)C(t1)={C1(t1)ift1≤μ,C2(t1)ift1>μ,
where, C1(t1) and C2(t1) are calculated as follows.Case 1 ((t1≤μ) (see Figure 1)).
In this section, the inventory model starting with no shortages is studied. The replenishment at the beginning of the cycle brings the inventory level up toS. Due to demand and deterioration, the inventory level gradually depletes during the period (0,t1) and falls to zero at t=t1. Thereafter, shortages occur during the period (t1,T), which are partially backlogged.
The backlogged demand is satisfied at the next replenishment. The inventory level,I(t), 0≤t≤T satisfies the following differential equations.
In this case, (2) becomes
(5)I′(t)+(θ-m(ξ))I(t)=-f(t),0≤t≤t1,I(t1)=0.
Equation (3) leads to the following two equations:
(6)I′(t)=-f(t)β(T-t),t1≤t≤μ,I(t1)=0,(7)I′(t)=-f(μ)β(T-t),μ≤t≤T,I(μ-)=I(μ+).
The solutions of (5)–(7) are, respectively,
(8)I(t)=e-(θ-m(ξ))t∫tt1f(x)e(θ-m(ξ))xdx,0≤t≤t1,(9)I(t)=-∫t1tf(x)β(T-x)dx,t1≤t≤μ,(10)I(t)=-f(μ)∫μtβ(T-x)dx-∫t1μf(x)β(T-x)dx,μ≤t≤T.
The total amount of deteriorated items during [0,t1] is
(11)D=∫0t1f(x)e(θ-m(ξ))xdx-∫0t1f(t)dt.
The cumulative inventory carried in the interval [0,t1] is found from (8) and is
(12)I1=∫0t1I(t)dt=∫0t1e-(θ-m(ξ))t[∫tt1f(x)e(θ-m(ξ))xdx]dt.
Due to (9) and (10), the time-weighted backorders due to shortages during the interval [t1,T] are
(13)I2=∫t1T[-I(t)]dt=∫t1μ[-I(t)]dt+∫μT[-I(t)]dt=∫t1μ(μ-t)f(t)β(T-t)dt+f(μ)∫μT[∫μtβ(T-x)dx]dt+∫μT[∫t1μf(x)β(T-x)dx]dt.
The amount of lost sales during [t1,T] is
(14)L=∫t1μ[1-β(T-t)]f(t)dt+f(μ)∫μT[1-β(T-t)]dt.
The cost C1(t1) in the time interval [0,T] is the sum of holding, shortage, deterioration, and opportunity costs and is given by
(15)C1(t1)=[ξ+c1{∫0t1e-(θ-m(ξ))t[∫tt1f(x)e(θ-m(ξ))xdx]dt}+c3{∫0t1f(x)e(θ-m(ξ))xdx-∫0t1f(t)dt}+c2{∫t1μ(μ-t)f(t)β(T-t)dt+f(μ)∫μT[∫μtβ(T-x)dx]dt+∫μT[∫t1μf(x)β(T-x)dx]dt}+c4{∫t1μ(1-β(T-t))f(t)dt+f(μ)∫μT(1-β(T-t))dt}].Figure 1
Inventory level for the model starting with no shortage over the cycle (caset1<μ).Case 2 ((t1>μ) (see Figure 2)).
In this case, (2) reduces to the following two equations:
(16)I′(t)+(θ-m(ξ))I(t)=-f(t),0≤t≤μ,I(μ-)=I(μ+),I′(t)+(θ-m(ξ))I(t)=-f(μ),μ≤t≤t1,I(t1)=0.
Equation (3) becomes
(17)I′(t)=-f(μ)β(T-t),t1≤t≤T,I(t1)=0.
Their solutions are, respectively,
(18)I(t)=e-(θ-m(ξ))t×[∫tμf(x)e(θ-m(ξ))xdx+f(μ)∫μt1e(θ-m(ξ))xdx],0≤t≤μ,(19)I(t)=f(μ)e-(θ-m(ξ))t∫tt1e(θ-m(ξ))xdx,μ≤t≤t1,(20)I(t)=-f(μ)∫tt1β(T-x)dx,t1≤t≤T.
The total amount of deteriorated items during [0,t1] is
(21)D=∫0μf(x)e(θ-m(ξ))xdx+f(μ)∫μt1e(θ-m(ξ))xdx-∫0μf(t)dt-∫μt1f(μ)dt.
The total inventory carried in the interval [0,t1] is found from (18) and (19) and is
(22)I1=∫0t1I(t)dt=∫0μI(t)dt+∫μt1I(t)dt=∫0μe-(θ-m(ξ))t×[∫tμf(x)e(θ-m(ξ))xdx+f(μ)∫μt1e(θ-m(ξ))xdx]dt+f(μ)∫μt1e-(θ-m(ξ))t[∫tt1e(θ-m(ξ))xdx]dt.
The time-weighted backorders due to shortages during the interval [t1,T] are
(23)I2=∫t1T[-I(t)]dt=f(μ)∫t1T[∫t1tβ(T-x)dx]dt=f(μ)∫t1T(T-x)β(T-x)dx.
The amount of lost sales during [t1,T] is
(24)L=f(μ)∫t1T[1-β(T-t)]dt.
The cost C2(t1) in the time interval [0,T] is the sum of holding, shortage, deterioration and opportunity costs and is given by
(25)C2(t1)=[ξ+c1{∫0μe-(θ-m(ξ))t×[∫tμf(x)e(θ-m(ξ))xdx+f(μ)∫μt1e(θ-m(ξ))xdx]dt+f(μ)∫μt1e-(θ-m(ξ))t[∫tt1e(θ-m(ξ))xdx]dt}+c3{∫0μf(x)e(θ-m(ξ))xdx+f(μ)×∫μt1e(θ-m(ξ))xdx-∫0μf(t)dt-∫μt1f(μ)dt}+c2{f(μ)∫t1T(T-x)β(T-x)dx}+c4{f(μ)∫t1T(1-β(T-t))dt}].Figure 2
Inventory level for the model starting with no shortage over the cycle (caset1>μ).


## 4. The Inventory Model WhenM1≤μ<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 withinM1, 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 timeM1).
In this case, the following subcases should be considered.Subcase 1.1 (
t
1
≤
M
1
≤
μ
<
T
). The purchasing cost is
(26)CA1,1(t1)=Cp(1-r)×[∫0t1f(x)e(θ-m(ξ))xdx+f(μ)×∫μTβ(T-x)dx+∫t1μf(x)β(T-x)dx].
The interest earned, IT1,1, during the period of positive inventory level is
(27)IT1,1(t1)=pIe∫0t1[∫0tf(x)dx]dt+pIe(M1-t1)∫0t1f(x)dx.
Since t1≤μ, the total cost in the time interval [0,T] is calculated using (15), (26), and (27):
(28)TC1,1(t1)=C1(t1)+CA1,1(t1)-IT1,1(t1).Subcase 1.2 (
M
1
<
t
1
≤
μ
<
T
). The purchasing cost is CA1,1 (relation (26)).The interest payable for the inventory not being sold after the due dateM1 is
(29)PT2,1(t1)=Cp(1-r)Ic∫M1t1e-(θ-m(ξ))t[∫tt1f(x)e(θ-m(ξ))xdx]dt.
The interest earned, IT2,1, is
(30)IT2,1(t1)=pIe∫0M1[∫0tf(x)dx]dt.
Since again t1≤μ, the total cost over [0,T] is calculated using the relations (15), (26), (29), and (30) and is
(31)TC1,2(t1)=C1(t1)+CA1,1(t1)+PT2,1(t1)-IT2,1(t1).Subcase 1.3 (
M
1
≤
μ
≤
t
1
≤
T
). The purchasing cost is
(32)CA2,1(t1)=Cp(1-r)×[∫0μf(x)e(θ-m(ξ))xdx+f(μ)×∫μt1e(θ-m(ξ))xdx+f(μ)∫t1Tβ(T-x)dx].
The interest earned, IT3,1, is
(33)IT3,1(t1)=pIe∫0M1[∫0tf(x)dx]dt.
The interest payable for the inventory not being sold after the due date M1 is
(34)PT3,1(t1)=Cp(1-r)Ic×[∫M1μe-(θ-m(ξ))t[∫tμf(x)e(θ-m(ξ))xdx]dt+f(μ)∫M1μe-(θ-m(ξ))t[∫μt1e(θ-m(ξ))xdx]dt+f(μ)∫μt1e-(θ-m(ξ))t[∫tt1e(θ-m(ξ))xdx]dt].
Since μ<t1, the total cost over (0,T) is again calculated from (25), (32)–(34) and is
(35)TC1,3(t1)=C2(t1)+CA3,1(t1)+PT3,1(t1)-IT3,1(t1).
The results obtained lead to the following total cost function:
(36)TC1(t1)={TC1,1(t1),t1≤M1≤μ<T,TC1,2(t1),M1<t1≤μ<T,TC1,3(t1),M1≤μ≤t1≤T.
So the problem is
(37)minTC1t1(t1).
Its solution requires, separately, studying each of the 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 ofTC1,1(t1) is(38)dTC1,1(t1)dt1={{c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4(1-β(T-t1))-pIe(M1-t1)+Cp(1-r)×(e(θ-m(ξ))t1-β(T-t1)){c1+c3(θ-m(ξ))}(θ-m(ξ))}f(t1)=0.
Since dTC1,1(0)/dt1<0 and dTC1,1(T)/dt1>0, (38) has at least one root. So if t1,1 is the root of (38) and if c2β(x)+c2xβ′(x)-c4β′(x)≥0, then this corresponds to minimum as
(39)(d2TC1,1(t1)dt12)t1=t1,1={{c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,1+{c2β(T-t1,1)+c2(T-t1,1)×β′(T-t1,1)-c4β′(T-t1,1)}+pIe+Cp(1-r)×{(θ-m(ξ))e(θ-m(ξ))t1,1+β′(T-t1)}}f(t1,1)>0.
Consequently, t1,1 is the unique unconstrained minimum of TC1,1(t1).The first-order condition for a minimum ofTC1,2(t1) is
(40)dTC1,2(t1)dt1=({c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4{1-β(T-t1)}+Cp(1-r)(e(θ-m(ξ))t1-β(T-t1))+Cp(1-r)Ic(θ-m(ξ))(e(θ-m(ξ))(t1-M1)-1))f(t1)=0.
If t1,2 is the root of (40) (this may or may not exist) and further if c2β(x)+c2xβ′(x)-c4β′(x)≥0, then
(41)(d2TC1,2(t1)dt12)t1=t1,2=({c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,2+Cp(1-r)Ice(θ-m(ξ))(t1,2-M1)+{c2β(T-t1,2)+c2(T-t1,2)×β′(T-t1,2)-c4β′(T-t1,2)}+Cp(1-r)×{(θ-m(ξ))e(θ-m(ξ))t1,2+β′(T-t1,2)})f(t1,2)>0,
and this t1,2 corresponds to unconstrained minimum of TC1,2(t1).The first-order condition for a minimum ofTC1,3(t1) is
(42)dTC1,3(t1)dt1=({c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4(1-β(T-t1))+Cp(1-r)(e(θ-m(ξ))t1-β(T-t1))+Cp(1-r)Ic(θ-m(ξ))(e(θ-m(ξ))(t1-M1)-1))f(μ)=0.
If t1,3 is the root of (42) (this may or may not exist) and c2β(x)+c2xβ′(x)-c4β′(x)≥0, then
(43)(d2TC1,3(t1)dt12)t1=t1,3=({c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,3+Cp(1-r)Ice(θ-m(ξ))(t1,3-M1)+Cp(1-r)((θ-m(ξ))e(θ-m(ξ))t1,3+β′(T-t1,3))+{c2β(T-t1,3)+c2(T-t1,3)β′(T-t1,3)-c4β′(T-t1,3)})f(μ)>0.
This t1,3 corresponds to unconstrained minimum of TC1,3(t1).Remark 1.
The functionTC1(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 timeM1.Step 1.
Find the global minimum ofTC1,1(t1), say t1,1,M1*, as follows.Substep 1.1. Compute t1,1,M1 from (38); 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 ofTC1,2(t1), say t1,2,M1*, as follows.Substep 2.1.Compute t1,2,M1from (40); if M1<t1,2,M1<μ, then set t1,2,M1*=t1,2,M1 and compute TC1,2(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 ofTC1,3(t1), say t1,3,M1*, as follows.Substep 3.1. Compute t1,3,M1 from (42); if μ<t1,3,M1, then set t1,3,M1*=t1,3,M1 and 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 themin{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 TC1(t1,M1*).Case 22 (payment is made at time M2).
When the payment is made at timeM2 the following cases should be considered.Subcase 2.1 (
t
1
≤
μ
<
M
2
<
T
). The purchasing cost is
(44)CA1,2(t1)=CA1,1(t1)(1-r).
The interest earned during the period of positive inventory level is
(45)IT1,2(t1)=pIe∫0t1[∫0tf(x)dx]dt+pIe(M2-t1)∫0t1f(x)dx.
Since t1≤μ, the total cost in the time interval [0,T] is calculated using (15), (44), and (45):
(46)TC2,1(t1)=C1(t1)+CA1,2(t1)-IT1,2(t1).Subcase 2.2 (
μ
<
t
1
≤
M
2
<
T
). The purchasing cost is
(47)CA2,2(t1)=CA2,1(t1)(1-r).
The interest earned, IT2,2(t1), is
(48)IT2,2(t1)=pIe[∫0μ[∫0tf(x)dx]dt+∫μM2[∫0μf(x)dx]dt+∫μt1[∫μtf(μ)dx]dt+∫t1M2[∫μt1f(μ)dx]dt].
Since again μ≤t1, the total cost over [0,T] is calculated using the relations (25), (47), and (48) and is
(49)TC2,2(t1)=C2(t1)+CA2,2(t1)-IT2,2(t1).Subcase 2.3 (
μ
≤
M
2
≤
t
1
≤
T
). The purchasing cost is CA2,2(t1).The interest earned,IT3,2, is
(50)IT3,2(t1)=pIe[∫0μ[∫0tf(x)dx]dt+∫μM2[∫0μf(x)dx]dt].
The interest payable for the inventory not being sold after the due date M2 is
(51)PT3,2(t1)=CpIcf(μ)[∫M2t1e-(θ-m(ξ))t[∫tt1e(θ-m(ξ))xdx]dt].
Since μ<t1, the total cost over [0,T] is again calculated from (25), (47), (50), and (51) and is
(52)TC2,3(t1)=C2(t1)+CA2,2(t1)+PT3,2(t1)-IT3,2(t1).
The results obtained lead to the following total cost function:
(53)TC2(t1)={TC2,1(t1),t1≤μ<M2<T,TC2,2(t1),μ<t1≤M2<T,TC2,3(t1),μ≤M2≤t1≤T.
So the problem is
(54)minTC2t1(t1).
Its solution, as in the previous case, requires, separately, studying each of the 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 forTC2,1(t1)  is
(55)dTC2,1(t1)dt1={{c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4(1-β(T-t1))-pIe(M2-t1)+Cp{e(θ-m(ξ))t1-β(T-t1)}{c1+c3(θ-m(ξ))}(θ-m(ξ))}f(t1)=0.
Since dTC2,1(0)/dt1<0 and dTC2,1(T)/dt1>0, (55) has at least one root. So, if t1,1 is the root of (55), this corresponds to minimum if c2β(x)+c2xβ′(x)-c4β′(x)≥0 as
(56)(d2TC2,1(t1)dt12)t1=t1,1={{c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,1+{c2β(T-t1,1)+c2(T-t1,1)β′(T-t1,1)-c4β′(T-t1,1)}+pIe+Cp{(θ-m(ξ))e(θ-m(ξ))t1,1+β′(T-t1)}}f(t1,1)>0.
So, t1,1 is the unconstrained minimum of TC2,1(t1).The first-order condition for a minimum ofTC2,2(t1) is
(57)dTC2,2(t1)dt1=({c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4(1-β(T-t1))+Cp{e(θ-m(ξ))t1-β(T-t1)}-pIe(M2-t1){c1+c3(θ-m(ξ))}(θ-m(ξ)))f(μ)=0.
If t1,2 is the root of (57) (this may or may not exist), this corresponds to unconstrained minimum of TC2,2(t1) if c2β(x)+c2xβ′(x)-c4β′(x)≥0 as
(58)(d2TC2,2(t1)dt12)t1=t1,2=({c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,2+{c2β(T-t1,2)+c2(T-t1,2)×β′(T-t1,2)-c4β′(T-t1,2)}+pIe+Cp{(θ-m(ξ))e(θ-m(ξ))t1,2+β′(T-t1,2)})f(μ)>0.
The first-order condition for a minimum of TC2,3(t1) is
(59)dTC2,3(t1)dt1=({c1+c3(θ-m(ξ))}(θ-m(ξ))(e(θ-m(ξ))t1-1)-c2(T-t1)β(T-t1)-c4(1-β(T-t1))+Cp(e(θ-m(ξ))t1-β(T-t1))+CpIc(θ-m(ξ))(e(θ-m(ξ))(t1-M2)-1))f(μ)=0.
If t1,3 is a root of (59) (this may or may not exist) and this corresponds to unconstrained minimum of TC2,3(t1) if c2β(x)+c2xβ′(x)-c4β′(x)≥0 as
(60)(d2TC2,3(t1)dt12)t1=t1,3=({c1+c3(θ-m(ξ))}e(θ-m(ξ))t1,3+{c2β(T-t1,3)+c2(T-t1,3)×β′(T-t1,3)-c4β′(T-t1,3)}+Cp{(θ-m(ξ))e(θ-m(ξ))t1,3+β′(T-t1,3)}+CpIce(θ-m(ξ))(t1,3-M2))f(μ)>0.Remark 2.
The functionTC2(t1) is not differentiable in M2.Then, the following procedure summarizes the previous results for the determination of the optimal replenishment policy, when payment is made at timeM2.Step 1.
Find the global minimum ofTC2,1(t1), say t1,1,M2*, as follows.Substep 1.1. Compute t1,1,M2 from (55); 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 ofTC2,2(t1), say t1,2,M2*, as follows.Substep 2.1. Compute t1,2,M2 from (57); 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 ofTC2,3(t1), say t1,3,M2*, as follows.Substep 3.1. Compute t1,3,M2 from (59); 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 themin{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 optimumt1 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 system parameters, is carried out. Here,m(ξ) is a function of the preservation technology cost ξ such that m(ξ)=θ(1-e-aξ), a≥0, and a is the simulation coefficient representing the percentage increase in m(ξ) per euro increase in ξ, which means m(ξ) is an increasing function bounded above by θ.The input parameters arec1=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, a=0.001, ξ=20€, T=0.5 years, f(t)=3e4.5t and β(x)=e-0.2x, M1=0.13 years, M2=0.43 years, p=15, Cp=10€, Ie=0.12, and Ic=0.15.


### 5.1. The Payment Is Made atM1

From (38), t1,1,M1=0.398924, which is not feasible as t1,1,M1>M1. Since TC1,1(0)=75.469 and TC1,1(M1)=71.9633, it follows that t1,1,M1*=M1. From (40), t1,2,M1=0.40288, which is not valid again as t1,2,M1>μ. Since TC1,1(M1)=TC1,2(M1)=71.9633 and TC1,2(μ)=67.0128, the optimal value for t1,2,M1*=μ. From (42), t1,3,M1=0.412557; 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*)=65.5819.Finally,TC1(t1,M1*)=min{TC1,1(M1),TC1,2(μ),TC1,3(t1,3,M1*)}=65.5819, and consequently t1,M1*=0.412557.


### 5.2. The Payment Is Made atM2

From (55), t1,1,M2=0.424279 which is not feasible as t1,1,M2>μ. Since TC2,1(0)=75.6719, TC2,1(μ)=66.2333, it follows that t1,1,M2*=μ. From (57), t1,2,M2=0.424279 which is valid again as μ<t1,2<M2 so t1,2,M2*=0.424279 and TC2,2(t1,2,M2*)=64.3494. From (59), t1,3,M2=0.424197; this value for t1 is not valid as t1,3<M2. Since TC2,3(M2)=64.5295, TC2,3(T)=65.2301 so t1,3,M2*=M2 and TC2,3(t1,3,M2*)=64.5295.Finally,TC2(t1,M2*)=min{TC2,1(μ),TC2,2(t1,2,M2*),TC2,3(M2)}=64.3494, and consequently t1,M2*=0.424279.So, asTC(t1*)=min{TC1(t1,M1*),TC2(t1,M2*)}=64.3494, the optimal t1 is t1*=t1,M2*=0.424279, which leads to a payment at M2.


## 5.1. The Payment Is Made atM1

From (38), t1,1,M1=0.398924, which is not feasible as t1,1,M1>M1. Since TC1,1(0)=75.469 and TC1,1(M1)=71.9633, it follows that t1,1,M1*=M1. From (40), t1,2,M1=0.40288, which is not valid again as t1,2,M1>μ. Since TC1,1(M1)=TC1,2(M1)=71.9633 and TC1,2(μ)=67.0128, the optimal value for t1,2,M1*=μ. From (42), t1,3,M1=0.412557; 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*)=65.5819.Finally,TC1(t1,M1*)=min{TC1,1(M1),TC1,2(μ),TC1,3(t1,3,M1*)}=65.5819, and consequently t1,M1*=0.412557.


## 5.2. The Payment Is Made atM2

From (55), t1,1,M2=0.424279 which is not feasible as t1,1,M2>μ. Since TC2,1(0)=75.6719, TC2,1(μ)=66.2333, it follows that t1,1,M2*=μ. From (57), t1,2,M2=0.424279 which is valid again as μ<t1,2<M2 so t1,2,M2*=0.424279 and TC2,2(t1,2,M2*)=64.3494. From (59), t1,3,M2=0.424197; this value for t1 is not valid as t1,3<M2. Since TC2,3(M2)=64.5295, TC2,3(T)=65.2301 so t1,3,M2*=M2 and TC2,3(t1,3,M2*)=64.5295.Finally,TC2(t1,M2*)=min{TC2,1(μ),TC2,2(t1,2,M2*),TC2,3(M2)}=64.3494, and consequently t1,M2*=0.424279.So, asTC(t1*)=min{TC1(t1,M1*),TC2(t1,M2*)}=64.3494, the optimal t1 is t1*=t1,M2*=0.424279, which leads to a payment at M2.


## 6. Sensitivity Analysis

Using the data of the previous example, a sensitivity analysis is carried out to explore the effect of change on the basic model’s parameters to the optimal policy (i.e.,t1 time of payment and optimal total cost). The results are presented in Tables 1 and 2, and some observations are summarized as follows.(1)
It is observed that as the holding costc1 decreases or increases the optimal time t1* increases or decreases, and the optimal total cost TC(t1*) decreases or increases, respectively.(2)
It is seen that as the shortage costc2 decreases or increases both the optimal time t1* and the optimal total cost TC(t1*) decrease or increase, respectively.(3)
It is examined that as the time (at which demand becomes constant)μ decreases or increases there is no effect on the optimal time t1* but the optimal total cost TC(t1*) decreases or increases, respectively.(4)
It is observed that as the deterioration costc3 decreases or increases the optimal time t1* slightly increases or decreases, and the optimal total cost TC(t1*) also slightly decreases or increases, respectively.(5)
It is seen that as the lost sales costc4 decrease or increase both the optimal time t1* and the optimal total cost TC(t1*) slightly decrease or increase, respectively.(6)
It is seen that as the decay rateθ decreases or increases the optimal time t1* increases or decreases and the optimal total cost TC(t1*) slightly decreases or increases, respectively.(7)
It is observed that as the preservation costξ decreases or increases the optimal time t1* and the optimal total cost TC(t1*) decrease or increase, respectively.(8)
It is noticed that as the rate of interest earnedIe decreases or increases the optimal time t1* decreases or increases, and the optimal total cost TC(t1*) slightly increases or decreases, respectively.(9)
It is noticed that as the selling pricep decreases or increases the optimal time t1* decreases or increases, but the optimal total cost TC(t1*) increases or decreases, respectively.(10)
It is seen that as the purchasing costCp decreases or increases the optimal time t1* increases or decreases, and the optimal total cost TC(t1*) decreases or increases, respectively.(11)
It is observed that as the preservation parametera decreases or increases the optimal time t1* decreases or increases, but the optimal total cost TC(t1*) increases or decreases, respectively.Table 1
The effect of changing the parameter (i) while keeping all other parameters unchanged.
Parameter (i)
Percentage of changes (%)
t
1
*
TC(t1*)
c
1
−20
0.436535
63.8099
−10
0.430325
64.0842
+10
0.418390
64.6058
+20
0.412653
64.8538
c
2
−20
0.412116
64.2352
−10
0.418648
64.2965
+10
0.429184
64.3955
+20
0.433494
64.4360
μ
−20
0.424279
57.5810
−10
0.424279
60.9523
+10
0.424279
67.7020
+20
0.424279
70.9180
c
3
−20
0.424299
64.3486
−10
0.424289
64.3490
+10
0.424269
64.3499
+20
0.424260
64.3503
c
4
−20
0.421352
64.3220
−10
0.422843
64.3360
+10
0.425662
64.3624
+20
0.426995
64.3749
θ
−20
0.424340
64.3468
−10
0.424310
64.3481
+10
0.424249
64.3508
+20
0.424218
64.3521
ξ
−20
0.424278
60.3495
−10
0.424278
62.3495
+10
0.424280
66.3494
+20
0.424280
68.3494
I
e
−20
0.424181
64.5541
−10
0.424230
64.4518
+10
0.424327
64.2471
+20
0.424374
64.1448
p
−20
0.424181
64.5541
−10
0.424230
64.4518
+10
0.424327
64.2471
+20
0.424374
64.1448
C
p
−20
0.425700
55.9219
−10
0.424996
60.1357
+10
0.423548
68.5630
+20
0.422804
72.7765Table 2
The effect of simulation coefficient (a) while keeping all other parameters unchanged.
Parameter (i)
Changing value
t
1
*
TC(t1*)
a
0.0005
0.424276
64.3496
0.005
0.424303
64.3484
0.01
0.424329
64.3473
0.02
0.424375
64.3453
0.05
0.424470
64.3412


## 7. Concluding Remarks

In this paper, we have developed an inventory model with ramp type demand rate, partially backlogged shortages, preservation technology, and two-level trade credit. The model is discussed under the two cases (1) when the payment is made at timeM1 to get the price discount and (2) when the payment is made at time M2 with no price discount. The effect of preservation technology is also taken into consideration throughout the model. The presented model is illustrated through numerical experiments with sensitivity. From the sensitivity table, it is observed that the changes in the parameters preservation technology cost (ξ), time at which demand becomes constant (μ), and purchasing cost (Cp) have greater effect on the system than the other parameters of the system. The convexity of the cost function is also exposed analytically.This model could be inclusive taking into consideration numerous replenishment cycles throughout the planning horizon.



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*Source: 126385-2013-02-25.xml*