# Optimal Transfer-Ordering Strategy for a Deteriorating Inventory in Declining Market

Authors: Nita H. Shah; Kunal T. Shukla
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

The retailer's optimal procurement quantity and the number of transfers from the warehouse to the display area are determined when demand is decreasing due to recession and items in inventory are subject to deterioration at a constant rate. The objective is to maximize the retailer's total profit per unit time. The algorithms are derived to find the optimal strategy by retailer. Numerical examples are given to illustrate the proposed model. It is observed that during recession when demand is decreasing, retailer should keep a check on transportation cost and ordering cost. The display units in the show room may attract the customer.


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## Body

## 1. Introduction

The management of inventory is a critical concern of the managers, particularly, during recession when demand is decreasing with time. The second most worrying issue is of transfer batching, the integration of production and inventory model, as well as the purchase and shipment of items. Goyal [1], for the first time, formulated single supplier-single retailer-integrated inventory model. Banerjee [2] derived a joint economic lot size model under the assumption that the supplier follows lot-for-lot shipment policy for the retailer. Goyal [3] extended Banerjee’s [2] model. It is assumed that numbers of shipments are equally sized and the production of the batch had to be finished before the start of the shipment. Lu [4] allowed shipments to occur during the production period. Goyal [5] derived a shipment policy in which, during production, a shipment is made as soon as the buyer is about to face stock out and all the produced stock manufactured up to that point is shipped out. Hill [6] developed an optimal two-stage lot sizing and inventory batching policies. Yang and Wee [7] developed an integrated multilot-size production inventory model for deteriorating items. Law and Wee [8] derived an integrated production-inventory model for ameliorating and deteriorating items using DCE approach. Yao et al. [9] argued the importance of supply chain parameters when vendor-buyer adopts joint policy. The interesting papers in this areas are by Wee [10], Hill [11, 12], Vishwanathan [13], Goyal and Nebebe [14], Chiang [15], Kim and Ha [16], Nieuwenhuyse and Vandaele [17], Siajadi et al. [18], and their cited references. The aforesaid articles are dealing with integrated Vendor-buyer inventory model when demand is deterministic and known constant.The aim of this paper is to determine the ordering and transfer policy which maximizes the retailer’s profit per unit time when demand is decreasing with time. It is assumed that on the receipt of the delivery of the items, retailer stocks some items in the showroom and rest of the items is kept in warehouse. The floor area of the showroom is limited and wellfurnished with the modern techniques. Hence, the inventory holding cost inside the showroom is higher as compared to that in warehouse. The problem is how often and how many items are to be transferred from the warehouse to the showroom which maximizes the retailer’s total profit per unit time. Here, demand is decreasing with time. This paper is organized as follows. Section2 deals with the assumptions and notations for the proposed model. In Section 3, a mathematical model is formulated to determine the ordering-transfer policy which maximizes the retailer’s profit per unit time. Section 4 deals with the establishment of the necessary conditions for an optimal solution. Using these conditions, the algorithms are developed. In Section 5, numerical examples are given. The sensitivity analysis of the optimal solution with respect to system parameter is carried out. The research article ends with conclusion in Section 5.


## 2. Mathematical Model

### 2.1. The Total Cost per Cycle in the Warehouse

The retailer ordersQ-units per order from a supplier and stocks these items in the warehouse. The q-units are transferred from the warehouse to the showroom until the inventory level in the warehouse reaches to zero. Hence Q = nq. The total cost per cycle during the cycle time T in the warehouse is the sum of (1), the ordering cost A, and (2) the inventory holding cost, hw[(n(n-1)/2)q]t1.


### 2.2. The Total Cost per Unit Cycle in the Showroom

Initially, the inventory level isL0≤L due to the unit’s transfer from the warehouse to the display area. The inventory level then depletes to R due to time-dependent demand and deterioration of units at the end of the retailer’s cycle time, “t1.” A graphical representation of the inventory system is exhibited in Figure 1.Figure 1
Combined inventory status for items in the warehouse and showroom.The differential equation representing inventory status at any instant of timet is given by(2.1)dI(t)dt=-D(t)-θI(t),0≤t≤t1
with boundary condition I(t1)=R. The solution of (2.1) is(2.2)I(t)=Reθ(t1-t)+a((eθ(t1-t)-1)(θ+b)θ2-b(t1eθ(t1-t)-t)θ);0≤t≤t1.
The total cost incurred during the cycle time t1 is the sum of the ordering cost, G and the inventory holding cost, where(2.3)inventory holding cost=hd∫0t1I(t)dt=hd(-Rθ+a(bθ2t12-2θ-2b-2θ2t12θ3))-hdeθt1(a(θbt1-θ-bθ3)-Rθ)
Using (2.2) and I(0)=q+R, we get(2.4)q=Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2θ2.
The revenue per cycle is(2.5)(P-C)q=(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)θ2.
Then inventory holding cost in the warehouse is
(2.6)hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)2θ2.
Hence, the total profit, ZP per cycle during the period [0, T] is(2.7)ZP=Revenue-[total cost in the warehouse]-[total cost in the showroom]=(n(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)θ2-A-hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)2θ2-nG-nhd(-Rθ+a(bθ2t12-2θ-2b-2θ2t12θ3))+nhdeθt1(a(θbt1-θ-bθ3)-Rθ)).
During period [0, T], there are n-transfers at every t1-time units. Hence, T = nt1. Therefore, the total profit per time unit is
(2.8)Z(n,R,t1)=ZPT=(n(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)/θ2-A-nG+hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)/2θ2-nhd(-R/θ+a((bθ2t12-2θ-2b-2θ2t1)/2θ3))+nhdeθt1(a((θbt1-θ-b)/θ3)-R/θ))nt1.


## 2.1. The Total Cost per Cycle in the Warehouse

The retailer ordersQ-units per order from a supplier and stocks these items in the warehouse. The q-units are transferred from the warehouse to the showroom until the inventory level in the warehouse reaches to zero. Hence Q = nq. The total cost per cycle during the cycle time T in the warehouse is the sum of (1), the ordering cost A, and (2) the inventory holding cost, hw[(n(n-1)/2)q]t1.


## 2.2. The Total Cost per Unit Cycle in the Showroom

Initially, the inventory level isL0≤L due to the unit’s transfer from the warehouse to the display area. The inventory level then depletes to R due to time-dependent demand and deterioration of units at the end of the retailer’s cycle time, “t1.” A graphical representation of the inventory system is exhibited in Figure 1.Figure 1
Combined inventory status for items in the warehouse and showroom.The differential equation representing inventory status at any instant of timet is given by(2.1)dI(t)dt=-D(t)-θI(t),0≤t≤t1
with boundary condition I(t1)=R. The solution of (2.1) is(2.2)I(t)=Reθ(t1-t)+a((eθ(t1-t)-1)(θ+b)θ2-b(t1eθ(t1-t)-t)θ);0≤t≤t1.
The total cost incurred during the cycle time t1 is the sum of the ordering cost, G and the inventory holding cost, where(2.3)inventory holding cost=hd∫0t1I(t)dt=hd(-Rθ+a(bθ2t12-2θ-2b-2θ2t12θ3))-hdeθt1(a(θbt1-θ-bθ3)-Rθ)
Using (2.2) and I(0)=q+R, we get(2.4)q=Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2θ2.
The revenue per cycle is(2.5)(P-C)q=(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)θ2.
Then inventory holding cost in the warehouse is
(2.6)hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)2θ2.
Hence, the total profit, ZP per cycle during the period [0, T] is(2.7)ZP=Revenue-[total cost in the warehouse]-[total cost in the showroom]=(n(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)θ2-A-hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)2θ2-nG-nhd(-Rθ+a(bθ2t12-2θ-2b-2θ2t12θ3))+nhdeθt1(a(θbt1-θ-bθ3)-Rθ)).
During period [0, T], there are n-transfers at every t1-time units. Hence, T = nt1. Therefore, the total profit per time unit is
(2.8)Z(n,R,t1)=ZPT=(n(P-C)(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)/θ2-A-nG+hwn(n-1)t1(Reθt1θ2+aeθt1θ+aeθt1b-aθ-ab-abt1eθt1θ-Rθ2)/2θ2-nhd(-R/θ+a((bθ2t12-2θ-2b-2θ2t1)/2θ3))+nhdeθt1(a((θbt1-θ-b)/θ3)-R/θ))nt1.


## 3. Necessary and Sufficient Condition for an Optimal Solution

The total profit per unit time of a retailer is a function of three variables, namely,n, R and t1:(3.1)∂2Z(n,R,t1)∂n2=-2An3t1<0.
Thus, the retailer’s total profit per unit time is a concave function of n for fixed R and t1.Next, to determine the optimum cycle time for showroom, for givenn, we first differentiate Z(n,R,t1) with respect to R. We get(3.2)∂Z(n,R,t1)∂R=(1-eθt1t1)(-(P-C)+hw(n-1)t12+hdθ).
Depending on the sign of (P-C)θ-hd three cases arise: Define Δ=(P-C)θ-hd.Case 1 (Δ<0).
IfΔ<0, then Z(n,R,t1) is a decreasing function of R for fixed R. It suggests that no transfer of units should be made from the warehouse to the showroom; so put R = 0 in Z(n,R,t1) and differentiate resultant expression with respect to t1. We have
(3.3)∂Z∂t1|R=0=0(a(P-C)(1-bt1)eθt1-(1/2)hw(n-1)aθ2t1(1-bt1)eθt1+(1/2)(hw(n-1)a((1-eθt1)(θ+b)+bθt1eθt1))/θ2-((hda/θ2)(bt1-1)(1-eθt1)))t1-(a(P-C)((1-eθt1)(θ+b)+bt1eθt1θ)/θ2+hw(n-1)a((1-eθt1)(θ+b)+bt1eθt1θ)t1/2θ2-A/n-G-(hda(bt1(2+θt1)/2θ2-(θ+b)(1+θt1)/θ3)-hda((bθt1-θ-b)eθt1/θ3)))t12=0.
The sufficiency condition is ∂2Z(n,R,t1)/∂t12<0, that is,
(3.4)12θ3nt13(-4naθ3t1Peθt1+4naθ3t1Ceθt1+4nθ2Paeθt1-4nθ2Caeθt1+4nθPabeθt1-4nθ2Pa+4nθ2Ca-4nGθ3-4Aθ3-4nθPab+4nθCab+4nhdaθ+4nhdab-4nθ2Pabt1eθt1-4nθCabeθt1+4nθ2Cabt1eθt1-4nhdaθeθt1-4nhdabeθt1+4nhdabt1θeθt1+2naθ4t12Peθt1+2naθ3t12Pbeθt1-2naθ4t13Pbeθt1-2naθ4t12Ceθt1-2naθ3t12Cbeθt1+2naθ4t13Cbeθt1-n2aθ4t13hweθt1+n2aθ3t13hwbeθt1+n2aθ4t14hwbeθt1+naθ4t13hweθt1-naθ3t13hwbeθt1-naθ4t14hwbeθt1-2naθ3t12hdeθt1-2naθ2t12hdbeθt1+2naθ3t13hdbeθt1+4naθ2t1hdeθt1)<0.
Thus, Z(n,t1), the total profit per unit time, is a concave function of t1 for fixed n. There exists a unique t1, denoted by t1*1 such that Z(n,t1*1) is maximum. Substituting t1*1 and R*=0 into (2.5) are obtain number of units to be transferred (say) q*1 for fixed n.Note.
Sinceq*1≤L for all q, q*1=L. If q*1>L, then obtain t1*1 using
(3.5)t1*1=1θln[1+Lθ2a(θ+b)].Case 2 (Δ=0).
In this case, we made (2.8) as
(3.6)Z(n,R,t1)=(hwReθt12+hwaeθt12θ+hwabeθt12θ2-hwa2θ-hwab2θ2-t1hwabeθt12θ-hwR2-Gt1-Ant1-nhwReθt12-nhwaeθt12θ-nhwabeθt12θ2+nhwa2θ+nhwab2θ2+nt1hwabeθt12θ+nhwR2+hdaθ-t1hdab2θ).
Here,
(3.7)∂Z(n,R,t1)∂R=-hw2(n-1)(eθt1-1)<0.
that is, Z(n,R,t1) is decreasing function of R for given n. So no transfer should be made from the warehouse to the showroom, that is, R = 0. So (3.6) becomes
(3.8)Z(n,t1)=(hwaeθt12θ+hwabeθt12θ2-hwa2θ-hwab2θ2-t1hwabeθt12θ-Gt1-Ant1-nhwaeθt12θ-nhwabeθt12θ2+nhwa2θ+nhwab2θ2+nt1hwabeθt12θ+hdaθ-t1hdab2θ).
The optimal value of t1*2 can be obtained by solving
(3.9)∂Z(n,t1)∂t1=(hwaeθt12-t1hwabeθt12+Gt12+Ant12-nhwaeθt12+hwt1nabeθt12-hdab2θ)=0.
The sufficiency condition is
(3.10)∂2Z(n,t1)∂t12=-(nhwaθeθt12-nabhweθt12-nabt1θhweθt12-aθhweθt12+abhweθt12+t1hwabθeθt12+2Gt13+2Ant13)<0,fort1=t1*2.
Then, Z(n,t1*2) is a concave function of t1*2 and hence Z(n,t1*2) is the maximum profit of the retailer. q*2 can be obtained by substituting value of t1*2 in (2.5).Note.
Sinceq*2≤L for all q, then q*2=L. If q*2>L, then obtain t1*2 using,
(3.11)t1*2=1θln[1+Lθ2a(θ+b)].Case 3 (Δ>0).
There are three subcases.Subcase 3.1.
((P-C)θ-hd)/θt1<hw(n-1)/2 and then ∂Z(n,R,t1)/∂R<0. It is same as Case 1.
The optimal transfer level of units in showroom is zero and there exists a uniquet1 (say) t1*3.1 such that Z(n,t1*3.1) is maximum.Note.
(1) t1*3.1≤2((P-C)θ-hd)/θt1hw(n-1) and then t1*3.1 is infeasible. (2) Because q≤L for all q, q*3.1=L. If q>L, then obtain t1*3.1 using (2.5). (3) The number of transfers from the warehouse to the showroom must be at least 2.Subcase 3.23.2.
((P-C)θ-hd)/θt1>hw(n-1)/2. Here, ∂Z(n,R,t1)/∂R>0. Therefore, raise the inventory level to the maximum allowable quantity. So from L=q+R and (2.5), we get
(3.12)R=Lθ2-aθeθt1-abeθt1+aθ+ab+abt1θeθt1θ2eθt1.
Then R is a function of t1. Substitute  (3.12) into (2.8). The resultant expression for the total profit per unit time is function of n and t1. The necessary condition for finding the optimal time t1*3.2 in showroom is
(3.13)∂Z(n,t1)∂t1=(Pabθt1eθt1-hdab2θ+Gt12+Ant12-(P-C)Lt12-(P-C)aθt12+hdaθ2t12+hdLθt12+nhwab2θ-(P-C)abθ2t12+hdabθ3t12-hwab2θ-nhwLθ2eθt1-CLt12eθt1-CLθt1eθt1+hwa2eθt1-hdLθt12eθt1-hdLt1eθt1+PLt12eθt1+PLθt1eθt1+Paθt12eθt1+Pat1eθt1+Pabθ2t12eθt1-Caθt12eθt1-Cat1eθt1-Cabθ2t12eθt1-Cabθt1eθt1-nhwa2eθt1-nhwab2θeθt1+hwLθ2eθt1+hwab2θeθt1-hdaθ2t12eθt1-hdaθt1eθt1-hdabθ3t12eθt1-hdabθ2t1eθt1).
The obtained t1=t1*3.2 maximizes the total profit, Z(n,t1*3.2), per unit time because
(3.14)∂2Z(n,t1)∂t12=(-2CLt13+hdLθt1eθt1-2PLt13eθt1-2PLθt12eθt1-PLθ2t1eθt1-2Paθt13eθt1-2Pat12eθt1-Paθt1eθt1-2Pabθ2t13eθt1+2Caθt13eθt1+2Cat12eθt1+Caθt1eθt1-2Gt13+2hdaθt12eθt1+hdabθt1eθt1+2Cabθ2t13eθt1+2Cabθt12eθt1+Cabt1eθt1+nhwaθ2eθt1+nhwab2eθt1-hwLθ22eθt1-2Cabθ2t13-2Ant13-hwab2eθt1+2hdaθ2t13eθt1+hdat1eθt1+2hdabθ3t13eθt1+2hdabθ2t13eθt1-2Caθt13+2Paθt13-2hdaθ2t13-2hdLθt13+2hdLt12eθt1-Pabt1eθt1-2hdabθ3t13+nhwLθ22eθt1+2CLt13eθt1+2CLθt12eθt1+CLθ2t1eθt1-hwaθ2eθt1+2hdLθt13eθt1-2Pabθt12eθt1+2Pabθ2t13+2PLt13)<0.Subcase 3.3.
((P-C)θ-hd)/θt1=hw(n-1)/2 and then
(3.15)t1*3.3=2((P-C)θ-hd)θhw(n-1).
Hence, one can obtain retransfer level of items in the showroom R*3.3 and optimal units q*3.3 transferred.Algorithm
Step 1.
Assign parametric values toA, G, hd, hw, P, C, a, b, θ, L.Step 2.
IfΔ<0, then go to Algorithm 3.1.Step 3.
IfΔ=0, then go to Algorithm 3.2.Step 4.
IfΔ>0, then go to Algorithm 3.3.Algorithm 3.1.
Step 1.
SetR = 0 and n = 1.Step 2.
Obtaint1*1 by solving (3.3) with Maple 11 (mathematical software) and q*1 from (2.5).Step 3.
Ifq*1<L, then t1*1 obtained in Step 2 is optimal; otherwise,
(3.16)t1*1=1θln[1+Lθ2a(θ+b)].Step 4.
ComputeZ(n,t1*1).Step 5.
Incrementn by 1.Step 6.
Continue Steps2 to 5 until Z(n,t1*1)<Z((n-1),t1*1).Algorithm 3.2.
Step 1.
SetR = 0 and n = 2.Step 2.
Obtaint1*2 from (3.8) and q*2 from (2.5).Step 3.
Ifq*2<L, then t1*2 obtained in Step 2 is optimal; otherwise,
(3.17)t1*2=1θln[1+Lθ2a(θ+b)].Step 4.
ComputeZ(n,t1*2).Step 5.
Incrementn by 1.Step 6.
Continue Steps2 to 5 until Z(n,t1*2)<Z((n-1),t1*2).Algorithm 3.3.
Step 1.
Setn = 2.Step 2.
Solve (3.3) to compute t1*3.1 and determine q*3.1 from (2.5) and R = 0.Step 3.
Ifq*3.1≤L, then t1*3.1 obtained in Step 2 is optimal; otherwise,
(3.18)t1*3.1=1θln[1+Lθ2a(θ+b)]
is optimal.Step 4.
If((P-C)θ-hd)/θt1<hw(n-1)/2  then Compute Z(n,t1*3.1), otherwise set Z(n,t1*3.1)=0.Step 5.
Solve (3.13) to compute t1*3.2.Step 6.
If((P-C)θ-hd)/θt1>hw(n-1)/2, then Substitute t1*3.2 into (3.12) to find R and Calculate Z(n,t1*3.2); otherwise set Z(n,t1*3.2)=0.Step 7.
Z(n,t1*3) = max{Z(n,t1*3.1),Z(n,t1*3.2)}.Step 8.
Incrementn by 1.Step 9.
Continue Steps2 to 8 until Z(n,t1*3)<Z((n-1),t1*3).


## 4. Numerical Examples

Example 4.1.
Consider the following parametric values in proper units: [a,θ,hd,hw,C,P] = [1000, 0.10, 0.6, 0.3, 1, 3]. Here, (P-C)θ-hd<0.
We apply Algorithm3.1. The variations in demand rate b, transfer cost G, ordering cost A, and maximum allowable units L are studied (see Tables 1, 2, 3, and 4).Table 1[Variations  forb][Fixed valuesL=150, A=90, G = 10, b=0.4]bnt1*1T*q*1Q*Z*0.4060.1380.830135.48812.941635.600.4560.1360.817132.85797.111629.220.5060.1330.804130.34782.041622.94Table 2[Variations  forG][Fixed valuesL=150, A=90, b=0.4]Gnt1*1T*q*1Q*Z*1090.1521.368148.49321336.4391600.1132070.1511.057147.53941032.7761560.0893060.1380.828135.1126810.67561490.671Table 3[Variations  forA][Fixed valuesL=150, G=10, b=0.4]Ant1*1T*q*1Q*Z*5060.1490.894145.631873.78611679.3776060.1460.876142.7661856.59661669.3397050.1440.72140.8545704.27271663.394Table 4[Variations  forL][Fixed valuesA=90, G=10, b=0.4]Lnt1*1T*q*1Q*Z*15060.1380.830135.48812.941635.6025050.1560.778151.90759.501636.6735050.1560.778151.90759.501636.67Example 4.2.
Consider the following parametric values in proper units: [a,θ,hd,hw,C,P] = [1000, 0.20, 0.40, 0.10, 1, 3]. Here, (P-C)θ-hd=0. Using Algorithm 3.2, variations in demand rate b, transferring cost G, ordering cost A, and maximum allowable number L on the decision variables and objective function are studied in Tables 5, 6, 7, and 8.Table 5[Variations  forb][Fixed valuesL=150, A=90, G=10, P=3, C=1, θ=0.2]bnt1*2T*q*2Q*Z*0.4100.1511.508148.431484.3051746.880.425100.1491.487146.141461.3931743.270.45100.1471.467143.941439.3981739.70Table 6[Variations  forG][Fixed valuesL=150, A=90, b=0.4, P=3, C=1, θ=0.2]Gnt1*2T*q*2Q*Z*10100.15081.508148.431484.3051746.881290.14931.3437147.00361323.0321734.1241480.14791.1832145.64711165.1761719.14Table 7[Variations  forA][Fixed valuesL=150, G=10, b=0.4, P=3, C=1, θ=0.2]Ant1*2T*q*2Q*Z*80100.15481.548152.32851523.2851753.25385100.15281.528150.3931503.931750.1390100.15081.508148.431484.311746.88Table 8[Variations  forL][Fixed valuesA=90, G=10, b=0.4, P=3, C=1, θ=0.2]Lnt1*2T*q*2Q*Z*100220.0992.18598.312162.861715.17150100.1511.508148.431484.311746.8817580.1701.358166.761334.121748.5520080.1701.358166.761334.121748.55Example 4.3.
Consider the following parametric values in proper units: [a, θ, hd, hw, C, P] = [1000, 0.40, 3, 1, 4, 12]. Here, (P-C)θ-hd>0. Using Algorithm 3.3, variations in demand rate; b, transferring cost G, ordering cost A, and maximum allowable number L on the decision variables and total profit per unit time are studied in Tables 9, 10, 11, and 12.Table 9[Variations  forb][Fixed valuesL=150, A=90, G=30, P=12, C=4, θ=0.40]bnt1*3T*q*3Q*Z*R0.4030.1510.452150.74452.227224.9100.4530.1450.436145.16435.477195.764.8450.5030.1410.422140.16420.487167.689.840Table 10[Variations  forG][Fixed valuesL=150, A=90, b=0.4, P=12, C=4, θ=0.4]Gnt1*3T*q*3Q*Z*R3030.1510.452150.74452.227224.9102030.1370.412138.01414.027294.2011.9931040.1010.405103.20412.787381.8246.804Table 11[Variations  forA][Fixed valuesL=150, b=0.4, G=30, P=12, C=4, θ=0.4]Ant1*3T*q*3Q*Z*R9030.1510.452150.74452.227224.9109530.1530.459152.87458.607214.22010030.1550.465154.97464.907203.680Table 12[Variations  forL][Fixed valuesA=90, b=0.4, G=30, P=12, C=4, θ=0.4]Ant1*3T*q*3Q*Z*R15030.15080.452150.74452.227224.91020030.15020.451153.04459.137231.6046.9625030.14960.449155.38466.137238.3994.62The following managerial issues are observed from Tables1–12.(1)
Increase in demand rate b decreasest1*, q*, and Z*. It is obvious that retailer’s total profit per unit time, cycle time in the warehouse, and procurement quantity from the supplier decrease as the demand decreases.(2)
Increase in transferring cost from the warehouse to the showroom increasest1*, q* and decreases Z*. Z* decreases because the number of transfer increases.(3)
Increase in ordering cost decreases cycle time in showroom and units transferred from warehouse to the showroom and retailer’s total profit per unit time. The cycle time in warehouse increases significantly.(4)
Increase in maximum allowable number in display area increasest1* and q* but no significant change is observed in the total profit per unit time of the retailer. The cycle time in warehouse and procurement quantity from the supplier decreases significantly.


## 5. Conclusions

In this article, an ordering transfer inventory model for deteriorating items is analyzed when the retailer owns showroom having finite floor space and the demand is decreasing with time. Algorithms are proposed to determine retailer’s optimal policy which maximizes his total profit per unit time. Numerical examples and the sensitivity analysis are given to deduce managerial insights.The proposed model can be extended to allow for time dependent deterioration. It is more realistic if damages during transfer from warehouse to showroom are incorporated.


## Assumptions

The following assumptions are used to derive the proposed model.(1)
The inventory system under consideration deals with a single item.(2)
The planning horizon is infinite.(3)
Shortages are not allowed. The lead time is negligible or zero.(4)
The maximum allowable item of displayed stock in the showroom isL.(5)
The time to transfer items from the warehouse to the showroom is negligible or zero.(6)
The units in inventory deteriorate at a constant rate “θ”, 0≤θ<1. The deteriorated units can neither be repaired nor replaced during the cycle time.(7)
The retailer ordersQ-units per order from a supplier and stocks these items in the warehouse. The items are transferred from the warehouse to the showroom in equal size of “q” units until the inventory level in the warehouse reaches to zero. This is known as retailer’s order-transfer policy.



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*Source: 506376-2010-02-08.xml*