An Inventory Model with Finite Replenishment Rate, Trade Credit Policy and Price-Discount Offer

Abstract

When some suppliers offer trade credit periods and price discounts to retailers in order to increase the demand of their products, retailers have to face different types of discount offers and credits within which they have to take a decision which is the best offer for them to make more profit. The retailers try to buy perfect-quality items at a reasonable price, and also they try to invest returns obtained by selling those items in such a manner that their business is not hampered. In this point of view, we consider an economic order quantity (EOQ) model for various types of time-dependent demand when delay in payment and price discount are permitted by suppliers to retailers. The models of various demand patterns are discussed analytically. Some numerical examples and graphical representations are considered to illustrate the model.

1. Introduction

Many classical inventory models assume that demand is constant. In present marketing environment, few items follow constant demand. Many product’s demands follow variable time-varying demand. The recent trend of the marketing system is to provide more buy opportunities to the retailer by the supplier by offering different discounts. To take the discount opportunity, retailers prefer to buy more beyond their capacity of buying. As a result, the supplier has the opportunity to sell more for better earning. This is the benefit of the supplier. The classical inventory model does not consider the delay time concept or variable demand. The proposed model considers time-varying demand and delay in payments along with finite replenishment rate.

The basic well-known square root formula for the EOQ of the item was formulated by Harris [1] based on constant demand. Donaldson [2] extended the constant demand to linear time-dependent demand model analytically with finite time horizon. Following Donaldson [2], significant contribution in this direction came out from researchers like Goyal [3], Goswami and Chaudhuri [4], Goyal et al. [5], and others. Hariga and Benkherouf [6] discussed an optimal and heuristic replenishment model for deteriorating items with an exponentially time-varying demand. Wee [7] studied a deterministic lot size inventory model for deteriorating items with shortage and decline market. Khanra and Chaudhuri [8] extended an inventory model with quadratic increasing demand over a finite time horizon and shortages. Sana and Chaudhuri [9] studied an inventory model with linear trend demand incorporating shortages. Cárdenas-Barrón [10] discussed the derivation of inventory models by using analytic geometry and algebra. Sarkar et al. [11] explained an inventory model with quadratic time-varying demand by considering Euler-Lagrange method.

It is common to all that every customer prefers to buy more at reduced price. Some researchers like Abad [12], and Kim and Hwang [13] developed the traditional quantity discount model. In the traditional EOQ model, it was assumed that the retailer pays the purchasing cost when he received the items from a supplier. In trade-credit policy, the supplier allows a certain fixed period to pay the purchasing cost. This fixed period which is settled by the supplier is called the credit period to the retailer. During this credit period, the supplier sells items to the retailer with different types of discounts to obtain more profit as early as possible during the credit period. Depending on this policy, Goyal [14] first developed an inventory model with permissible delay in payments. Aggarwal and Jaggi [15] developed an inventory model with an exponentially deteriorating rate by considering permissible delay in payments. Chu et al. [16] extended Goyal’s [14] model by considering the case of deterioration. Jamal et al. [17] extended an inventory model with shortages.

Teng [18] developed an EOQ model for a retailer to order small lot size in order to take the benefit of permissible delay in payments. Arcelus et al. [19] developed an inventory model by considering the retailer’s maximizing profit and inventory policies for vendor’s trade promotion offer of price/credit on the purchase of perishable items. Huang [20] extended an inventory model of retailer’s inventory system as a cost minimization model to determine the retailer’s optimum inventory cycle time and optimal order quantity. Huang [21] developed an economic production quantity (EPQ) model of retailer’s inventory system to investigate the optimal retailer’s decisions under two levels of trade credit policy. Cárdenas-Barrón [22] extended optimal ordering policies in response to a discount offer. Teng et al. [23] explained optimal ordering decisions with returns and excess inventory. Sarkar [24] discussed an inventory model with delay in payments in the presence of imperfect production. Sarkar [25] developed an inventory model with delay in payments and time-varying deterioration rate. Forghani et al. [26] explained an inventory model in the single period inventory system with price adjustment.

This paper considers an inventory model for credit periods and price-discount offers with different types of time varying demand and constant supply rate 𝐾 up to time 𝑡=𝑡₁. During [0,𝑡₁], inventory piles up by adjusting the demand. The accumulated inventory level at time 𝑡₁ depletes gradually to meet the demand, and the level reaches zero level at time 𝑇(𝑡₁≤𝑇). The agreement between the supplier and the retailer is such that total purchasing cost of whole amount (𝐾𝑡₁) would be paid within the time 𝑅(𝑅>𝑡₁) with purchasing cost at discount rate 𝜌. The different delay periods with different discount rates on the purchasing cost are permitted by the supplier to the retailer. During the credit period, the retailer can earn interest by selling items whereas interest of purchasing cost is charged against the delay of excess time of credit of payment period by the retailer to the supplier.

The rest of the paper is designed as follows. The mathematical model is presented in Section 2. In Section 3, numerical examples are given. Finally, concluding remarks are explained in Section 4. See Table 1 for the comparison of this model with previous works.

Table 1 

Comparisons of this model with previous works.

2. Mathematical Model

We consider the following notation to develop the model.𝑇^∗: the optimal length of inventory cycle (decision variable)𝑡^∗₁:the optimal duration of replenishment (decision variable)𝑄₁(𝑡):on-hand inventory at time 𝑡(0≤𝑡≤𝑡₁)𝑄₂(𝑡):on-hand inventory at time 𝑡(𝑡₁≤𝑡≤𝑇)𝐷(𝑡):time-varying demand rate𝐾:constant replenishment/supply rate𝑅: variable delay period𝑅_𝑖:𝑖th permissible delay period𝜌_𝑖:discount rate on purchasing cost at 𝑖th permissible delay period𝐶₁:ordering cost per order𝐶₂:unit holding cost per unit time, excluding interest charge𝐶₃:purchasing cost per unit𝐶₄:maximum retail price per unit𝑃:selling price per unit𝐼_𝑐: rate of interest gaining due to the credit balance𝐼_𝑓:rate of interest due to financing inventory𝑇:length of the inventory cycle𝑡₁: duration of the replenishment𝑍_1𝑖:average profit of the system when 𝑇≥𝑅_𝑖𝑍_2𝑖:average profit of the system when 𝑇≤𝑅_𝑖.

The following assumptions are considered to develop this model.(1)The inventory system involves only single type of product.(2)The demand rate is constant or time dependent (quadratic, linear, and exponential).(3)Different discount rates on the purchasing cost for different delay periods are considered.(4)Replenishment rate is instantaneously infinite, but its size is finite.(5)Time horizon is infinite.(6)Lead time is negligible.(7)Neither shortage nor backlogging is considered.

The cycle starts with zero inventory at supply rate 𝐾. The replenishment or supply continues up to time 𝑡₁. During the time span [0,𝑡₁], inventory piles up by adjusting the demand in the market. This accumulated inventory level at time 𝑡₁ depletes gradually to meet the demand and it reaches zero level at time 𝑡=𝑇(𝑇>𝑡₁). Generally, the supplier offers delay period 𝑅(𝑅>𝑡₁) to the retailer to pay the total purchasing cost (𝐶₃𝐾𝑡₁) of items. For different delay periods 𝑅_𝑖(𝑖=1,2,3), different discounts 𝜌_𝑖(𝑖=1,2,3) of purchasing cost are offered to the retailer by the supplier. In this direction, we consider the purchasing cost of different delay periods as follows: 𝐶₃=𝐶₄1−𝜌₁when𝑅=𝑅₁,𝐶₄1−𝜌₂when𝑅=𝑅₂,𝐶₄1−𝜌₃when𝑅=𝑅₃,∞when𝑅>𝑅₃, (1) where 𝑅^′_𝑖s are the 𝑖th permissible delay to settle the purchasing cost at which the discount rate to the retailer is 𝜌_𝑖. Also 𝐶₃ tends to be ∞ at 𝑅>𝑅₃. That is, at infinite purchasing cost, the retailer never purchases any item from the supplier. Indirectly, the supplier would not supply the product to the retailer while delay period 𝑅 exceeds 𝑅₃. In our model two cases may arise.

Case 1 (𝑇≥𝑅). That is, inventory cycle length 𝑇 is larger or equal to the credit period 𝑅 (see Figure 1). When 𝑇≥𝑅, there are some profits based on credit balance during the delay period and there is some interest charged due to financing inventory during [𝑅,𝑇].

Figure 1 

Inventory versus time (Case 1  𝑇≥𝑅).

Case 2 (𝑇≤𝑅). That is, inventory cycle length 𝑇 is smaller or equal to the credit period 𝑅 (see Figure 2). When 𝑇≤𝑅, there are some profits based on credit balance during the delay period and there is no interest charged due to financing inventory.

Figure 2 

Inventory versus time (Case 2  𝑇≥𝑅).

The governing differential equations of this model are 𝑑𝑄₁(𝑡)𝑑𝑡=𝐾−𝐷with𝑄₁(0)=0,0≤𝑡≤𝑡₁,𝑑𝑄₂(𝑡)𝑑𝑡=−𝐷with𝑄₂(𝑇)=0,𝑡₁≤𝑡≤𝑇. (2)

From (2), we obtain 𝑄₁(𝑡)=𝐾𝑡−^𝑡₀𝐷(𝑡)𝑑𝑡,0≤𝑡≤𝑡₁,𝑄₂(𝑡)=^𝑇_𝑡𝐷(𝑡)𝑑𝑡,𝑡₁≤𝑡≤𝑇. (3)(4)

By utilizing the continuity at 𝑡₁, 𝑄₁(𝑡₁)=𝑄₂(𝑡₁), we have 𝐾𝑡₁−^𝑡1₀𝐷(𝑡)𝑑𝑡=^𝑇_𝑡1𝐷(𝑡)𝑑𝑡, (5)

that is, 𝑡₁=1𝐾^𝑇₀𝐷(𝑡)𝑑𝑡. (6)

Now we formulate Cases 1 and 2 using (3) to (6).

Case 1 (𝑇≥𝑅). When the inventory cycle (𝑇) is larger or equal to the credit period 𝑅, the holding cost excluding the interest charges is 𝐶₂{∫^𝑡1₀𝑄₁(𝑡)𝑑𝑡+∫^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡}.
The profit gains due to credit balance during the delay period [0,𝑅] are 𝐼_𝑐𝑃∫^𝑅₀(𝑅−𝑡)𝐷(𝑡)𝑑𝑡.
The interest charged for financing inventory during [𝑅,𝑇] is 𝐼_𝑓𝐶₃∫^𝑇_𝑅𝑄₂(𝑡)𝑑𝑡.
Therefore, the total profit is (see Figure 1) 𝑃_1𝑖=𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃^𝑡1₀𝑅_𝑖−𝑡𝐷(𝑡)𝑑𝑡+^𝑅𝑖_𝑡1𝑅_𝑖−𝑡𝐷(𝑡)𝑑𝑡−𝐶₂^𝑡1₀𝑄₁(𝑡)𝑑𝑡+^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡−𝐼_𝑓𝐶₃^𝑇_𝑅𝑖𝑄₂(𝑡)𝑑𝑡−𝐶₁for𝑖∈{1,2,3}. (7)
Hence, the average profit is 𝑍_1𝑖=𝑃_1𝑖𝑇=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃^𝑡1₀𝑅_𝑖−𝑡𝐷(𝑡)𝑑𝑡+^𝑅𝑖_𝑡1𝑅_𝑖−𝑡𝐷(𝑡)𝑑𝑡−𝐶₂^𝑡1₀𝑄₁(𝑡)𝑑𝑡+^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡−𝐼_𝑓𝐶₃^𝑇_𝑅𝑖𝑄₂(𝑡)𝑑𝑡−𝐶₁for𝑖∈{1,2,3}. (8)

Case 2 (𝑇≤𝑅). When the inventory cycle (𝑇) is smaller or equal to the credit period 𝑅, the holding cost excluding the interest charges is 𝐶₂{∫^𝑡1₀𝑄₁(𝑡)𝑑𝑡+∫^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡}.
The profit gains due to credit balance during the time [0,𝑅] are 𝐼_𝑐𝑃[∫^𝑇₀(𝑇−𝑡)𝐷(𝑡)𝑑𝑡+𝐾𝑡₁(𝑅−𝑇)].
Therefore, the total profit is (see Figure 2) 𝑃_2𝑖=𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃^𝑡1₀()𝐷(𝑡)+𝑇−𝑡𝑑𝑡_𝑇𝑡1(𝑇−𝑡)𝐷(𝑡)𝑑𝑡+𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂^𝑡1₀𝑄₁(𝑡)𝑑𝑡+^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡−𝐶₁for𝑖∈{1,2,3}. (9)
Hence, the average profit is 𝑍_2𝑖=𝑃_2𝑖𝑇=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃^𝑡1₀()𝐷(𝑡)+𝑇−𝑡𝑑𝑡_𝑇𝑡1(𝑇−𝑡)𝐷(𝑡)𝑑𝑡+𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂^𝑡1₀𝑄₁(𝑡)𝑑𝑡+^𝑇_𝑡1𝑄₂(𝑡)𝑑𝑡−𝐶₁for𝑖∈{1,2,3}. (10)
Now our objective is to maximize 𝑍=Sup{𝑍_1𝑖,𝑍_2𝑖} for 𝑖∈{1,2,3} and obtain the optimal replenishment period 𝑡^∗₁ and inventory cycle length 𝑇^∗. We discuss various time-dependent or constant demands by utilizing this general formula.

2.1. Quadratic Demand Pattern

The quadratic time-dependent demand is of the form 𝐷(𝑡)=𝑎+𝑏𝑡+𝑐𝑡² where 𝑎>0, 𝑏>0, and 𝑐>0. This trend of demand is applied to the products like essential commodities and seasonal goods.

From (2), we have 𝑄₁(𝑡)=𝐾𝑡−𝑎𝑡−𝑏𝑡²2−𝑐𝑡³3,𝑄₂(𝑡)()+𝑏=𝑎𝑇−𝑡2𝑇²−𝑡²+𝑐3𝑇³−𝑡³. (11) Using these 𝑄₁(𝑡) and 𝑄₂(𝑡) in (8) and (10), we obtain 𝑍_1𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6+𝑐𝑅⁴_𝑖12−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2+𝑏𝑇³3+𝑐𝑇⁴4−𝑎𝑇𝑡₁−𝑏𝑇²𝑡₁2−𝑐𝑇³𝑡₁3−𝐼_𝑓𝐶₃𝑎𝑇²2+𝑏𝑇³3+𝑐𝑇⁴4−𝑎𝑇𝑅_𝑖−𝑅²_𝑖2−𝑏2𝑇²𝑅_𝑖−𝑅³_𝑖3−𝑐3𝑇³𝑅_𝑖−𝑅⁴_𝑖4−𝐶₁,𝑍_2𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑇²2+𝑏𝑇³6+𝑐𝑇⁴12+𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2+𝑏𝑇³3+𝑐𝑇⁴4−𝑎𝑇𝑡₁−𝑏𝑇²𝑡₁2−𝑐𝑇³𝑡₁3−𝐶₁. (12) Using (6) in the above objective functions, we obtain 𝑍_1𝑖=𝑃−𝐶₃𝑎+𝑏𝑇2+𝑐𝑇²3+𝐼_𝑐𝑃𝑇𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6+𝑐𝑅⁴_𝑖12−𝐶₂1𝑎2𝐾²𝑏𝑇+²𝑇³4+𝑐²𝑇⁵9+𝑎𝑏𝑇²+𝑏𝑐𝑇⁴3+2𝑎𝑐𝑇³3+𝑎𝑇2+𝑏𝑇²3+𝑐𝑇³4−𝑎²𝑇𝐾−𝑎𝑏𝑇²𝐾−2𝑎𝑐𝑇³−𝑏3𝐾²𝑇³−4𝐾𝑏𝑐𝑇⁴−𝑐3𝐾²𝑇⁵9𝐾−𝐼_𝑓𝐶₃𝑎𝑇2+𝑏𝑇²3+𝑐𝑇³4−𝑎𝑅_𝑖+𝑎𝑅²_𝑖−2𝑇𝑏𝑇𝑅_𝑖2+𝑏𝑅³_𝑖−6𝑇𝑐𝑇²𝑅_𝑖3+𝑐𝑅⁴_𝑖−𝐶12𝑇𝑇¹,𝑍_2𝑖=𝑃−𝐶₃𝑎+𝑏𝑇2+𝑐𝑇²3+𝐼_𝑐𝑃𝑎𝑅_𝑖+𝑏𝑇𝑅_𝑖2+𝑐𝑇²𝑅_𝑖3−𝑎𝑇2−𝑏𝑇²3−𝑐𝑇³4−𝐶₂1𝑎2𝐾²𝑏𝑇+²𝑇³4+𝑐²𝑇⁵9+𝑎𝑏𝑇²+𝑏𝑐𝑇⁴3+2𝑎𝑐𝑇³3+𝑎𝑇2+𝑏𝑇²3+𝑐𝑇³4−𝑎²𝑇𝐾−𝑎𝑏𝑇²𝐾−2𝑎𝑐𝑇³−𝑏3𝐾₂𝑇³−4𝐾𝑏𝑐𝑇⁴−𝑐3𝐾²𝑇⁵−𝐶9𝐾𝑇¹. (13)

To obtain the optimal cycle length, we construct two lemmas as follows.

Lemma 1. When (𝑑𝑍_1𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[𝑅_𝑖,∞), then 𝑍_1𝑖 has a maximum value at 𝑇=𝑇^∗ if [4𝜉₁(𝑐)(𝑇^∗)⁶+𝜉₂(𝑏,𝑐)(𝑇^∗)⁵+𝜉₃(𝑎,𝑏,𝑐)(𝑇^∗)⁴+𝜉₄(𝑎,𝑏,𝑐)(𝑇^∗)³−2𝜉₆(𝑎,𝑏,𝑐)]<0. Otherwise, 𝑍_1𝑖 has a maximum value at unique 𝑇^∗={𝑇∣(𝑐/3)𝑇³+(𝑏/2)𝑇²+𝑎𝑇−𝐾𝑅_𝑖=0,𝑇>0} if [𝜉₁(𝑐)𝑅⁶_𝑖+𝜉₂(𝑏,𝑐)𝑅⁵_𝑖+𝜉₃(𝑎,𝑏,𝑐)𝑅⁴_𝑖+𝜉₄(𝑎,𝑏,𝑐)𝑅³_𝑖+𝜉₅(𝑎,𝑏)𝑅²_𝑖+𝜉₆(𝑎,𝑏,𝑐)]>0 holds.

Proof. See Appendix A.

Lemma 2. When (𝑑𝑍_2𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[0,𝑅_𝑖], then 𝑍_2𝑖 has a maximum value at 𝑇=𝑇^∗ if [5(𝑇^∗)⁶𝜉₁(𝑐)+(𝑇^∗)⁵𝜉₂(𝑏,𝑐)+2(𝑇^∗)⁴𝜉₇(𝑎,𝑏,𝑐)+(𝑇^∗)³𝜉₈(𝑎,𝑏,𝑐)−2𝐶₁]<0. Otherwise, 𝑍_2𝑖 has a maximum value at 𝑇=𝑅_𝑖 if [(𝑅_𝑖)⁶𝜉₁(𝑐)+(𝑅_𝑖)⁵𝜉₂(𝑏,𝑐)+(𝑅_𝑖)⁴𝜉₇(𝑎,𝑏,𝑐)+(𝑅_𝑖)³𝜉₈(𝑎,𝑏,𝑐)+(𝑅_𝑖)²𝜉₉(𝑎,𝑏,𝑐)+𝐶₁]>0.

Proof. See Appendix A.  

2.2. Linear Time-Dependent Demand Pattern

We consider linear time-dependent demand which can be obtained if we substitute 𝑐=0 in the form of quadratic demand. We get 𝐷(𝑡)=𝑎+𝑏𝑡, 𝑎>0, 𝑏>0 (see for instances Donaldson [2], Goyal [3], and Goswami and Chaudhuri [4]). Generally, in some computer games, computer android applications, we obtain this type of linear time-dependent demand.

Now from (2), we have 𝑄₁(𝑡)=𝐾𝑡−𝑎𝑡−𝑏𝑡²2, (14) and 𝑄₂(𝑡)=𝑎(𝑇−𝑡)+(𝑏/2)(𝑇²−𝑡²), respectively.

Using 𝑄₁(𝑡) and 𝑄₂(𝑡) in (8) and (10), we obtain 𝑍_1𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2+𝑏𝑇³3−𝑎𝑇𝑡₁−𝑏𝑇²𝑡₁2−𝐼_𝑓𝐶₃𝑎𝑇²2+𝑏𝑇³3−𝑎𝑇𝑅_𝑖−𝑅²_𝑖2−𝑏2𝑇²𝑅_𝑖−𝑅³_𝑖3−𝐶₁,𝑍_2𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑇²2+𝑏𝑇³6+𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2+𝑏𝑇³3−𝑎𝑇𝑡₁−𝑏𝑇²𝑡₁2−𝐶₁. (15)

Using (6) in the above objective functions, we have 𝑍_1𝑖=𝑃−𝐶₃𝑎+𝑏𝑇2+𝐼_𝑐𝑃𝑇𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6−𝐶₂1𝑎2𝐾²𝑏𝑇+²𝑇³4+𝑎𝑏𝑇²+𝑎𝑇2+𝑏𝑇²3−𝑎²𝑇𝐾−𝑎𝑏𝑇²𝐾−𝑏²𝑇³4𝐾−𝐼_𝑓𝐶₃𝑎𝑇2+𝑏𝑇²3−𝑎𝑅_𝑖+𝑎𝑅²_𝑖−2𝑇𝑏𝑇𝑅_𝑖2+𝑏𝑅³_𝑖−𝐶6𝑇𝑇¹,𝑍_2𝑖=𝑃−𝐶₃𝑎+𝑏𝑇2+𝐼_𝑐𝑃𝑎𝑅_𝑖+𝑏𝑇𝑅_𝑖2−𝑎𝑇2−𝑏𝑇²3−𝐶₂1𝑎2𝐾²𝑏𝑇+²𝑇³4+𝑎𝑏𝑇²+𝑎𝑇2+𝑏𝑇²3−𝑎²𝑇𝐾−𝑎𝑏𝑇²𝐾−𝑏²𝑇³−𝐶4𝐾𝑇¹. (16)

To obtain the optimal cycle length, we construct the following lemmas.

Lemma 3. When (𝑑𝑍_1𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[𝑅_𝑖,∞), then 𝑍_1𝑖 has a maximum value at 𝑇=𝑇^∗ if [2(𝑇^∗)⁴𝜉₃(𝑎,𝑏,0)+(𝑇^∗)³𝜉₄(𝑎,𝑏,0)−2𝜉₆(𝑎,𝑏,0)]<0. Otherwise 𝑍_1𝑖 has a global maximum value at 𝑇=((−𝑎)+(𝑎²+2𝐾𝑏𝑅_𝑖))/𝑏 if 𝑅_𝑖≤2(𝐾−𝑎)/𝑏 holds.

Proof. See Appendix B.

Lemma 4. When (𝑑𝑍_2𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[0,𝑅_𝑖], then 𝑍_2𝑖 has a maximum value at 𝑇=𝑇^∗ if [2(𝑇^∗)⁴𝜉₃(𝑎,𝑏,0)+(𝑇^∗)³𝜉₈(𝑎,𝑏,0)−2𝐶₁]<0. Otherwise 𝑍_2𝑖 has a maximum value at 𝑇=𝑅_𝑖 if [(𝑅_𝑖)⁴𝜉₃(𝑎,𝑏,0)+(𝑅_𝑖)³𝜉₈(𝑎,𝑏,0)+(𝑅_𝑖)²𝜉₅(𝑎,𝑏)+𝐶₁]>0.

Proof. See Appendix B.  

2.3. Constant Demand Pattern

We consider the demand as constant which can be found by substituting 𝑏=0 in the linear demand. We obtain 𝐷=𝑎, 𝑎>0. See for instances Harris [1] and Cárdenas-Barrón [10, 22]. This type of demand is usually found in product’s life cycle.

Now from (2), we have 𝑄₁(𝑡)=(𝐾−𝑎)𝑡 and 𝑄₂(𝑡)=𝑎(𝑇−𝑡), respectively.

Using 𝑄₁(𝑡) and 𝑄₂(𝑡) in (8) and (10), we get 𝑍_1𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑅²_𝑖2−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2−𝑎𝑇𝑡₁−𝐼_𝑓𝐶₃𝑎𝑇²2−𝑎𝑇𝑅_𝑖−𝑅²_𝑖2−𝐶₁,𝑍_2𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁+𝐼_𝑐𝑃𝑎𝑇²2+𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂𝐾𝑡²₁2+𝑎𝑇²2−𝑎𝑇𝑡₁−𝐶₁, (17)(18) respectively. Using (6) in the above functions, we obtain 𝑍_1𝑖=𝑃−𝐶₃𝐼𝑎+^𝑐𝑃𝑎𝑅²_𝑖2𝑇−𝐶₂𝑎𝑇2−𝑎²𝑇2𝐾−𝐼_𝑓𝐶₃𝑎𝑇2−𝑎𝑅_𝑖+𝑎𝑅²_𝑖−𝐶2𝑇𝑇¹,𝑍_2𝑖=𝑃−𝐶₃(𝑎)+𝐼_𝑐𝑃𝑎𝑅_𝑖−𝑎𝑇2−𝐶₂𝑎𝑇2−𝑎²𝑇−𝐶2𝐾𝑇¹. (19) To obtain optimal cycle length, we formulate the following lemmas.

Lemma 5. If the conditions 𝑅²_𝑖<𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}/(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)<𝐾²𝑅²_𝑖/𝑎² and 𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}>0 hold, then 𝑍_1𝑖 has a global maximum at 𝑇=𝑇^∗=[𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}/(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)]^1/2.

Proof. See Appendix C.

Lemma 6. If 0<2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)<𝑅²_𝑖 hold, then 𝑍_2𝑖 has a global maximum at 𝑇=𝑇^∗=[2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)]^1/2.

Proof. See Appendix C.  

2.4. Exponential Demand Pattern

We consider another important type of demand as exponential type, that is, 𝐷(𝑡) varies exponentially with time 𝑡. In this case, the demand rate increases very fast as in seasonal goods, new computer parts like RAM and data storage device. For this type of demand rate, we consider 𝐷(𝑡)=𝑎exp(𝑏𝑡) where 𝑎>0, 𝑏>0.

From (2), we have 𝑄₁(𝑡)𝑎=𝐾𝑡+𝑏(,𝑄1−exp𝑏𝑡)₂(𝑡)=𝑎𝑏exp(𝑏𝑇)−exp(𝑏𝑡). (20) Using these 𝑄₁(𝑡) and 𝑄₂(𝑡) in (8) and (10), we obtain 𝑍_1𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁−𝐼_𝑐𝑃𝑎𝑅_𝑖𝑏+𝑎𝑏²1−exp𝑏𝑅_𝑖−𝐶₂𝐾𝑡²₁2+𝑎𝑏𝑡₁+1𝑏()−()+𝑇exp𝑏𝑇exp𝑏𝑇𝑏−𝑡₁exp(𝑏𝑇)−𝐼_𝑓𝐶₃𝑎𝑏𝑇−𝑅_𝑖()−()exp𝑏𝑇exp𝑏𝑇−exp𝑏𝑅_𝑖𝑏−𝐶₁,𝑍_2𝑖=1𝑇𝑃−𝐶₃𝐾𝑡₁−𝐼_𝑐𝑃𝑎𝑇𝑏+𝑎𝑏²1−exp(𝑏𝑇)−𝐾𝑡₁𝑅_𝑖−𝑇−𝐶₂𝐾𝑡²₁2+𝑎𝑏𝑡₁+1𝑏()−()+𝑇exp𝑏𝑇exp𝑏𝑇𝑏−𝑡₁exp(𝑏𝑇)−𝐶₁. (21) From (6), we have 𝑡₁=1𝐾^𝑇₀()𝑎𝑎exp𝑏𝑡𝑑𝑡=().𝐾𝑏exp𝑏𝑇−1 (22)

Substituting this in 𝑍_1𝑖 and 𝑍_2𝑖, we have 𝑍_1𝑖=𝑎𝑏𝑇𝑃−𝐶₃exp(𝑏𝑇)−1−𝐼_𝑐𝑃𝑅_𝑖+1𝑏1−exp𝑏𝑅_𝑖−𝐶₂1𝑏()−()+𝑇exp𝑏𝑇exp𝑏𝑇𝑏+𝑎exp(𝑏𝑇)−()𝐾𝑏𝑎exp2𝑏𝑇−𝑎2𝐾𝑏2𝐾𝑏−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖()−()exp𝑏𝑇exp𝑏𝑇−exp𝑏𝑅_𝑖𝑏−𝑏𝐶₁𝑎,𝑍_2𝑖=𝑎𝑏𝑇𝑃−𝐶₃exp(𝑏𝑇)−1−𝐼_𝑐𝑃1−exp(𝑏𝑇)𝑏+𝑅_𝑖+𝑇−𝑅_𝑖exp(𝑏𝑇)−𝐶₂1𝑏()−()+𝑇exp𝑏𝑇exp𝑏𝑇𝑏+𝑎exp(𝑏𝑇)−()𝐾𝑏𝑎exp2𝑏𝑇−𝑎2𝐾𝑏−2𝐾𝑏𝑏𝐶₁𝑎. (23)

Lemma 7. When (𝑑𝑍_1𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[𝑅_𝑖,∞), then 𝑍_1𝑖 has a maximum value at 𝑇=𝑇^∗ if (𝑑²𝑍_1𝑖/𝑑𝑇²)|_𝑇=𝑇^∗<0. Otherwise, 𝑍_1𝑖 has a unique maximum value 𝑇=𝑇^∗=(1/𝑏)[ln(1+𝐾𝑏𝑅_𝑖/𝑎)], if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0.

Proof. See Appendix D.

Lemma 8. When (𝑑𝑍_2𝑖/𝑑𝑇)|_𝑇=𝑇^∗=0 exists for 𝑇^∗∈[0,𝑅_𝑖], then 𝑍_2𝑖 has a maximum value at 𝑇=𝑇^∗ if (𝑑²𝑍_2𝑖/𝑑𝑇²)|_𝑇=𝑇^∗<0. Otherwise, 𝑍_2𝑖 has a maximum value 𝑇=𝑅_𝑖, if (𝑑𝑍_2𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0

Proof. See Appendix D. 

3. Numerical Examples

Example  A. We consider the following parametric values in appropriate units: 𝐶₁=$250 per order, 𝐶₂=$20/unit/year, 𝑅₁=120/365 year, 𝑅₂=150/365 year, 𝑅₃=180/365 year, 𝜌₁=10%, 𝜌₂=5%, 𝜌₃=2%, 𝑎=100 units, 𝑏=50 units, 𝑐=10 units, 𝐷(𝑡)=𝑎+𝑏𝑡+𝑐𝑡², 𝐶₄=$120/units, 𝑃=$180/units, 𝐼_𝑓=0.16/$/year, 𝐼_𝑐=0.13/$/year, and 𝐾=500 units.

Then the optimal solutions are {𝑍₁₁=$7522.33, 𝑇^∗=0.85 year, 𝑡^∗₁=0.21year}, {𝑍₁₂=$6980.84, 𝑇^∗=0.72 year, 𝑡^∗₁=0.17year},  {𝑍₁₃=$6760.68, 𝑇^∗=0.66 year, 𝑡^∗₁=0.15year}, {𝑍₂₁=$7174.54, 𝑇^∗=0.33 year, 𝑡^∗₁=0.07year}, {𝑍₂₂=$6852.25, 𝑇^∗=0.41 year, 𝑡^∗₁=0.91year}, {𝑍₂₃=$6725.72, 𝑇^∗=0.49 year, 𝑡^∗₁=0.11year}.

Among the above optimal solutions, the optimal solution is {𝑍₁₁=$7522.33, 𝑇^∗=0.85 year, 𝑡^∗₁=0.21 year} global maximum. The average profit function is highly nonlinear. Figure 3 shows the concavity of the functions and comparisons of the cost functions.

Figure 3 

The figures of the average profit functions are in descending order as 𝑍11, 𝑍21, 𝑍12, 𝑍22, 𝑍13, and 𝑍23.

Example  B. We consider the following parametric values in appropriate units: 𝐶₁=$250 per order, 𝐶₂=$20/unit/year, 𝑅₁=120/365 year, 𝑅₂=150/365 year, 𝑅₃=180/365 year, 𝜌₁=10%, 𝜌₂=5%, 𝜌₃=2%, 𝑎=100 units, 𝑏=50 units, 𝐷(𝑡)=𝑎+𝑏𝑡, 𝐶₄=$120/units, 𝑃=$180/units, 𝐼_𝑓=0.16/$/year, 𝐼_𝑐=0.13/$/year, and 𝐾=500 units.

Then the optimal solutions are {𝑍₁₁=$7404.27, 𝑇^∗=0.70 year, 𝑡^∗₁=0.17year}, {𝑍₁₂=$6895.63, 𝑇^∗=0.62 year, 𝑡^∗₁=0.14year}, {𝑍₁₃=$6689.02, 𝑇^∗=0.58 year, 𝑡^∗₁=0.13year}, {𝑍₂₁=$7149.18, 𝑇^∗=0.33 year, 𝑡^∗₁=0.07year}, {𝑍₂₂=$6816.19, 𝑇^∗=0.41 year, 𝑡^∗₁=0.09year}, and {𝑍₂₃=$6677, 𝑇^∗=0.49 year, 𝑡^∗₁=0.11year}.

Among the above optimal solutions, the optimal solution is {𝑍₁₁=$7404.27, 𝑇^∗=0.70 year, 𝑡^∗₁=0.17year} global maximum. The average profit function is highly nonlinear. Figure 4 shows the concavity of the functions and comparisons of the cost functions.

Figure 4 

The figures of the average profit functions are in descending order as 𝑍11, 𝑍21, 𝑍12, 𝑍22, 𝑍13, and 𝑍23.

Example  C. We consider the following parametric values in appropriate units: 𝐶₁=$250 per order, 𝐶₂=$20/unit/year, 𝑅₁=120/365year, 𝑅₂=150/365year, 𝑅₃=180/365year, 𝜌₁=10%, 𝜌₂=5%, 𝜌₃=2%, 𝑎=100 units, 𝐷(𝑡)=𝑎, 𝐶₄=$120/unit,  𝑃=$180/unit,  𝐼_𝑓=0.16/$/year,  𝐼_𝑐=0.13/$/year, and 𝐾=500.

Then the optimal solutions are {𝑍₁₁=$6566.5, 𝑇^∗=0.36year,  𝑡^∗₁=0.07year}, {𝑍₁₂=$6160.63, 𝑇^∗=0.35year,  𝑡^∗₁=0.07 year},  {𝑍₁₃=$6004.87, 𝑇^∗=0.33 year,  𝑡^∗₁=0.07year}, {𝑍₂₁=$6561.20, 𝑇^∗=0.33 year,  𝑡^∗₁=0.07year}, {𝑍₂₂=$6143.73, 𝑇^∗=0.41year,  𝑡^∗₁=0.08year}, and {𝑍₂₃=$5915.57, 𝑇^∗=0.49year,  𝑡^∗₁=0.10year}.

From the above optimal solutions, the optimal solution is {𝑍₁₁=$6566.5, 𝑇^∗=0.36year,  𝑡^∗₁=0.07year} global maximum. We obtain a closed type formula for 𝑇^∗, and Figure 5 shows the concavity of the functions and comparisons of the cost functions.

Figure 5 

The figures of the average profit functions are in descending order as 𝑍11, 𝑍21, 𝑍12, 𝑍22, 𝑍13, and 𝑍23.

Example  D. We consider the following parametric values in appropriate units: 𝐶₁=$250 per order, 𝐶₂=$10/unit/month, 𝑅₁=1 month, 𝑅₂=2  months, 𝑅₃=3  months, 𝜌₁=20%, 𝜌₂=15%, 𝜌₃=10%, 𝑎=50 units, 𝑏=0.16 units, 𝐷(𝑡)=𝑎exp(𝑏𝑡), 𝐶₄=$200/unit, 𝑃=$265/unit, 𝐼_𝑓=0.16$/month, 𝐼_𝑐=0.13$/month, and 𝐾=500 units.

Then the optimal solutions are {𝑍₁₁=$5840.84, 𝑇^∗=5.86  months, 𝑡^∗₁=0.97 month}, {𝑍₁₂=$5227.55, 𝑇^∗=4.51  months, 𝑡^∗₁=0.66 month}, {𝑍₁₃=$4712.47, 𝑇^∗=3.32  months, 𝑡^∗₁=0.44 month}, {𝑍₂₁=$5270.15, 𝑇^∗=1 month, 𝑡^∗₁=0.11 month}, {𝑍₂₂=$5081.96, 𝑇^∗=2  months, 𝑡^∗₁=0.24 month}, and {𝑍₂₃=$4710.04, 𝑇^∗=3  months, 𝑡^∗₁=0.39 month}.

From the above optimal solutions, the optimal solution is {𝑍₁₁=$5840.84, 𝑇^∗=5.86  months, 𝑡^∗₁=0.97 month} global maximum. The average profit function is highly non-linear. Figure 6 shows the concavity of the functions and comparisons of the cost functions.

Figure 6 

The figures of the average profit functions are in descending order as 𝑍11, 𝑍21, 𝑍12, 𝑍22, 𝑍13, and 𝑍23.

Sensitivity Analysis. The sensitivity analysis of the key parameters is given in Table 2 for quadratic demand, Table 3 for linear demand, Table 4 for constant demand, and Table 5 for exponential demand. (i)If ordering cost increases, then material handlling cost, shipping cost, and placing order’s cost increase; as a result the total relevant profit decreases. (ii)If the unit holding cost per unit item increases, the total profit of the system decreases. (iii)Increasing value of maximum retailer price increases the total purchasing cost of the whole system which decreases the total profit. (iv)Increasing value of selling price per item increases the total selling price per lot which reduces the total cost of the whole system. Therefore, the total profit of the system increases. (v)If we increase the replenishment rate then the holding cost increases which results the decreasing value of the total profit.

Table 2 

Sensitivity analysis for the quadratic demand pattern.

Table 3 

Sensitivity analysis for the linear demand pattern.

Table 4 

Sensitivity analysis for the constant demand pattern.

Table 5 

Sensitivity analysis for the exponential demand pattern.

4. Concluding Remarks

In most EOQ models, the retailer pays the purchasing cost of items to the supplier as soon as the items are received. In the competitive marketing environment, the supplier offers the retailer a delay period called the trade credit period to stimulate the retailer to buy more items. Before the end of the trade credit period, the retailer can sell his products, accumulate revenue, and earn interest. Besides it, a higher interest is charged if the payment is not settled by the end of the trade credit period. Several papers discussing the topic have been mentioned in the literature that investigates inventory problems under various conditions. Many articles in this direction consider the replenishment rate to be instantaneously infinite, but its size is finite and the lot is placed in the beginning of the cycle. Some models consider finite replenishment/supply rate for constant demand rate. Our models consider the demand of the products as an increasing function of time. From the view point of the supplier, price discount on the purchasing cost of the items by retailer is given at a different delay period to motivate the retailer to buy more. Generally, suppliers consider maximum delay period, after which they would not take any risk of getting back money from retailers. As a result, the purchasing cost 𝐶₃ is infinite when 𝑅 is greater than 𝑅₃; that is, retailer never purchases items at infinite cost. In addition, we establish eight effective and easy-to-use lemmas and figures to help the retailer to take optimal strategy for his marketing policy. We solved the model analytically. We found global optimal solutions for different types of demand patterns. We illustrate the numerical results graphically. As far as the knowledge of authors goes, such type of model has not yet been discussed in the EOQ literature. The application of our model in a different field of inventory models where the assumptions are valid. Further extension of the model may be generalized by considering shortages, deterioration, and stochastic demand. One immediate extension of this model is to consider partial backlogging.

Appendices

A. Derivation of Lemmas 1 and 2

Differentiating (13), the equations become 𝑑𝑍_1𝑖=𝑑𝑇𝑃−𝐶₃𝑏2+2𝑐𝑇3−𝐼_𝑐𝑃𝑇²𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6+𝑐𝑅⁴_𝑖12−𝐶₂1𝑎2𝐾²+3𝑏²𝑇²4+5𝑐²𝑇⁴9++2𝑎𝑏𝑇4𝑏𝑐𝑇³3+2𝑎𝑐𝑇²+𝑎2+2𝑏𝑇3+3𝑐𝑇²4−𝑎²𝐾−2𝑎𝑏𝑇𝐾−2𝑎𝑐𝑇²𝐾−3𝑏²𝑇²−4𝐾4𝑏𝑐𝑇³−3𝐾5𝑐²𝑇⁴9𝐾−𝐼_𝑓𝐶₃𝑎2+2𝑏𝑇3+3𝑐𝑇²4−𝑎𝑅²_𝑖2𝑇²−𝑏𝑅_𝑖2−𝑏𝑅³_𝑖6𝑇²−2𝑐𝑇𝑅_𝑖3−𝑐𝑅⁴_𝑖12𝑇²+𝐶₁𝑇²=1𝑇²𝜉₁(𝑐)𝑇⁶+𝜉₂(𝑏,𝑐)𝑇⁵+𝜉₃(𝑎,𝑏,𝑐)𝑇⁴+𝜉₄(𝑎,𝑏,𝑐)𝑇³+𝜉₅(𝑎,𝑏)𝑇²+𝜉₆(𝑎,𝑏,𝑐), (A.1) where 𝜉₁(𝑐)=5𝑐²𝐶₂18𝐾,𝜉₂(𝑏,𝑐)=2𝑏𝑐𝐶₂,𝜉3𝐾₃(𝑎,𝑏,𝑐)=2𝑎𝑐+3𝑏²4𝐶₂−𝐶2𝐾²+𝐼_𝑓𝐶₃3𝑐4,𝜉₄(𝑎,𝑏,𝑐)=𝑃−𝐶₃2𝑐3+𝐶₂𝑎𝑏𝐾−𝐶₂+𝐼_𝑓𝐶₃2𝑏3+2𝐼_𝑓𝐶₃𝑐𝑅_𝑖3,𝜉₅(𝑎,𝑏)=𝑃−𝐶₃𝑏2+𝐶₂𝑎²𝐾−𝐶₂+𝐼_𝑓𝐶₃𝑎2+𝐼_𝑓𝐶₃𝑏𝑅_𝑖2,𝜉₆(𝑎,𝑏,𝑐)=𝐶₁+𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6+𝑐𝑅⁴_𝑖𝐼12^𝑓𝐶₃−𝐼_𝑐𝑃,𝑑²𝑍_1𝑖𝑑𝑇²=𝑃−𝐶₃2𝑐3+2𝐼_𝑐𝑃𝑇³𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6+𝑐𝑅⁴_𝑖12−𝐶₂12𝐾3𝑏²𝑇2+20𝑐²𝑇³9+2𝑎𝑏+4𝑏𝑐𝑇²+4𝑎𝑐𝑇+2𝑏3+3𝑐𝑇2−2𝑎𝑏𝐾−4𝑎𝑐𝑇𝐾−3𝑏²𝑇−2𝐾4𝑏𝑐𝑇²𝐾−20𝑐²𝑇³9𝐾−𝐼_𝑓𝐶₃2𝑏3+3𝑐𝑇2+𝑎𝑅²_𝑖𝑇³+𝑏𝑅³_𝑖3𝑇³−2𝑐𝑅_𝑖3+𝑐𝑅⁴_𝑖6𝑇³−2𝐶₁𝑇³=1𝑇³4𝜉₁(𝑐)𝑇⁶+𝜉₂(𝑏,𝑐)𝑇⁵+𝜉₃(𝑎,𝑏,𝑐)𝑇⁴+𝜉₄(𝑎,𝑏,𝑐)𝑇³−2𝜉₆(𝑎,𝑏,𝑐). (A.2)

Now, 𝑑𝑍_2𝑖=𝑑𝑇𝑃−𝐶₃𝑏2+2𝑐𝑇3−𝐼_𝑐𝑃2𝑐𝑇𝑅_𝑖3+𝑏𝑅_𝑖2−𝑎2−2𝑏𝑇3−3𝑐𝑇²4−𝐶₂1𝑎2𝐾²+3𝑏²𝑇²4+5𝑐²𝑇⁴9+2𝑎𝑏𝑇+4𝑏𝑐𝑇³3+2𝑎𝑐𝑇²+𝑎2+2𝑏𝑇3+3𝑐𝑇²4−𝑎²𝐾−2𝑎𝑏𝑇𝐾−2𝑎𝑐𝑇²𝐾−3𝑏²𝑇²−4𝐾4𝑏𝑐𝑇³−3𝐾5𝑐²𝑇⁴+𝐶9𝐾𝑇¹²=1𝑇²𝜉₁(𝑐)𝑇⁶+𝜉₂(𝑏,𝑐)𝑇⁵+𝜉₇(𝑎,𝑏,𝑐)𝑇⁴+𝜉₈(𝑎,𝑏,𝑐)𝑇³+𝜉₉(𝑎,𝑏,𝑐)𝑇²+𝐶₁, (A.3) where 𝜉₁(𝑐)=5𝑐²𝐶₂18𝐾,𝜉₂(𝑏,𝑐)=2𝑏𝑐𝐶₂,𝜉3𝐾₇(𝑎,𝑏,𝑐)=2𝑎𝑐+3𝑏²4𝐶₂−𝐶2𝐾²+𝐼_𝑐𝑃3𝑐4,𝜉₈(𝑎,𝑏,𝑐)=𝑃−𝐶₃2𝑐3+𝐶₂𝑎𝑏𝐾−𝐶₂+𝐼_𝑐𝑃2𝑏3+2𝐼_𝑐𝑃𝑐𝑅_𝑖3,𝜉₉(𝑎,𝑏)=𝑃−𝐶₃𝑏2+𝐶₂𝑎²𝐾−𝐶₂+𝐼_𝑐𝑃𝑎2+𝐼_𝑐𝑃𝑏𝑅_𝑖2,𝑑²𝑍_2𝑖𝑑𝑇²=𝑃−𝐶₃2𝑐3−𝐼_𝑐𝑃2𝑐𝑅_𝑖3−2𝑏3−3𝑐𝑇2−𝐶₂12𝐾3𝑏²𝑇2+20𝑐²𝑇³9+2𝑎𝑏+4𝑏𝑐𝑇²++4𝑎𝑐𝑇2𝑏3+3𝑐𝑇2−2𝑎𝑏𝐾−4𝑎𝑐𝑇𝐾−3𝑏²𝑇−2𝐾4𝑏𝑐𝑇²𝐾−20𝑐²𝑇³−9𝐾2𝐶₁𝑇³=1𝑇³5𝜉₁(𝑐)𝑇⁶+𝜉₂(𝑏,𝑐)𝑇⁵+2𝜉₇(𝑎,𝑏,𝑐)𝑇⁴+𝜉₈(𝑎,𝑏,𝑐)𝑇³−2𝐶₁. (A.4)

To obtain the optimal cycle length, proof of two lemmas are as follows.

Proof of Lemma 1. The first part of the lemma is obvious from the above calculations of 𝑑𝑍_1𝑖/𝑑𝑇 and 𝑑²𝑍_1𝑖/𝑑𝑇². For the second part, if 𝑍_1𝑖 does not have any stationary points in [𝑅_𝑖,∞), then 𝑍_1𝑖 is either monotonic increasing or monotonic decreasing function of 𝑇∈[𝑅_𝑖,∞). Here, 𝑑𝑍_1𝑖/𝑑𝑇→∞ as 𝑇→∞ because 𝜉₁(𝑐)>0. Hence, 𝑍_1𝑖 will be monotonic increasing if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0 as 𝑍_1𝑖 does not have stationary points in [𝑅_𝑖,∞). Here, 𝑡₁=(1/𝐾)(𝑎𝑇+(𝑏/2)𝑇²+(𝑐/3)𝑇³), 𝑡₁ increase with increasing value of 𝑇. As in our models 𝑡₁≤𝑅_𝑖; hence, for the feasibility of the model, (𝑡₁)^∗=𝑅_𝑖 and 𝑇^∗ is obtained from (𝑐/3)(𝑇^∗)³+(𝑏/2)(𝑇^∗)²+𝑎(𝑇^∗)=𝐾𝑅_𝑖. By Descarte’s rule, (𝑐/3)(𝑇^∗)³+(𝑏/2)(𝑇^∗)²+𝑎(𝑇^∗)−𝐾𝑅_𝑖=0 may have atmost one positive root and two negative roots. Therefore, 𝑍_1𝑖 has a maximum value at 𝑇^∗={𝑇∣(𝑐/3)𝑇³+(𝑏/2)𝑇²+𝑎𝑇−𝐾𝑅_𝑖=0,𝑇>0} if (1/𝑅²_𝑖)[𝜉₁(𝑐)𝑅⁶_𝑖+𝜉₂(𝑏,𝑐)𝑅⁵_𝑖+𝜉₃(𝑎,𝑏,𝑐)𝑅⁴_𝑖+𝜉₄(𝑎,𝑏,𝑐)𝑅³_𝑖+𝜉₅(𝑎,𝑏)𝑅²_𝑖+𝜉₆(𝑎,𝑏,𝑐)]>0. Here 𝑇^∗ is unique if it exists, by Descarte’s rule, hence the proof.

Proof of Lemma 2. The first part of the lemma is obvious (see the expressions of 𝑑𝑍_1𝑖/𝑑𝑇 and 𝑑²𝑍_1𝑖/𝑑𝑇²). For the second part, if 𝑍_2𝑖 does not have any stationary points in [0,𝑅_𝑖], then 𝑍_2𝑖 is either monotonic increasing or monotonic decreasing function of 𝑇∈[0,𝑅_𝑖]. Here 𝑑𝑍_2𝑖/𝑑𝑇→∞ as 𝑇→0 because 𝐶₁>0. Hence, 𝑍_2𝑖 is monotonic increasing if (𝑑𝑍_2𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0 as 𝑍_2𝑖 does not have stationary points in [0,𝑅_𝑖]. Hence 𝑍_2𝑖 has a maximum value at 𝑇=𝑅_𝑖 if [(𝑅_𝑖)⁶𝜉₁(𝑐)+(𝑅_𝑖)⁵𝜉₂(𝑏,𝑐)+(𝑅_𝑖)⁴𝜉₇(𝑎,𝑏,𝑐)+(𝑅_𝑖)³𝜉₈(𝑎,𝑏,𝑐)+(𝑅_𝑖)²𝜉₉(𝑎,𝑏,𝑐)+𝐶₁]>0, hence the proof.

B. Derivation of Lemmas 3 and 4

Now differentiating (16), we obtain 𝑑𝑍_1𝑖=𝑑𝑇𝑃−𝐶₃𝑏2−𝐼_𝑐𝑃𝑇²𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6−𝐶₂1𝑎2𝐾²+3𝑏²𝑇²4+𝑎+2𝑎𝑏𝑇2+2𝑏𝑇3−𝑎²𝐾−2𝑎𝑏𝑇𝐾−3𝑏²𝑇²4𝐾−𝐼_𝑓𝐶₃𝑎2+2𝑏𝑇3−𝑎𝑅²_𝑖2𝑇²−𝑏𝑅_𝑖2−𝑏𝑅³_𝑖6𝑇²+𝐶₁𝑇²=1𝑇²𝜉₃(𝑎,𝑏,0)𝑇⁴+𝜉₄(𝑎,𝑏,0)𝑇³+𝜉₅(𝑎,𝑏)𝑇²+𝜉₆(𝑎,𝑏,0), (B.1) where 𝜉₃(𝑎,𝑏,0)=3𝑏²4𝐶₂,𝜉2𝐾₄()=𝐶𝑎,𝑏,0²𝑎𝑏𝐾−𝐶₂+𝐼_𝑓𝐶₃2𝑏3,𝜉₅()=𝑎,𝑏𝑃−𝐶₃𝑏2+𝐶₂𝑎²𝐾−𝐶₂+𝐼_𝑓𝐶₃𝑎2+𝐼_𝑓𝐶₃𝑏𝑅_𝑖2,𝜉₆(𝑎,𝑏,0)=𝐶₁+𝑎𝑅²2+𝑏𝑅³6𝐼_𝑓𝐶₃−𝐼_𝑐𝑃,𝑑²𝑍_1𝑖𝑑𝑇²=2𝐼_𝑐𝑃𝑇³𝑎𝑅²_𝑖2+𝑏𝑅³_𝑖6−𝐶₂12𝐾3𝑏²𝑇2++2𝑎𝑏2𝑏3−2𝑎𝑏𝐾−3𝑏²𝑇2𝐾−𝐼_𝑓𝐶₃2𝑏3+𝑎𝑅²_𝑖𝑇³+𝑏𝑅³_𝑖3𝑇³−2𝐶₁𝑇³=1𝑇³2𝜉₃(𝑎,𝑏,0)𝑇⁴+𝜉₄(𝑎,𝑏,0)𝑇³−2𝜉₆(𝑎,𝑏,0). (B.2) Now, 𝑑𝑍_2𝑖=𝑑𝑇𝑃−𝐶₃𝑏2−𝐼_𝑐𝑃𝑏𝑅_𝑖2−𝑎2−2𝑏𝑇3−𝐶₂1𝑎2𝐾²+3𝑏²𝑇²4+𝑎+2𝑎𝑏𝑇2+2𝑏𝑇3−𝑎²𝐾−2𝑎𝑏𝑇𝐾−3𝑏²𝑇²+𝐶4𝐾𝑇¹²=1𝑇²𝜉₃(𝑎,𝑏,0)𝑇⁴+𝜉₈(𝑎,𝑏,0)𝑇³+𝜉₅(𝑎,𝑏)𝑇²+𝐶₁, (B.3) where 𝜉₈()=𝐶𝑎,𝑏,0²𝑎𝑏𝐾−𝐶₂+𝐼_𝑐𝑃2𝑏3,𝑑²𝑍_2𝑖𝑑𝑇²=𝐼_𝑐𝑃2𝑏3−𝐶₂12𝐾3𝑏²𝑇2++2𝑎𝑏2𝑏3−2𝑎𝑏𝐾−3𝑏²𝑇−2𝐾2𝐶₁𝑇³=1𝑇³2𝜉₃(𝑎,𝑏,0)𝑇⁴+𝜉₈(𝑎,𝑏,0)𝑇³−2𝐶₁. (B.4)

Proof of Lemma 3. The first part of the lemma is obvious from the formulas of 𝑑𝑍_1𝑖/𝑑𝑇 and 𝑑²𝑍_1𝑖/𝑑𝑇². For the second part, if 𝑍_1𝑖 does not have any stationary points in [𝑅_𝑖,∞), then 𝑍_1𝑖 is either a monotonic increasing or monotonic decreasing function of 𝑇∈[𝑅_𝑖,∞). Here, 𝑑𝑍_1𝑖/𝑑𝑇→∞ as 𝑇→∞ because 𝜉₃(𝑎,𝑏,0)>0. Hence, 𝑍_1𝑖 is monotonic increasing if [(𝑅_𝑖)⁴𝜉₃(𝑎,𝑏,0)+(𝑅_𝑖)³𝜉₄(𝑎,𝑏,0)+(𝑅_𝑖)²𝜉₅(𝑎,𝑏)+𝜉₆(𝑎,𝑏,0)]>0 as 𝑍_1𝑖 does not have stationary points in [𝑅_𝑖,∞). Here 𝑡₁=(1/𝐾)(𝑎𝑇+(𝑏/2)𝑇²) and 𝑡₁ increases with increasing value of 𝑇. According to our model, 𝑡₁<𝑅_𝑖; hence for the feasibility of the model, 𝑡^∗₁=𝑅_𝑖 and 𝑇^∗ is obtained from (𝑏/2)(𝑇^∗)²+𝑎(𝑇^∗)=𝐾𝑅_𝑖. By Descarte’s rule (𝑏/2)(𝑇^∗)²+𝑎(𝑇^∗)−𝐾𝑅_𝑖=0 may have atmost one positive root and one negative root. Now the positive root of the above equation is 𝑇^∗=(−𝑎+𝑎²+2𝐾𝑏𝑅_𝑖)/𝑏. Here 𝑇^∗ is unique by Descarte’s rule. For feasibility of our model, 𝑇^∗ must be greater than or equal to 𝑅_𝑖. Therefore, 𝑇^∗≥𝑅_𝑖⇒−𝑎+𝑎²+2𝐾𝑏𝑅_𝑖≥𝑏𝑅_𝑖⇒𝑎²+2𝐾𝑏𝑅_𝑖≥𝑎+𝑏𝑅_𝑖⇒𝑎²+2𝐾𝑏𝑅_𝑖≥(𝑎+𝑏𝑅_𝑖)²⇒𝑎²+2𝐾𝑏𝑅_𝑖−𝑎²−2𝑎𝑏𝑅_𝑖−𝑏²(𝑅_𝑖)²≥0⇒2(𝐾−𝑎)/𝑏≥𝑅_𝑖. Thus 𝑍_1𝑖 has a unique maximum value at 𝑇=(−𝑎+𝑎²+2𝐾𝑏𝑅_𝑖)/𝑏 if 𝑅_𝑖≤2(𝐾−𝑎)/𝑏 holds, hence the proof.

Proof of Lemma 4. The first part of the lemma is obvious (see the expressions of 𝑑𝑍_1𝑖/𝑑𝑇 and 𝑑²𝑍_1𝑖/𝑑𝑇²). For the second part, if 𝑍_2𝑖 does not have any stationary points in [0,𝑅_𝑖], then 𝑍_2𝑖 is either monotonic increasing or monotonic decreasing function of 𝑇∈[0,𝑅_𝑖]. Here, 𝑑𝑍_2𝑖/𝑑𝑇→∞ as 𝑇→0 because 𝐶₁>0. So, 𝑍_2𝑖 is monotonic increasing if (1/(𝑅_𝑖)²)[(𝑅_𝑖)⁴𝜉₃(𝑎,𝑏,0)+(𝑅_𝑖)³𝜉₈(𝑎,𝑏,0)+(𝑅_𝑖)²𝜉₅(𝑎,𝑏)+𝐶₁]>0 as 𝑍_2𝑖 does not have stationary points in [0,𝑅_𝑖]. Hence, 𝑍_2𝑖 has a maximum value when 𝑇=𝑅_𝑖 if [(𝑅_𝑖)⁴𝜉₃(𝑎,𝑏,0)+(𝑅_𝑖)³𝜉₈(𝑎,𝑏,0)+(𝑅_𝑖)²𝜉₅(𝑎,𝑏)+𝐶₁]>0. Hence the proof.

C. Derivation of Lemmas 5 and 6

Differentiating (19), we have 𝑑𝑍_1𝑖=−𝐼𝑑𝑇^𝑐𝑃𝑇²𝑎𝑅²_𝑖2−𝐶₂𝑎2−𝑎²2𝐾−𝐼_𝑓𝐶₃𝑎2−𝑎𝑅²_𝑖2𝑇²+𝐶₁𝑇²,𝑑²𝑍_1𝑖𝑑𝑇²=𝐼_𝑐𝑃𝑇³𝑎𝑅²_𝑖−𝐼_𝑓𝐶₃𝑎𝑅²_𝑖𝑇³−2𝐶₁𝑇³,𝑑𝑍_2𝑖=𝐼𝑑𝑇^𝑐𝑃𝑎2−𝐶₂𝑎2−𝑎²+𝐶2𝐾𝑇¹²,𝑑²𝑍_2𝑖𝑑𝑇²=−2𝐶₁𝑇³. (C.1)

Proof of Lemma 5. We have 𝑑𝑍_1𝑖/𝑑𝑇=[−(𝐼_𝑐𝑃/𝑇²){(𝑎𝑅²_𝑖)/2}−𝐶₂{𝑎/2−𝑎²/2𝐾}−(𝐼_𝑓𝐶₃){𝑎/2−𝑎𝑅²_𝑖/2𝑇²}+𝐶₁/𝑇²] and 𝑑²𝑍_1𝑖/𝑑𝑇²=[(𝐼_𝑐𝑋/𝑇³){𝑎𝑅²_𝑖}−𝐼_𝑓𝐶₃{𝑎𝑅²/𝑇³}−2𝐶₁/𝑇³]. For maximization, 𝑑𝑍_1𝑖/𝑑𝑇=0. This implies that 𝑇=[𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}/(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)]^1/2=𝑇^∗(say). At 𝑇=𝑇^∗,  𝑑²𝑍_1𝑖/𝑑𝑇²=−(1/𝐾)[(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)/𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}]^1/2. If 𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}>0, then 𝑑²𝑍_1𝑖/𝑑𝑇²<0. Again for the feasibility of the model, 𝑡₁=𝑎𝑇/𝐾<𝑅_𝑖⇒𝑇<(𝐾𝑅_𝑖)/𝑎 and 𝑇>𝑅_𝑖. Combining these, we have 𝑅_𝑖<𝑇<𝐾𝑅_𝑖/𝑎. That is, 𝑅²_𝑖<𝑇²<(𝐾𝑅_𝑖)²/𝑎². Hence at 𝑇=𝑇^∗=[𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}/(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)]^1/2, 𝑍_1𝑖 has a maximum value if 𝑅²_𝑖<𝐾{2𝐶₁+𝑎𝑅²_𝑖(𝐼_𝑓𝐶₃−𝐼_𝑐𝑃)}/(𝐶₂𝑎(𝐾−𝑎)+𝐼_𝑓𝐶₃𝑎𝐾)<𝐾²𝑅²_𝑖/𝑎² hold, hence the proof.

Proof of Lemma 6. We have 𝑑𝑍_2𝑖/𝑑𝑇=[𝐼_𝑐𝑃(𝑎/2)−𝐶₂{𝑎/2−𝑎²/2𝐾}+𝐶₁/𝑇²] and 𝑑²𝑍_2𝑖/𝑑𝑇²=[−2𝐶₁/𝑇³]. For optimization 𝑑𝑍_2𝑖/𝑑𝑇=0. This gives 𝑇=[2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)]^1/2. For the real values of 𝑇, 𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾>0 and also for the feasibility of the model, 𝑇<𝑅_𝑖 which imply that 𝑇²<(𝑅_𝑖)²⇒0<2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)<𝑅²_𝑖. The second derivative of 𝑍_2𝑖 at 𝑇=𝑇^∗ is always negative. Therefore 𝑍_2𝑖 has a global maximum at 𝑇=𝑇^∗=[2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)]^1/2 if 0<2𝐶₁𝐾/(𝐶₂𝑎(𝐾−𝑎)−𝐼_𝑐𝑃𝑎𝐾)<𝑅²_𝑖 hold, hence the lemma.

D. Derivation of Lemmas 7 and 8

Now differentiating 𝑍_1𝑖 and 𝑍_2𝑖 with respect to 𝑇, we obtain 𝑑𝑍_1𝑖=𝑎𝑑𝑇𝑏𝑇𝑃−𝐶₃()−()𝑏exp𝑏𝑇exp𝑏𝑇−1𝑇+𝐼_𝑐𝑃𝑇𝑅_𝑖+1𝑏1−exp𝑏𝑅_𝑖−𝐶₂𝑎exp(𝑏𝑇)𝐾−𝑎exp(2𝑏𝑇)𝐾()−()+𝑇𝑏exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑏𝑇𝑎exp2𝑏𝑇−12𝑇𝐾𝑏+𝑎𝑏𝑇()+()2𝐾𝑏𝑇−exp𝑏𝑇exp𝑏𝑇𝑏𝑇−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖()−𝑏exp𝑏𝑇𝑇−𝑅_𝑖exp(𝑏𝑇)𝑇+exp(𝑏𝑇)−exp𝑏𝑅_𝑖+𝑏𝑇𝑏𝐶₁.𝑎𝑇 (D.1) Using 𝑑𝑍_1𝑖/𝑑𝑇=0, 𝑑²𝑍_1𝑖𝑑𝑇²=𝑎𝑏𝑇𝑃−𝐶₃×𝑏²()−()()exp𝑏𝑇𝑇𝑏exp𝑏𝑇−exp𝑏𝑇+1𝑇²−𝐼_𝑐𝑃𝑇²𝑅_𝑖+1𝑏1−exp𝑏𝑅_𝑖−𝐶₂𝑎𝑏exp(𝑏𝑇)𝐾−2𝑎𝑏exp(2𝑏𝑇)𝐾+𝑇𝑏²()−()exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑇𝑎exp𝑏𝑇𝐾𝑏𝑇²+𝑎exp(2𝑏𝑇)−()𝐾𝑇𝑎exp2𝑏𝑇2𝑇²+1𝐾𝑏𝑏𝑇²−𝑎2𝐾𝑏𝑇²+exp(𝑏𝑇)𝑇−exp(𝑏𝑇)𝑏𝑇²−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖𝑏²()−𝑅exp𝑏𝑇_𝑖𝑇𝑏exp(𝑏𝑇)−exp(𝑏𝑇)𝑇²+𝑇𝑏exp(𝑏𝑇)−exp(𝑏𝑇)+exp𝑏𝑅_𝑖𝑏𝑇²−𝑏𝐶₁𝑎𝑇². (D.2) From Case 2, we have 𝑑𝑍_2𝑖=𝑎𝑑𝑇𝑏𝑇𝑃−𝐶₃()−()𝑏exp𝑏𝑇exp𝑏𝑇−1𝑇−𝐼_𝑐𝑃𝑇𝑇𝑏exp(𝑏𝑇)−𝑅_𝑖()−1𝑏exp𝑏𝑇(+𝑅𝑏𝑇1−exp𝑏𝑇)_𝑖exp(𝑏𝑇)𝑇−𝑅_𝑖𝑇−exp(𝑏𝑇)−𝐶₂𝑎exp(𝑏𝑇)𝐾−𝑎exp(2𝑏𝑇)𝐾()−()+𝑇𝑏exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑏𝑇𝑎exp2𝑏𝑇−12𝑇𝐾𝑏+𝑎𝑏𝑇()+()2𝐾𝑏𝑇−exp𝑏𝑇exp𝑏𝑇+𝑏𝑇𝑏𝐶₁.𝑎𝑇 (D.3) Using 𝑑𝑍_2𝑖/𝑑𝑇=0, 𝑑²𝑍_2𝑖𝑑𝑇²=𝑎𝑏𝑇𝑃−𝐶₃×𝑏²()−()()exp𝑏𝑇𝑇𝑏exp𝑏𝑇−exp𝑏𝑇+1𝑇²−𝐼_𝑐𝑃𝑇𝑏²exp(𝑏𝑇)−𝑅_𝑖𝑏²()−()exp𝑏𝑇exp𝑏𝑇𝑇+1𝑏𝑇²−exp(𝑏𝑇)𝑏𝑇²+𝑅_𝑖𝑏exp(𝑏𝑇)𝑇−𝑅_𝑖exp(𝑏𝑇)𝑇²+𝑅_𝑖𝑇²−𝐶₂𝑎𝑏exp(𝑏𝑇)𝐾−2𝑎𝑏exp(2𝑏𝑇)𝐾+𝑇𝑏²()−()exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑇𝑎exp𝑏𝑇𝐾𝑏𝑇²+𝑎exp(2𝑏𝑇)−()𝐾𝑇𝑎exp2𝑏𝑇2𝑇²+1𝐾𝑏𝑏𝑇²−𝑎2𝐾𝑏𝑇²+exp(𝑏𝑇)𝑇−exp(𝑏𝑇)𝑏𝑇²−𝑏𝐶₁𝑎𝑇². (D.4)

Proof of Lemma 7. The first part of the lemma is obvious. For the second part, if 𝑍_1𝑖 does not have any stationary points in [𝑅_𝑖,∞), then 𝑍_1𝑖 is either monotonic increasing or monotonic decreasing function of 𝑇∈[𝑅_𝑖,∞). Here, 𝑑𝑍_1𝑖/𝑑𝑇→∞ as 𝑇→∞ because 𝑑𝑍_1𝑖=𝑎𝑑𝑇𝑏𝑇𝑃−𝐶₃()−()𝑏exp𝑏𝑇exp𝑏𝑇−1𝑇+𝐼_𝑐𝑃𝑇𝑅_𝑖+1𝑏1−exp𝑏𝑅_𝑖−𝐶₂𝑎exp(𝑏𝑇)𝐾−𝑎exp(2𝑏𝑇)𝐾()−()+𝑇𝑏exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑏𝑇𝑎exp2𝑏𝑇−12𝑇𝐾𝑏+𝑎𝑏𝑇()+()2𝐾𝑏𝑇−exp𝑏𝑇exp𝑏𝑇𝑏𝑇−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖𝑏exp(𝑏𝑇)−𝑇−𝑅_𝑖exp(𝑏𝑇)𝑇+exp(𝑏𝑇)−exp𝑏𝑅_𝑖+𝑏𝑇𝑏𝐶₁=()𝑎𝑇𝑎exp𝑏𝑇𝑏𝑇𝑃−𝐶₃1𝑏−𝑇−𝐶₂𝑎𝐾𝑎+𝑇𝑏−1𝐾𝑏𝑇−1+𝑏𝑇−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖𝑅𝑏+^𝑖𝑇+1+𝑎𝑏𝑇𝑏𝑇𝑃−𝐶₃𝑇+𝐼_𝑐𝑃𝑇𝑅_𝑖+1𝑏−𝐶₂𝑎−12𝐾𝑏𝑇𝑏𝑇+𝐼_𝑓𝐶₃exp𝑏𝑅_𝑖+𝑎𝑏𝑇𝑏𝑇−𝐶₂−𝑎𝐾+𝑎()+𝐶2𝑇𝐾𝑏exp2𝑏𝑇¹𝑇². (D.5) Using L’Hospital’s rule lim_𝑇→∞𝑑𝑍_1𝑖𝑑𝑇=lim_𝑇→∞𝑎exp(𝑏𝑇)𝑏𝑇²×𝑃−𝐶₃(𝑏𝑇−1)−𝐶₂𝑎𝑇𝐾+𝑇²𝑎𝑏−1𝐾𝑏−𝑇+𝑏−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖𝑇𝑏+𝑅_𝑖+1𝑏+lim_𝑇→∞𝑎𝑏𝑇²𝑃−𝐶₃+𝐼_𝑐𝑃𝑅_𝑖+1𝑏−𝐶₂𝑎−12𝐾𝑏𝑏+𝐼_𝑓𝐶₃exp𝑏𝑅_𝑖𝑏+lim_𝑇→∞𝑎𝑏𝑇²exp(2𝑏𝑇)−𝐶₂−𝑎𝑇𝐾+𝑎2𝐾𝑏+lim_𝑇→∞𝐶₁𝑇²=𝑎𝑏lim_𝑇→∞𝐹(𝑇)𝑇²+𝑎𝑏lim_𝑇→∞𝐺(𝑇)𝑇²=𝑎𝑏lim_𝑇→∞𝐹(𝑇)+𝐺(𝑇)𝑇²≃𝑎𝑏lim_𝑇→∞𝐺(𝑇)𝑇²|𝐺(𝑇>|𝐹(𝑇=𝑎since)|)|𝑏lim_𝑇→∞𝐺^′(𝑇)=𝑎2𝑇2𝑏lim_𝑇→∞𝐺^′′(𝑇)⟶+∞, (D.6) where 𝐹(𝑇)=exp(𝑏𝑇)𝑃−𝐶₃(𝑏𝑇−1)−𝐶₂𝑎𝑇𝐾+𝑇²𝑎𝑏−1𝐾𝑏−𝑇+𝑏−𝐼_𝑓𝐶₃𝑇−𝑅_𝑖𝑇𝑏+𝑅_𝑖+1𝑏+𝑃−𝐶₃𝑇+𝐼_𝑐𝑃𝑅_𝑖+1𝑏−𝐶₂𝑎−12𝐾𝑏𝑏+𝐼_𝑓𝐶₃exp𝑏𝑅_𝑖𝑏+𝐶₁𝑇²,𝐺(𝑇)=−𝑎𝐶₂().2𝐾𝑏exp2𝑏𝑇)(1−2𝑏𝑇 (D.7) Now, 𝐺(𝑇)=−𝑎𝐶₂()2𝐾𝑏exp2𝑏𝑇)(1−2𝑏𝑇⟹𝐺^′(𝑇)=2𝑎𝑏𝑇𝐶₂𝐾exp(2𝑏𝑇)⟹𝐺^′′(𝑇)=(2𝑏+1)2𝑎𝑏𝐶₂𝐾exp(2𝑏𝑇)⟹𝐺^′′(𝑇)⟶+∞. (D.8) Hence, 𝑍_1𝑖 is monotonic increasing if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0 as 𝑍_1𝑖 does not have stationary points in [𝑅_𝑖,∞). Here, 𝑡₁=(𝑎/𝐾𝑏)(exp(𝑏𝑇)−1) and 𝑡₁ increases with increasing value of 𝑇. As in our model, 𝑡₁<𝑅_𝑖; hence, for the feasibility of the model, 𝑡^∗₁=𝑅_𝑖 and 𝑇^∗ is obtained from (𝑎/𝑏)(exp(𝑏𝑇^∗)−1)=𝐾𝑅_𝑖. Therefore, 𝑍_1𝑖 has a maximum value at 𝑇^∗=(1/𝑏)ln(1+𝐾𝑏𝑅_𝑖/𝑎) if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0. Here 𝑇^∗ is unique, hence the proof.

Proof of Lemma 8. The first part of the lemma is obvious. For the second part, if 𝑍_2𝑖 does not have any stationary points in [0,𝑅_𝑖], then 𝑍_2𝑖 is either monotonic increasing or monotonic decreasing function of 𝑇∈[0,𝑅_𝑖]. Here 𝑑𝑍_2𝑖/𝑑𝑇→∞ as 𝑇→0 because 𝑑𝑍_2𝑖=𝑎𝑑𝑇𝑏𝑇𝑃−𝐶₃()−()𝑏exp𝑏𝑇exp𝑏𝑇−1𝑇−𝐼_𝑐𝑃𝑇𝑇𝑏exp(𝑏𝑇)−𝑅_𝑖()−1𝑏exp𝑏𝑇()+𝑅𝑏𝑇1−exp𝑏𝑇^𝑖exp(𝑏𝑇)𝑇−𝑅_𝑖𝑇−exp(𝑏𝑇)−𝐶₂𝑎exp(𝑏𝑇)𝐾−𝑎exp(2𝑏𝑇)𝐾()−()+𝑇𝑏exp𝑏𝑇𝑎exp𝑏𝑇+()𝐾𝑏𝑇𝑎exp2𝑏𝑇−12𝑇𝐾𝑏+𝑎𝑏𝑇()+()2𝐾𝑏𝑇−exp𝑏𝑇exp𝑏𝑇+𝑏𝑇𝑏𝐶₁.𝑎𝑇 (D.9) Using L’Hospital’s Rule, we obtain lim_𝑇→0𝑑𝑍_2𝑖𝑑𝑇=lim_𝑇→0𝑎𝑏𝑇³×𝑃−𝐶₃𝑏𝑇exp(𝑏𝑇)−exp(𝑏𝑇)−1𝑇−𝐼_𝑐𝑃𝑇𝑇𝑏exp(𝑏𝑇)−𝑅_𝑖()−1𝑏exp𝑏𝑇𝑏1−exp(𝑏𝑇)+𝑅_𝑖exp(𝑏𝑇)−𝑅_𝑖−𝑇exp(𝑏𝑇)−𝐶₂𝑇𝑎exp(𝑏𝑇)−𝑎exp(2𝑏𝑇)𝐾()−()()+𝑇𝑏exp𝑏𝑇2𝑎exp𝑏𝑇−𝑎exp2𝑏𝑇−12𝐾𝑏𝑏+𝑎()+()2𝐾𝑏−𝑇exp𝑏𝑇exp𝑏𝑇𝑏+𝑏𝐶₁𝑇𝑎=lim_𝑇→0𝐻(𝑇)𝑇³=lim_𝑇→0𝐻^′(𝑇)3𝑇²⟶+∞, (D.10) where 𝐻(𝑇)=𝑎𝑏𝑃−𝐶₃×𝑏𝑇exp(𝑏𝑇)−exp(𝑏𝑇)−1𝑇−𝐼_𝑐𝑃𝑇𝑇𝑏exp(𝑏𝑇)−𝑅_𝑖()−1𝑏exp𝑏𝑇𝑏1−exp(𝑏𝑇)+𝑅_𝑖exp(𝑏𝑇)−𝑅_𝑖−𝑇exp(𝑏𝑇)−𝐶₂𝑇𝑎exp(𝑏𝑇)−𝑎exp(2𝑏𝑇)𝐾()−()()+𝑇𝑏exp𝑏𝑇2𝑎exp𝑏𝑇−𝑎exp2𝑏𝑇−12𝐾𝑏𝑏+𝑎()+()2𝐾𝑏−𝑇exp𝑏𝑇exp𝑏𝑇𝑏+𝑏𝐶₁𝑇𝑎. (D.11) Now, 𝐻^′(𝑇)=𝑎𝑏𝑃−𝐶₃×𝑏𝑇exp(𝑏𝑇)−exp(𝑏𝑇)+1+𝑇²𝑏²exp(𝑏𝑇)−𝐼_𝑐𝑃𝑇𝑏exp(𝑏𝑇)+𝑇²𝑏²exp(𝑏𝑇)−𝑅_𝑖𝑇𝑏²exp(𝑏𝑇)−𝐶₂𝑇𝑎exp(𝑏𝑇)−𝑎exp(2𝑏𝑇)𝐾−exp(𝑏𝑇)−𝑇²𝑎𝑏exp(𝑏𝑇)−2𝑎𝑏exp(2𝑏𝑇)𝐾+2𝑏exp(𝑏𝑇)+𝑇³𝑏²()−()()exp𝑏𝑇2𝑎exp𝑏𝑇−𝑎exp2𝑏𝑇−12𝐾𝑏𝑏+𝑎+()2𝐾𝑏exp𝑏𝑇𝑏+𝑏𝐶₁𝑎. (D.12) Here, lim_𝑇→0𝐻^′(𝑇)=𝐶₁ so that 𝑍_1𝑖 is monotonic increasing if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0 as 𝑍_1𝑖 does not have stationary points in [0,𝑅_𝑖]. Here, 𝑡₁=(𝑎/𝐾𝑏)(exp(𝑏𝑇)−1), and 𝑡₁ increases with increasing value of 𝑇. According to our model, 𝑡₁<𝑅_𝑖; hence, for the feasibility of the model, 𝑡^∗₁=𝑅_𝑖 and 𝑇^∗ is obtained from (𝑎/𝑏)(exp(𝑏𝑇^∗)−1)=𝐾𝑅_𝑖. Therefore, 𝑍_1𝑖 has a maximum value at 𝑇^∗=(1/𝑏)ln(1+𝐾𝑏𝑅_𝑖/𝑎) if (𝑑𝑍_1𝑖/𝑑𝑇)|_{𝑇=𝑅𝑖}>0. Here, 𝑇^∗ is unique, hence the proof.

Acknowledgments

The authors want to acknowledge the reviewers for their constructive comments to revise the paper. The first author would like to express his heartiest gratitude to his parents, wife, and son for their valuable support during the research. This research is supported by the University Grant Commission, Delhi, India, from the research grant of the Minor Project (no. 41-1433/2012(SR)). The corresponding author wants to acknowledge the infrastructural assistance available at Vidyasagar University, West Bengal, India.