Effect of Variable Inflation on Supply Chain Management in FMCG Sector with Single Supplier-Multiple Retailers and Trade Credit

Inflation is the result of a greater supply of money than its demand. Greater demand than supply, leads to inflationary conditions in an economy. Buzacott [1] was the first researcher who wrote something about inflation in inventory models by formulating a minimum cost model for a single item in inventory and studied the effect of inflation. Guria et al. [2] framed an inventory policy for an item with inflation and demand depending on the selling price by adding a new parameter of immediate part payments to the wholesaler. In 1985, Goyal [3] gave an EOQ model with permissible delays in payments. Later in 1995, Aggarwal and Jaggi [4] extended their work and gave a relation between sales revenue and delay period along with the rate of interest. Supplier-Retailer coordination/setup is taken up by innumerable researchers. Esmaeili and Nasrabadi [5] in 2020 talked about the single vendor-multiple retailers set up.

Thusly, the main contribution of this article is to examine the role of variable inflation rates in inventory management. The main reason behind considering the variable inflation in each replenishment cycle is the competitive and fast changing scenario exiting in the present world and as a result, the prices rise faster as compared to earlier. A supplier with multiple retailers’ setups is considered. Equipoise of total inventory cost for multiple retailers in a given competitive environment is calculated. The supplier and multiple retailers' trade credits are considered and the process is controlled by the supplier showing the well-coordinated supply chain management. As per our literature review, the variable inflation per cycle is not taken into account by any researcher.

In this piece of writing, the effect of inflation is studied at every step. The main reason behind doing all this is to see how inflation affects the prices or cost, when inflation rises every time, the inventory get replenished. We have considered a linear rise in inflation which no one has taken before this, making our work a novel piece. The coordination between the supplier and the retailer has been taken to a different level. Using the coordination between the two, the retailer's cost equipoise has been worked out. The biggest advantage to the market (more precisely to the consumer) from this would be that the price of a product would remain the same with every retailer, that too when the deep analysis and effect of inflation has been considered. Retailer’s cost equipoise is an unconventional concept in inventory management.

The supplier takes an order from a retailer at time t₀=0, the inventory level reaches zero at t₁ and then the inventory is refilled instantaneously. No shortages or deterioration rate is considered, as the targeted field is FMCG. Inventory and time-dependent demand along with inflation-based demand are considered. The model and its graphical representation are shown in Figure 1, where t₀ is zero and the horizon is finite. I_i+1(t) depicts the inventory level in (i+1)th cycle.

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Figure 1. Graphical representation of the model

Case 1: When there is no coordination between the supplier and the retailers.

The differential equation of the inventory level in (i+1)th cycle will be given as follows:

$\begin{gathered}\frac{d I_{i+1}(t)}{d t}+k I_{i+1}(t)=-\left(b_1 t+b_2 e^{\alpha_i t}\right), t_i \leq t \leq  t_{i+1}\end{gathered}$         (1)

where, boundary conditions are I_i+1(t_i+1)=0 and I_i(t_i)=Q_i+1. Therefore, the solution of Eq. (1) is.

$\begin{gathered}I_{i+1}(t)=e^{-k t} \int_t^{t_{i+1}} e^{k u}\left(b_1 u+b_2 e^{\alpha_i u}\right) d u \,\,t_i \leq  t \leq t_{i+1}\end{gathered}$          (2)

Refer to Appendix A for the solution to Eq. (1).

The order quantity for (i+1)th cycle is:

$Q_{i+1}=I_{i+1}\left(t_i\right)=e^{-k t_i} \int_{t_i}^{t_{i+1}} e^{k t}\left(b_1 t+b_2 e^{\alpha_i t}\right) d t$

$such \,that$, $u \in\left[t, t_{i+1}\right]$ and $t_i \leq t \leq u \leq t_{i+1}$            (3)

The ordering cost of the retailer is:

$Y\left(\sum_{j=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right)\right)\right)$        (4)

Refer to Appendix B for the solution to Eq. (4).

The holding cost of the retailer is:

$\begin{aligned} & \left(\sum_{j=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right) * \frac{1}{2 k^3} b_1\left(2-2 e^{k\left(-t_j+t_{1+j}\right)}-2 k t_j\right.\right.\right. \\ & \left.+k^2 t_j^2+2 e^{k\left(-t_j+t_{1+j}\right)} k t_{1+j}-k^2 t_{1+j}^2\right) \\ & \left.+\frac{b_2\left(\frac{e^{-k t_j+t_{1+j} \alpha_{1+j}}\left(-e^{k t_j}+e^{k t_{1+j}}\right)}{k}+\frac{e^{t_j \alpha_{1+j}}-e^{t_{1+j} \alpha_{1+j}}}{\alpha_{1+j}}\right)}{k+\alpha_{1+j}}\right)\end{aligned}$              (5)

Refer to Appendix C for the solution to Eq. (5).

The purchasing cost of the retailer is:

$\begin{aligned} & \left(W\left(\sum_{j=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right) *\left(b_1\left(\frac{e^{k\left(t_{j+1}-t_j\right)} * t_{j+1}}{k}\right.\right.\right.\right.\right.  -\frac{e^{k\left(t_{j+1}-t_j\right)}}{k^2}-\frac{t_j}{k}+\frac{1}{\left.k^2\right)} \\ & \left.\left.\left.+b_2\left(\frac{e^{\alpha_{j+1} * t_{j+1}} * e^{k\left(t_{j+1}-t_j\right)}-e^{t_j^* \alpha_{j+1}}}{k+\alpha_{j+1}}\right)\right)\right)\right) \\ & \end{aligned}$                (6)

Refer to Appendix D for the solution to Eq. (6).

The retailer’s total cost will be the sum of ordering cost, holding cost, and purchasing cost.

Adding Eqns. (4), (5), and (6), we get:

$\begin{aligned} & R_I  =W\left(\sum_{i=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right) *\left(b_1\left(\frac{e^{k\left(t_{j+1}-t_j\right)} * t_{j+1}}{k}\right.\right.\right.\right.  \left.-\frac{e^{k\left(t_{j+1}-t_j\right)}}{k^2}-\frac{t_j}{k}+\frac{1}{k^2}\right)  \left.\left.+b_2\left(\frac{e^{\alpha_{j+1} * t_{j+1}} * e^{k\left(t_{j+1}-t_j\right)}-e^{t_j^* \alpha_{j+1}}}{k+\alpha_{j+1}}\right)\right)\right) \\ & +Y\left(\sum_{j=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right)\right)\right)  +h\left(\sum_{i=0}^{n-1}\left(\left(e^{\sum_{i=0}^j \alpha_{i+1}\left(t_{i+1}-t_i\right)}\right) * \frac{1}{2 k^3} b_1\left(2-2 e^{k\left(-t_j+t_{1+j}\right.}\right)\right.\right.  \left.-2 k t_j+k^2 t_j^2+2 e^{k\left(-t_j+t_{1+j}\right)} k t_{1+j}-k^2 t_{1+j}^2\right) \\ & \left.+\frac{b_2\left(\frac{e^{-k t_j+t_{1+j} \alpha_{1+j}}\left(-e^{k t_j}+e^{k t_{1+j}}\right)}{k}+\frac{e^{t_j \alpha_{1+j}}-e^{t_{1+j} \alpha_{1+j}}}{\alpha_{1+j}}\right)}{k+\alpha_{1+j}}\right) \\ & \end{aligned}$                (7)

Let $n^*, t_1^*, t_2^*, \ldots, t_{n-1}^*$ be the optimal solution for the minimum total cost of the retailer (R_I). Under this first case, of no coordination supplier’s cost will be affected by the retailer’s replenishment rate. The supplier’s total cost will be the sum of set up cost and manufacturing cost.

Set-up cost is a further sum of Labour Cost (LC) and Machinery Cost (MC).

$\begin{aligned} S_I\left(n^*, t_0, t_1^*, t_2^*, \ldots\right. & \left.t_{n-1}^*\right) =\left(n^*-1\right)(L C+M C)  +\sum_{j=0}^{n^*-1} c * Q_{j+1}^*\end{aligned}$        (8)

The total order quantity during the planning horizon is:

$Q_I=\sum_{j=0}^{n^*-1} Q_{j+1}^*$             (9)

Algorithm 1 for Case 1:  (In case of no coordination)

Step 1: To find the optimal schedule for ordering from the retailer.

Step 1 (i): Assign n=2, 3, 4, and so on.

Step 1 (ii): Solve Eq. (10) for n=2, 3, 4 and so on.

Step 1 (iii): Derived values of t₀, t₁, t₂, and so on will be used to calculate the value of the total cost of the retailer from Eq. (7).

Step 2: Find the optimal replenishment cycle.

Step 3: Use the optimal replenishment cycle value derived in the previous step to calculate the optimal cost of the supplier from Eq. (8) and the total quantity from Eq. (9).

Case 2: When there is coordination among the supplier and multiple retailers.

Credulous arrangement for competitive market model

Our system for Case 2 constitutes a set of retailers with different costs of the product. The intent behind fabricating this arrangement is to reach an equipoise, in inventory cost of retailers under the assistance of the supplier (at two phases, to achieve equipoise and thenceforth deciding credit period).

Let M_i, i=1,2,3,4,…. be the set of nabe retailers with dissimilar inventory costs at the initial step. Here we presume, that each retailer has goods and in the initial phase has different inventory costs. Under the defined array, a pair of retailers will be undertaken that have maximum and minimum inventory costs of the product. The inventory cost of each retailer will be designated as $C_{R_i}$, i=1,2,3,… to calculate the inventory cost of each retailer, we follow below mentioned procedure.

The inventory cost of the retailers is calculated from Eq. (7). The unit cost of the product is given by dividing the solutions obtained from Eq. (7) by the solutions of Eq. (9). This particular unit cost of the product will be different for all the retailers in the market (keeping in mind the variable holding cost, ordering cost, and purchasing cost of the retailer initially). The supplier will contact the retailers and will give them trade credits for bringing their cost up to equipoise in the competitive free market.

Proffer design

In this section, we consider an arrangement most appropriate for FMCG competitive market. Under this, unabridged procedure, RATProffer is initiated, in which each retailer will always proffer another nabe retailer with the lowest cost of the product, where cost is always an integer.

RATProffer

Each retailer, $C_{R_i}, i=1,2,3, \ldots$. proffer $\quad C_{R_{i j}}=C_j+$ $\left(\frac{C_i-C_j}{z}\right)$ where $z \geq 1, \quad i, j \in M_i$, to another retailer $C_{R_i}, i=$ $1,2,3, \ldots \ldots \in M_i$ by using the formula defined by the above-mentioned formula. For calculating equipoise of Inventory Cost for Multiple Retailers in a given Competitive Environment the following algorithm will be followed:

Algorithm 2 for Case 2:

Step 1: Under the coordination case, use the value derived in step 1 (c) (from Algorithm 1) of the total cost of the retailer and the value of total quantity from Step 1 (e) (from Algorithm 1) divide the total cost of a retailer with the quantity of the product to evaluate the per unit cost of the product for one retailer.

Step 2: Repeat Step 1 for 4 retailers.

Step 3: Consider the values per unit cost of the product for Retailers as C₁, C₂, C₃, C₄.

Step 3 (a): Find the largest among the four values of C₁, C₂, C₃, C₄.

Step 3 (b): Find the smallest among the four values of C₁, C₂, C₃, C₄.

Step 3 (c): When C₁ is largest and C₂ is smallest, Replace C₂ with S, and calculate S+(C₁-S)/z and replace C₁= C₂=S.

Step 3 (d): Repeat Step 3 (c) when C₁ is largest and C₃ is smallest by replacing C₂ with C₃.

Step 3 (e): Repeat Step 3 (c) when C₁ is largest and C₄ is smallest by replacing C₂ with C₄.

Step 3 (f): When C₂ is largest and C₁ is smallest, calculate VC₁ C₂=C₁+((C₂-C₁)/z) and assign C₁=C₂=VC₁ C₂.

Step 3 (g): Repeat Step 3 (e) when C₂ is largest and C₃ is smallest by replacing C₁ with C₃.

Step 3 (h): When $C_3$  is largest and $C_1$  is smallest, calculate $V C_1 C_3=C_1+\left(\frac{C_3-C_1}{z}\right)$ and replace C₁=C₃=VC₁ C₃

Step 3 (i): Repeat Step 3 (h) when C₃ is largest and C₂ is smallest by replacing C₁ with C₂.

Step 3 (j): If C₁=C₂=C₃=C₄ then the equipoise is achieved and the program ends, if no then go to Step 3 (a).

Once the equipoise is attained, it meant that all retailers now have the same inventory cost for a product, no matter how many different costs a retailer had at the preliminary stage. The retailer whose cost was greater than the equipoise earlier, this meant that he would be given compensation in terms of interest through the trade credit facility offered by the supplier whereas the retailer whose cost was prior less than the equipoise, would automatically benefit from the increased equipoise cost.

Definition 1: (Tenable Layout). A layout is tenable if the cost of the product for different retailers is the same.

Definition 2: (CostPact). A pact between two retailers during the execution of RATProffer.

Theorem 1: The layout RATProffer is without any stalemate point. This implies that there prevails some inventory cost in M_i considering the layout to be tenable.

Proof: Refer to Appendix E.

Theorem 2: The layout in due course will reach equipoise.

Proof: Refer to Appendix F.

To determine the optimality of the replenishment schedule.

To resolve the optimality of the replenishment schedule, from Eq. (7) of the total cost of the retailer we will calculate $\frac{\partial R_I}{\partial t_i}$  and then put $\frac{\partial R_I}{\partial t_i}=0$ where i=1, 2, 3, ……n-1. Since we are working on multiple retailers set up, the optimality will be for all the retailers individually. We will be proving for R_I2 and will be the same for all the other retailers.

$\begin{aligned} & \frac{\partial \mathrm{R}_{\mathrm{I} 2}}{\partial \mathrm{t}_{\mathrm{i}}}=\mathrm{h}\left\{\left(\alpha_{\mathrm{i}}-\alpha_{1+\mathrm{i}}\right) *\left(\mathrm{e}^{\sum_{\mathrm{i}=0}^{\mathrm{j}} \alpha_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)}\right) *\right.  \left(\frac{1}{2 \mathrm{k}^3} \mathrm{~b}_1\left(2-2 \mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{j}}+\mathrm{t}_{1+\mathrm{j}}\right)}-2 \mathrm{kt}_{\mathrm{j}}+\mathrm{k}^2 \mathrm{t}_{\mathrm{j}}^2+\right.\right.  \left.2 e^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{j}}+\mathrm{t}_{1+\mathrm{j}}\right)} \mathrm{kt}_{1+\mathrm{j}}-\mathrm{k}^2 \mathrm{t}_{1+\mathrm{j}}^2\right)+ \\ & \left.\frac{b_2\left(\frac{e^{-k_j+t_{1+j} \alpha_{1+j}}\left(-e^{k t_j}+e^{k t_1+j}\right)}{k}+\frac{e^{t_j \alpha_{1+j}}-e^{t_{1+j} \alpha_{1+j}}}{\alpha_{1+j}}\right)}{k+\alpha_{1+j}}\right)+  \left(\mathrm{e}^{\sum_{\mathrm{i}=0}^{\mathrm{j}} \alpha_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)}\right) * \frac{\left(-\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{-1+\mathrm{i}}+\mathrm{t}_{\mathrm{i}}\right)} \mathrm{k}+\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{i}}+\mathrm{t}_{1+\mathrm{i}}\right)} \mathrm{k}\right) \mathrm{b}_1}{\mathrm{k}^3}\\&+  \frac{\mathrm{b}_1\left(-1+e^{\mathrm{k}\left(-\mathrm{t}_{-1+i}+\mathrm{t}_{\mathrm{i}}\right)}+\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{-1+i}+\mathrm{t}_{\mathrm{i}}\right)} \mathrm{kt}_{\mathrm{i}}-\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{i}}+\mathrm{t}_{1+\mathrm{i}}\right)} \mathrm{kt}_{1+\mathrm{i}}\right)}{\mathrm{k}^2}+  \end{aligned}$

$\begin{aligned} & \frac{\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{-1+\mathrm{i}}+\mathrm{t}_{\mathrm{i}}\right)+\mathrm{t}_{\mathrm{i}} \alpha_{\mathrm{i}_2}}}{\mathrm{k}+\alpha_{\mathrm{i}}}+\frac{\mathrm{e}^{\mathrm{t}_{\mathrm{i}} \alpha_{\mathrm{i}} \mathrm{b}_2\left(-\frac{1}{\mathrm{k}}+\frac{\left.\mathrm{e}^{\mathrm{k}(-\mathrm{t}}-1+\mathrm{i}+\mathrm{t}_{\mathrm{i}}\right)}{\mathrm{k}}-\frac{1}{\alpha_{\mathrm{i}}}\right) \alpha_{\mathrm{i}}}}{\mathrm{k}+\alpha_{\mathrm{i}}}+\left.\frac{\mathrm{e}^{\mathrm{t}_{\mathrm{i}} \alpha_{1+\mathrm{i}} \mathrm{b}_2}}{\mathrm{k}+\alpha_{1+\mathrm{i}}}-\frac{\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{i}}+\mathrm{t}_{1+\mathrm{i}}\right)+\mathrm{t}_{1+\mathrm{i}} \alpha_{1+\mathrm{i}} \mathrm{b}_2}}{\mathrm{k}+\alpha_{1+\mathrm{i}}}\right\}+  \\&\mathrm{Y}\left\{\mathrm{e}^{\sum_{\mathrm{i}=0}^{\mathrm{i}-1} \alpha_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)} *\left(\alpha_{\mathrm{i}}-\alpha_{\mathrm{i}+1}\right)\right\}+ \mathrm{e}^{\sum_{\mathrm{i}=0}^{\mathrm{i}} \alpha_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)} *\left(\alpha_{\mathrm{i}}-\alpha_{\mathrm{i}+1}\right) *  \left(b_1\left(\frac{e^{k\left(t_{i+1}-t_i\right)_* t_{i+1}}}{k}-\frac{e^{k\left(t_{i+1}-t_i\right)}}{k^2}-\frac{t_i}{k}+\frac{1}{k^2}\right)+\right. \\ & \end{aligned}$

$\begin{aligned} & \mathrm{b}_2\left(\frac{\left.\mathrm{e}^{\alpha_{i+1} * \mathrm{t}_{i+1} * \mathrm{e}^{\mathrm{k}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)}-\mathrm{e}^{\mathrm{t}_{\mathrm{i}} * \alpha_{i+1}}}\right)}{k+\alpha_{i+1}}\right)+  \mathrm{e}^{\sum_{\mathrm{i}=0}^{\mathrm{i}} \alpha_{i+1}\left(\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}\right)} *\left(\mathrm{~b}_1\left(-\frac{1}{\mathrm{k}}+\frac{\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{i}}+\mathrm{t}_{1+\mathrm{i}}\right)}}{\mathrm{k}}-\right.\right.  \left.\mathrm{e}^{\mathrm{k}\left(-\mathrm{t}_{\mathrm{i}}+\mathrm{t}_{1+\mathrm{i}}\right)} * \mathrm{t}_{1+\mathrm{i}}\right)+ \end{aligned}$

$\frac{b_2\left(-e^{k\left(-t_i+t_{1+i}\right)+t_{1+i^*} \alpha_{1+i}} k-e^{\left.t_i * \alpha_{1+i} \alpha_{1+i}\right)}\right.}{k+\alpha_{1+i}}))=0$      (10)    

By using Eq. (10) the values of t_i’s are obtained and finally, we calculate the total cost of the retailers. Next, we prove, Optimality of the replenishment cycle in Theorem 3.

Theorem 3: The unique solution to a nonlinear system of Eq. (10) derived by obtaining the first derivative of R_I2is the optimal replenishment cycle length for a given n.

Proof: Refer to Appendix G for proof.

For sensitivity analysis out of four retailers, we are randomly choosing R_I2 (as for all the other retailers' the process will be the same). In this section Tables and graphs will help to acknowledge the relationship between different parameters and the total cost of the retailer.

7.1 Sensitivity analysis takeaways

From Table 5, the following interpretations are made:

1. When there is an increase in the value of Y, there is an increase in the value of the total cost of the retailer.
2. An increase in the value of W has a positive relationship with the value of the total cost of the retailer as well.
3. As the value of b₁ increases, so is the value of the total cost of the retailer and with a fall in the value of b_1, retailer’s cost experiences a fall.
4. It can be manifested that there is a direct relation between b₁ and the retailer’s total cost.
5. The increase in the parameters of LC, MC, and c increases the cost of the supplier.
6. The decrease in the parameters of LC, MC, and c has a direct relation to the total cost of the supplier.

Explanations of figures contained in the sensitivity analysis section

1. In Figure 4, the total cost of 4 retailers (depicted in Table 4), is plotted to see how the values act. It demonstrates well that the values first decrease and then gradually increase. Giving convexity of the total cost of four different retailers.
2. The value of the total cost of supplier SR_I2 and how it is affected by the change in values of c (purchasing cost of the supplier) is displayed in Figure 5. Both have direct relations.
3. The effect on the total cost of supplier SR_I2 by variations in the value of Labour Cost (LC) and Machinery Cost (MC) is demonstrated in Figure 6, as both these costs increase so is the cost of SR_I2 also rises, and vice versa.
4. The more the ordering cost (Y) of the retailer more will be the total cost, this positive relation between Y and R_I2 can be established in Figure 7.
5. Figure 8, shows the relation between R_I2 and SR_I2 with the scaling constant of the inflation function (b₂). As the value of b2 increases so the value of other two also increases signifying a direct relation that as the inflation increases, the cost of the supplier and the retailer increases as well.
6. As the value of wholesale price for the retailer rises so is the total inventory of the retailer, this directly proportional relationship between the two variables (R_I2 and W) is seen in Figure 9.
7. The relation between R_I2 and SR_I2 with the increasing demand rate (b₁) is shown in Figure 10. The relation is directly proportional.

Table 5. Sensitivity analysis of different parameters

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Figure 4. Graphical representation of Table 2

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Figure 5. Effect on value of SR_I2 due to change in c

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Figure 6. Effect on value of SR_I2 due to change in LC/MC

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Figure 7. Effect on value of R_I2 due to change in Y

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Figure 8. Effect on value of R_I2 and SR_I2 due to change in b₂

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Figure 9. Effect on value of R_I2 due to change in W

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Figure 10. Effect on value of R_I2 and SR_I2 due to change in b₁

Our research carefully studies the market changes caused by inflation. It’s observed that inflation changes randomly so we can't keep inflation the same during the whole time frame of the model. Keeping this vibrant nature of inflation as the kernel of our research we created this near-to-market reality model. A linear rise in inflation in each replenishment cycle, results in harnessing every minute swing in inflation.

Thereafter calculating the values of the total cost of four different retailers (as shown in Table 2, by applying Algorithm 1), we used Algorithm 2, to calculate equipoise. Equipoise of the inventory cost of the retailers is calculated in coordination with the supplier. After attaining the equipoise in the cost of a product, the profits will automatically flow towards the retailers in the form of new equipoise cost (if equipoise cost is greater than the earlier cost), if the case is reversed, the supplier will offer a credit period to the remaining retailers. In this way, the supplier too will gain as he/she will now allow credit terms to the limited retailers only.

The theorem to prove that the layout under Case 2 is tenable is mentioned. Even the layout is free from any kind of stalemate point is also demonstrated in theorem 2 with the help of Lemma 1. Theorem 3 is proved to express the optimality of the total cost of the retailer.

In the managerial stance, the new model will allow managers greater leeway in making decisions. The supplier should take necessary precautions, especially in Case 2, as he/she will be solely responsible for managing the equipoise. The trade-credit period has been reduced considerably and the number of retailers to whom it will be given is also reduced, a profitable affair for suppliers furthermore.

The inventory model of equipoise is overriding all the factors which were affecting the inventory cost of the different retailers in the nabe environment to be towering high than each other. The equipoise cost will be a beneficial game for the consumer as well. Concluding we can affirm that the equipoise concept is profitable for all the contenders in the market.

For the prospects for future research, we propose the following pointers.

(i). Shortages can be incorporated and partially backlogged.

(ii). Extension can be done taking into consideration fuzzy demand.

(iii). Case 2, can be expanded for multiple suppliers-multiple retailers.