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

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.

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. Section 2 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 orders 𝑄-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, ℎ_𝑤[(𝑛(𝑛−1)/2)𝑞]𝑡₁.

2.2. The Total Cost per Unit Cycle in the Showroom

Initially, the inventory level is 𝐿₀≤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 time 𝑡 is given by

𝑑𝐼(𝑡)𝑑𝑡=−𝐷(𝑡)−𝜃𝐼(𝑡),0≤𝑡≤𝑡₁ (2.1) with boundary condition 𝐼(𝑡₁)=𝑅. The solution of (2.1) is

𝐼(𝑡)=𝑅𝑒^{𝜃(𝑡1−𝑡)}+𝑎((e^{𝜃(𝑡1−𝑡)}−1)(𝜃+𝑏)𝜃²−𝑏(𝑡₁𝑒^{𝜃(𝑡1−𝑡)}−𝑡)𝜃);0≤𝑡≤𝑡₁. (2.2) The total cost incurred during the cycle time 𝑡₁ is the sum of the ordering cost, G and the inventory holding cost, where

inventoryholdingcost=ℎ_𝑑^𝑡1₀𝐼(𝑡)𝑑𝑡=ℎ_𝑑𝑅(−𝜃+𝑎(𝑏𝜃²𝑡²₁−2𝜃−2𝑏−2𝜃²𝑡₁2𝜃³))−ℎ_𝑑𝑒^𝜃𝑡1(𝑎(𝜃𝑏𝑡₁−𝜃−𝑏𝜃³𝑅)−𝜃) (2.3) Using (2.2) and 𝐼(0)=𝑞+𝑅, we get

𝑞=𝑅𝑒^𝜃𝑡1𝜃²+𝑎𝑒^𝜃𝑡1𝜃+𝑎𝑒^𝜃𝑡1𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡₁𝑒^𝜃𝑡1𝜃−𝑅𝜃²𝜃². (2.4) The revenue per cycle is

(𝑃−𝐶)𝑞=(𝑃−𝐶)(𝑅𝑒^𝜃𝑡1𝜃²+𝑎𝑒^𝜃𝑡1𝜃+𝑎𝑒^𝜃𝑡1𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡₁𝑒^𝜃𝑡1𝜃−𝑅𝜃²)𝜃². (2.5) Then inventory holding cost in the warehouse is ℎ_𝑤𝑛(𝑛−1)𝑡₁(𝑅𝑒^𝜃𝑡1𝜃²+𝑎𝑒^𝜃𝑡1𝜃+𝑎𝑒^𝜃𝑡1𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡₁𝑒^𝜃𝑡1𝜃−𝑅𝜃²)2𝜃². (2.6) Hence, the total profit, 𝑍𝑃 per cycle during the period [0, 𝑇] is

=𝑍𝑃=Revenue−[totalcostinthewarehouse]−[totalcostintheshowroom]𝑛(𝑃−𝐶)(𝑅𝑒^𝜃𝑡1𝜃²+𝑎𝑒^𝜃𝑡1𝜃+𝑎𝑒^𝜃𝑡1𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡₁𝑒^𝜃𝑡1𝜃−𝑅𝜃²)𝜃²−ℎ−𝐴_𝑤𝑛(𝑛−1)𝑡₁(𝑅𝑒^𝜃𝑡1𝜃²+𝑎𝑒^𝜃𝑡1𝜃+𝑎𝑒^𝜃𝑡1𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡₁𝑒^𝜃𝑡1𝜃−𝑅𝜃²)2𝜃²−𝑛𝐺−𝑛ℎ_𝑑𝑅(−𝜃+𝑎(𝑏𝜃²𝑡²₁−2𝜃−2𝑏−2𝜃²𝑡₁2𝜃³))+𝑛ℎ_𝑑𝑒^𝜃𝑡1(𝑎(𝜃𝑏𝑡₁−𝜃−𝑏𝜃³𝑅)−𝜃). (2.7) During period [0, 𝑇], there are 𝑛-transfers at every 𝑡₁-time units. Hence, 𝑇 = 𝑛𝑡₁. Therefore, the total profit per time unit is 𝑍(𝑛,𝑅,𝑡₁)=𝑍𝑃𝑇=(^{𝑛(𝑃−𝐶)(𝑅𝑒1𝜃𝑡𝜃2+𝑎𝑒1𝜃𝑡𝜃+𝑎𝑒1𝜃𝑡𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡1𝑒1𝜃𝑡𝜃−𝑅𝜃2)/𝜃2−𝐴−𝑛𝐺}_{+ℎ𝑤𝑛(𝑛−1)𝑡1(𝑅𝑒}^1𝜃𝑡_𝜃²_+𝑎𝑒^1𝜃𝑡_{𝜃+𝑎𝑒}^1𝜃𝑡_{𝑏−𝑎𝜃−𝑎𝑏−𝑎𝑏𝑡1𝑒}^1𝜃𝑡_{𝜃−𝑅𝜃}²_)/2𝜃²_{−𝑛ℎ𝑑(−𝑅/𝜃+𝑎((𝑏𝜃2𝑡21−2𝜃−2𝑏−2𝜃2𝑡1)/2𝜃3))+𝑛ℎ𝑑𝑒1𝜃𝑡(𝑎((𝜃𝑏𝑡1−𝜃−𝑏)/𝜃3)−𝑅/𝜃)})𝑛𝑡₁. (2.8)

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:

𝜕²𝑍(𝑛,𝑅,𝑡₁)𝜕𝑛²=−2𝐴𝑛³𝑡₁<0. (3.1) 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 given 𝑛, we first differentiate 𝑍(𝑛,𝑅,𝑡₁) with respect to 𝑅. We get

𝜕𝑍(𝑛,𝑅,𝑡₁)𝜕𝑅=(1−𝑒^𝜃𝑡1𝑡₁ℎ)(−(𝑃−𝐶)+^𝑤(𝑛−1)𝑡₁2+ℎ_𝑑𝜃). (3.2) Depending on the sign of (𝑃−𝐶)𝜃−ℎ_𝑑 three cases arise: Define Δ=(𝑃−𝐶)𝜃−ℎ_𝑑.

Case 1 (Δ<0). If Δ<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 𝜕𝑍𝜕𝑡₁|_𝑅=0(=0_{𝑎(𝑃−𝐶)(1−𝑏𝑡1)𝑒1𝜃𝑡−(1/2)ℎ𝑤(𝑛−1)𝑎𝜃2𝑡1(1−𝑏𝑡1)𝑒1𝜃𝑡+(1/2)(ℎ𝑤(𝑛−1)𝑎((1−𝑒1𝜃𝑡)(𝜃+𝑏)+𝑏𝜃𝑡1𝑒1𝜃𝑡))/𝜃2−((ℎ𝑑𝑎/𝜃2)(𝑏𝑡1−1)(1−𝑒1𝜃𝑡))})𝑡₁−(^{𝑎(𝑃−𝐶)((1−𝑒1𝜃𝑡)(𝜃+𝑏)+𝑏𝑡1𝑒1𝜃𝑡𝜃)/𝜃2+ℎ𝑤(𝑛−1)𝑎((1−𝑒1𝜃𝑡)(𝜃+𝑏)+𝑏𝑡1𝑒1𝜃𝑡𝜃)𝑡1/2𝜃2}_{−𝐴/𝑛−𝐺−(ℎ𝑑𝑎(𝑏𝑡1(2+𝜃𝑡1)/2𝜃2−(𝜃+𝑏)(1+𝜃𝑡1)/𝜃3)−ℎ𝑑𝑎((𝑏𝜃𝑡1−𝜃−𝑏)𝑒1𝜃𝑡/𝜃3))})𝑡²₁=0. (3.3) The sufficiency condition is 𝜕²𝑍(𝑛,𝑅,𝑡₁)/𝜕𝑡²₁<0, that is, 12𝜃³𝑛𝑡³₁−4𝑛𝑎𝜃³𝑡₁𝑃𝑒^𝜃𝑡1+4𝑛𝑎𝜃³𝑡₁𝐶𝑒^𝜃𝑡1+4𝑛𝜃²𝑃𝑎𝑒^𝜃𝑡1−4𝑛𝜃²𝐶𝑎𝑒^𝜃𝑡1+4𝑛𝜃𝑃𝑎𝑏𝑒^𝜃𝑡1−4𝑛𝜃²𝑃𝑎+4𝑛𝜃²𝐶𝑎−4𝑛𝐺𝜃³−4𝐴𝜃³−4𝑛𝜃𝑃𝑎𝑏+4𝑛𝜃𝐶𝑎𝑏+4𝑛ℎ_𝑑𝑎𝜃+4𝑛ℎ_𝑑𝑎𝑏−4𝑛𝜃²𝑃𝑎𝑏𝑡₁𝑒^𝜃𝑡1−4𝑛𝜃𝐶𝑎𝑏𝑒^𝜃𝑡1+4𝑛𝜃²𝐶𝑎𝑏𝑡₁𝑒^𝜃𝑡1−4𝑛ℎ_𝑑𝑎𝜃𝑒^𝜃𝑡1−4𝑛ℎ_𝑑𝑎𝑏𝑒^𝜃𝑡1+4𝑛ℎ_𝑑𝑎𝑏𝑡₁𝜃𝑒^𝜃𝑡1+2𝑛𝑎𝜃⁴𝑡²₁𝑃𝑒^𝜃𝑡1+2𝑛𝑎𝜃³𝑡²₁𝑃𝑏𝑒^𝜃𝑡1−2𝑛𝑎𝜃⁴𝑡³₁𝑃𝑏𝑒^𝜃𝑡1−2𝑛𝑎𝜃⁴𝑡²₁𝐶𝑒^𝜃𝑡1−2𝑛𝑎𝜃³𝑡²₁𝐶𝑏𝑒^𝜃𝑡1+2𝑛𝑎𝜃⁴𝑡³₁𝐶𝑏𝑒^𝜃𝑡1−𝑛²𝑎𝜃⁴𝑡³₁ℎ_𝑤𝑒^𝜃𝑡1+𝑛²𝑎𝜃³𝑡³₁ℎ_𝑤𝑏𝑒^𝜃𝑡1+𝑛²𝑎𝜃⁴𝑡⁴₁ℎ_𝑤𝑏𝑒^𝜃𝑡1+𝑛𝑎𝜃⁴𝑡³₁ℎ_𝑤𝑒^𝜃𝑡1−𝑛𝑎𝜃³𝑡³₁ℎ_𝑤𝑏𝑒^𝜃𝑡1−𝑛𝑎𝜃⁴𝑡⁴₁ℎ_𝑤𝑏𝑒^𝜃𝑡1−2𝑛𝑎𝜃³𝑡²₁ℎ_𝑑𝑒^𝜃𝑡1−2𝑛𝑎𝜃²𝑡²₁ℎ_𝑑𝑏𝑒^𝜃𝑡1+2𝑛𝑎𝜃³𝑡³₁ℎ_𝑑𝑏𝑒^𝜃𝑡1+4𝑛𝑎𝜃²𝑡₁ℎ_𝑑𝑒^𝜃𝑡1<0. (3.4) 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. Since 𝑞^∗1≤𝐿 for all 𝑞, 𝑞^∗1=𝐿. If 𝑞^∗1>𝐿, then obtain 𝑡₁^∗1 using 𝑡₁^∗1=1𝜃ln[1+𝐿𝜃²𝑎(𝜃+𝑏)]. (3.5)

Case 2 (Δ=0). In this case, we made (2.8) as 𝑍(𝑛,𝑅,𝑡₁ℎ)=^𝑤𝑅𝑒^𝜃𝑡12+ℎ_𝑤𝑎𝑒^𝜃𝑡1+ℎ2𝜃^𝑤𝑎𝑏𝑒^𝜃𝑡12𝜃²−ℎ_𝑤𝑎−ℎ2𝜃^𝑤𝑎𝑏2𝜃²−𝑡₁ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡1−ℎ2𝜃_𝑤𝑅2−𝐺𝑡₁−𝐴𝑛𝑡₁−𝑛ℎ_𝑤𝑅𝑒^𝜃𝑡12−𝑛ℎ_𝑤𝑎𝑒^𝜃𝑡1−2𝜃𝑛ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡12𝜃²+𝑛ℎ_𝑤𝑎+2𝜃𝑛ℎ_𝑤𝑎𝑏2𝜃²+𝑛𝑡₁ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡1+2𝜃𝑛ℎ_𝑤𝑅2+ℎ_𝑑𝑎𝜃−𝑡₁ℎ_𝑑𝑎𝑏2𝜃. (3.6) Here, 𝜕𝑍(𝑛,𝑅,𝑡₁)ℎ𝜕𝑅=−^𝑤2(𝑛−1)(𝑒^𝜃𝑡1−1)<0. (3.7) 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 𝑍(𝑛,𝑡₁ℎ)=^𝑤𝑎𝑒^𝜃𝑡1+ℎ2𝜃^𝑤𝑎𝑏𝑒^𝜃𝑡12𝜃²−ℎ_𝑤𝑎−ℎ2𝜃^𝑤𝑎𝑏2𝜃²−𝑡₁ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡1−𝐺2𝜃𝑡₁−𝐴𝑛𝑡₁−𝑛ℎ_𝑤𝑎𝑒^𝜃𝑡1−2𝜃𝑛ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡12𝜃²+𝑛ℎ_𝑤𝑎+2𝜃𝑛ℎ_𝑤𝑎𝑏2𝜃²+𝑛𝑡₁ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡1+ℎ2𝜃^𝑑𝑎𝜃−𝑡₁ℎ_𝑑𝑎𝑏2𝜃. (3.8) The optimal value of 𝑡₁^∗2 can be obtained by solving 𝜕𝑍(𝑛,𝑡₁)𝜕𝑡₁=ℎ_𝑤𝑎𝑒^𝜃𝑡12−𝑡₁ℎ_𝑤𝑎𝑏𝑒^𝜃𝑡12+𝐺𝑡²₁+𝐴𝑛𝑡²₁−𝑛ℎ_𝑤𝑎𝑒^𝜃𝑡12+ℎ_𝑤𝑡₁𝑛𝑎𝑏𝑒^𝜃𝑡12−ℎ_𝑑𝑎𝑏2𝜃=0. (3.9) The sufficiency condition is 𝜕²𝑍(𝑛,𝑡₁)𝜕𝑡²₁=−𝑛ℎ_𝑤𝑎𝜃𝑒^𝜃𝑡12−𝑛𝑎𝑏ℎ_𝑤𝑒^𝜃𝑡12−𝑛𝑎𝑏𝑡₁𝜃ℎ_𝑤𝑒^𝜃𝑡12−𝑎𝜃ℎ_𝑤𝑒^𝜃𝑡12+𝑎𝑏ℎ_𝑤𝑒^𝜃𝑡12+𝑡₁ℎ_𝑤𝑎𝑏𝜃𝑒^𝜃𝑡12+2𝐺𝑡³₁+2𝐴𝑛𝑡³₁<0,for𝑡₁=𝑡₁^∗2. (3.10) 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. Since 𝑞^∗2≤𝐿 for all 𝑞, then 𝑞^∗2=𝐿. If 𝑞^∗2>𝐿, then obtain 𝑡₁^∗2 using, 𝑡₁^∗2=1𝜃ln[1+𝐿𝜃²𝑎(𝜃+𝑏)]. (3.11)

Case 3 (Δ>0). There are three subcases.Subcase 3.1. ((𝑃−𝐶)𝜃−ℎ_𝑑)/𝜃𝑡₁<ℎ_𝑤(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 unique 𝑡₁ (say) 𝑡₁^∗3.1 such that 𝑍(𝑛,𝑡₁^∗3.1) is maximum.Note. (1) 𝑡₁^∗3.1≤2((𝑃−𝐶)𝜃−ℎ_𝑑)/𝜃𝑡₁ℎ_𝑤(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 3.2 3.2. ((𝑃−𝐶)𝜃−ℎ_𝑑)/𝜃𝑡₁>ℎ_𝑤(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 𝑅=𝐿𝜃²−𝑎𝜃𝑒^𝜃𝑡1−𝑎𝑏𝑒^𝜃𝑡1+𝑎𝜃+𝑎𝑏+𝑎𝑏𝑡₁𝜃𝑒^𝜃𝑡1𝜃²𝑒^𝜃𝑡1. (3.12) 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 𝜕𝑍(𝑛,𝑡₁)𝜕𝑡₁=𝑃𝑎𝑏𝜃𝑡₁𝑒^𝜃𝑡1−ℎ_𝑑𝑎𝑏+𝐺2𝜃𝑡²₁+𝐴𝑛𝑡²₁−(𝑃−𝐶)𝐿𝑡²₁−(𝑃−𝐶)𝑎𝜃𝑡²₁+ℎ_𝑑𝑎𝜃²𝑡²₁+ℎ_𝑑𝐿𝜃𝑡²₁+𝑛ℎ_𝑤𝑎𝑏−2𝜃(𝑃−𝐶)𝑎𝑏𝜃²𝑡²₁+ℎ_𝑑𝑎𝑏𝜃³𝑡²₁−ℎ_𝑤𝑎𝑏−2𝜃𝑛ℎ_𝑤𝐿𝜃2𝑒^𝜃𝑡1−𝐶𝐿𝑡²₁𝑒^𝜃𝑡1−𝐶𝐿𝜃𝑡₁𝑒^𝜃𝑡1+ℎ_𝑤𝑎2𝑒^𝜃𝑡1−ℎ_𝑑𝐿𝜃𝑡²₁𝑒^𝜃𝑡1−ℎ_𝑑𝐿𝑡₁𝑒^𝜃𝑡1+𝑃𝐿𝑡²₁𝑒^𝜃𝑡1+𝑃𝐿𝜃𝑡₁𝑒^𝜃𝑡1+𝑃𝑎𝜃𝑡²₁𝑒^𝜃𝑡1+𝑃𝑎𝑡₁𝑒^𝜃𝑡1+𝑃𝑎𝑏𝜃²𝑡²₁𝑒^𝜃𝑡1−𝐶𝑎𝜃𝑡²₁𝑒^𝜃𝑡1−𝐶𝑎𝑡₁𝑒^𝜃𝑡1−𝐶𝑎𝑏𝜃²𝑡²₁𝑒^𝜃𝑡1−𝐶𝑎𝑏𝜃𝑡₁𝑒^𝜃𝑡1−𝑛ℎ_𝑤𝑎2𝑒^𝜃𝑡1−𝑛ℎ_𝑤𝑎𝑏2𝜃𝑒^𝜃𝑡1+ℎ_𝑤𝐿𝜃2𝑒^𝜃𝑡1+ℎ_𝑤𝑎𝑏2𝜃𝑒^𝜃𝑡1−ℎ_𝑑𝑎𝜃²𝑡²₁𝑒^𝜃𝑡1−ℎ_𝑑𝑎𝜃𝑡₁𝑒^𝜃𝑡1−ℎ_𝑑𝑎𝑏𝜃³𝑡²₁𝑒^𝜃𝑡1−ℎ_𝑑𝑎𝑏𝜃²𝑡₁𝑒^𝜃𝑡1. (3.13) The obtained 𝑡₁=𝑡₁^∗3.2 maximizes the total profit, 𝑍(𝑛,𝑡₁^∗3.2), per unit time because 𝜕²𝑍(𝑛,𝑡₁)𝜕𝑡²₁=−2𝐶𝐿𝑡³₁+ℎ_𝑑𝐿𝜃𝑡₁𝑒^𝜃𝑡1−2𝑃𝐿𝑡³₁𝑒^𝜃𝑡1−2𝑃𝐿𝜃𝑡²₁𝑒^𝜃𝑡1−𝑃𝐿𝜃²𝑡₁𝑒^𝜃𝑡1−2𝑃𝑎𝜃𝑡³₁𝑒^𝜃𝑡1−2𝑃𝑎𝑡²₁𝑒^𝜃𝑡1−𝑃𝑎𝜃𝑡₁𝑒^𝜃𝑡1−2𝑃𝑎𝑏𝜃²𝑡³₁𝑒^𝜃𝑡1+2𝐶𝑎𝜃𝑡³₁𝑒^𝜃𝑡1+2𝐶𝑎𝑡²₁𝑒^𝜃𝑡1+𝐶𝑎𝜃𝑡₁𝑒^𝜃𝑡1−2𝐺𝑡³₁+2ℎ_𝑑𝑎𝜃𝑡²₁𝑒^𝜃𝑡1+ℎ_𝑑𝑎𝑏𝜃𝑡₁𝑒^𝜃𝑡1+2𝐶𝑎𝑏𝜃²𝑡³₁𝑒^𝜃𝑡1+2𝐶𝑎𝑏𝜃𝑡²₁𝑒^𝜃𝑡1+𝐶𝑎𝑏𝑡₁𝑒^𝜃𝑡1+𝑛ℎ_𝑤𝑎𝜃2𝑒^𝜃𝑡1+𝑛ℎ_𝑤𝑎𝑏2𝑒^𝜃𝑡1−ℎ_𝑤𝐿𝜃²2𝑒^𝜃𝑡1−2𝐶𝑎𝑏𝜃²𝑡³₁−2𝐴𝑛𝑡³₁−ℎ_𝑤𝑎𝑏2𝑒^𝜃𝑡1+2ℎ_𝑑𝑎𝜃²𝑡³₁𝑒^𝜃𝑡1+ℎ_𝑑𝑎𝑡₁𝑒^𝜃𝑡1+2ℎ_𝑑𝑎𝑏𝜃³𝑡³₁𝑒^𝜃𝑡1+2ℎ_𝑑𝑎𝑏𝜃²𝑡³₁𝑒^𝜃𝑡1−2𝐶𝑎𝜃𝑡³₁+2𝑃𝑎𝜃𝑡³₁−2ℎ_𝑑𝑎𝜃²𝑡³₁−2ℎ_𝑑𝐿𝜃𝑡³₁+2ℎ_𝑑𝐿𝑡²₁𝑒^𝜃𝑡1−𝑃𝑎𝑏𝑡₁𝑒^𝜃𝑡1−2ℎ_𝑑𝑎𝑏𝜃³𝑡³₁+𝑛ℎ_𝑤𝐿𝜃²2𝑒^𝜃𝑡1+2𝐶𝐿𝑡³₁𝑒^𝜃𝑡1+2𝐶𝐿𝜃𝑡²₁𝑒^𝜃𝑡1+𝐶𝐿𝜃²𝑡₁𝑒^𝜃𝑡1−ℎ_𝑤𝑎𝜃2𝑒^𝜃𝑡1+2ℎ_𝑑𝐿𝜃𝑡³₁𝑒^𝜃𝑡1−2𝑃𝑎𝑏𝜃𝑡²₁𝑒^𝜃𝑡1+2𝑃𝑎𝑏𝜃²𝑡³₁+2𝑃𝐿𝑡³₁<0. (3.14)Subcase 3.3. ((𝑃−𝐶)𝜃−ℎ_𝑑)/𝜃𝑡₁=ℎ_𝑤(𝑛−1)/2 and then 𝑡₁^∗3.3=2((𝑃−𝐶)𝜃−ℎ_𝑑)𝜃ℎ_𝑤(𝑛−1). (3.15) 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 to 𝐴, 𝐺, ℎ_𝑑, ℎ_𝑤, 𝑃, 𝐶, 𝑎, 𝑏, 𝜃, 𝐿.Step 2. If Δ<0, then go to Algorithm 3.1.Step 3. If Δ=0, then go to Algorithm 3.2.Step 4. If Δ>0, then go to Algorithm 3.3.

Algorithm 3.1. Step 1. Set 𝑅 = 0 and 𝑛 = 1.Step 2. Obtain 𝑡₁^∗1 by solving (3.3) with Maple 11 (mathematical software) and 𝑞^∗1 from (2.5).Step 3. If 𝑞^∗1<𝐿, then 𝑡₁^∗1 obtained in Step 2 is optimal; otherwise, 𝑡₁^∗1=1𝜃ln[1+𝐿𝜃²𝑎(𝜃+𝑏)]. (3.16)Step 4. Compute 𝑍(𝑛,𝑡₁^∗1).Step 5. Increment 𝑛 by 1.Step 6. Continue Steps 2 to 5 until 𝑍(𝑛,𝑡₁^∗1)<𝑍((𝑛−1),𝑡₁^∗1).

Algorithm 3.2. Step 1. Set 𝑅 = 0 and 𝑛 = 2.Step 2. Obtain 𝑡₁^∗2 from (3.8) and 𝑞^∗2 from (2.5).Step 3. If 𝑞^∗2<𝐿, then 𝑡₁^∗2 obtained in Step 2 is optimal; otherwise, 𝑡₁^∗2=1𝜃ln[1+𝐿𝜃²𝑎(𝜃+𝑏)]. (3.17)Step 4. Compute 𝑍(𝑛,𝑡₁^∗2).Step 5. Increment 𝑛 by 1.Step 6. Continue Steps 2 to 5 until 𝑍(𝑛,𝑡₁^∗2)<𝑍((𝑛−1),𝑡₁^∗2).

Algorithm 3.3. Step 1. Set 𝑛 = 2.Step 2. Solve (3.3) to compute 𝑡₁^∗3.1 and determine 𝑞^∗3.1 from (2.5) and 𝑅 = 0.Step 3. If 𝑞^∗3.1≤𝐿, then 𝑡₁^∗3.1 obtained in Step 2 is optimal; otherwise, 𝑡₁^∗3.1=1𝜃ln[1+𝐿𝜃²𝑎(𝜃+𝑏)] (3.18) is optimal.Step 4. If ((𝑃−𝐶)𝜃−ℎ_𝑑)/𝜃𝑡₁<ℎ_𝑤(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−ℎ_𝑑)/𝜃𝑡₁>ℎ_𝑤(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. Increment 𝑛 by 1.Step 9. Continue Steps 2 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,ℎ_𝑑,ℎ_𝑤,𝐶,𝑃] = [1000, 0.10, 0.6, 0.3, 1, 3]. Here, (𝑃−𝐶)𝜃−ℎ_𝑑<0.
We apply Algorithm 3.1. 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 

Table 2 

Table 3 

Table 4 

Example 4.2. Consider the following parametric values in proper units: [𝑎,𝜃,ℎ_𝑑,ℎ_𝑤,𝐶,𝑃] = [1000, 0.20, 0.40, 0.10, 1, 3]. Here, (𝑃−𝐶)𝜃−ℎ_𝑑=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 

Table 6 

Table 7 

Table 8 

Example 4.3. Consider the following parametric values in proper units: [𝑎, 𝜃, ℎ_𝑑, ℎ_𝑤, 𝐶, 𝑃] = [1000, 0.40, 3, 1, 4, 12]. Here, (𝑃−𝐶)𝜃−ℎ_𝑑>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 

Table 10 

Table 11 

Table 12 

The following managerial issues are observed from Tables 1–12.

(1)Increase in demand rate b decreases 𝑡^∗₁, 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 increases 𝑡^∗₁, 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 increases 𝑡^∗₁ 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 is 𝐿.(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 orders 𝑄-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.

Notations

𝐿:	The maximum allowable number of displayed units in the showroom
𝐼(𝑡):	The inventory level at any instant of time 𝑡 in the showroom, 𝐼(𝑡)≤𝐿
𝐷(𝑡):	The demand rate at time 𝑡. Consider 𝐷(𝑡)=𝑎(1−𝑏𝑡) where 𝑎,𝑏>0, 𝑎≫𝑏. 𝑎 denotes constant demand and 0<𝑏<1 denotes the rate of change of demand due to recession
𝜃:	Constant rate deterioration, 0≤𝜃<1
ℎ𝑤:	The unit inventory carrying cost per annum in the warehouse
ℎ𝑑:	The unit inventory carrying cost per annum in the showroom, with ℎ𝑑>ℎ𝑤
𝑃:	The unit selling price of the item
𝐶:	The unit purchase cost, with 𝐶<𝑃
𝐴:	The ordering cost per order
𝐺:	The known fixed cost per transfer from the warehouse to the showroom
𝑇:	The cycle time in the warehouse, (a decision variable)
𝑛:	The integer number of transfers from the warehouse to the showroom per order (a decision variable)
𝑡1:	The cycle time in the showroom (a decision variable)
𝑄:	The optimum procurement units from a supplier (decision variable)
𝑞:	The number of units per transfer from the warehouse to the showroom, 0≤𝑞≤𝐿 (a decision variable)
𝑅:	The inventory level of items in the showroom regarding the transfer of 𝑞-units from the warehouse to the showroom.

Acknowledgment

The authors are thankful to anonymous reviewers for constructive comments and suggestions.