A Replenishment Policy for an Inventory Model with Price-Sensitive Demand with Linear and Quadratic Back Order in a Finite Planning Horizon

A Replenishment Policy for an Inventory Model with Price-Sensitive Demand with Linear and Quadratic Back Order in a Finite Planning Horizon

Nitin Kumar Mishra Renuka S. Namwad* Ranu

Department of Mathematics, Lovely Professional University, Phagwara 144411, India

Corresponding Author Email:
Page:
1537-1546
|
DOI:
https://doi.org/10.18280/mmep.110614
30 December 2023
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Revised:
3 March 2024
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Accepted:
15 March 2024
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Available online:
22 June 2024
| Citation

OPEN ACCESS

Abstract:

In this research article, we proposed an inventory model for the replenishment policy. The focus of our research article is on the companies that frequently deal with backorders. An advanced inventory model considering back order has been proposed the results highlight the dynamic nature of the system, with optimal values achieved in different cycles. In this model, replenishment policy is given and to lower the economic ordering cost, we used new parameters such as price-sensitive demand, complete back ordering, and backorder is taken as a quadratic function as well as linear backorder with shortages in a finite planning horizon. The result is discussed for both backorders (linear and quadratic), to minimize the total cost obtained using the Hessian matrix to be positive definite. Software ‘MATHEMATICA VERSION 12’ has been used for the solution of the proposed model by using numerical iterative method. For different parameters, different tables are provided. The outcomes of the sensitivity analysis with the help of tables and graphs are depicted. Finally, we have discussed the conclusion and practical implications.

Keywords:

price-sensitive demand, shortage, linear backorder, quadratic backorder, supply chain, finite planning horizon

1. Introduction

While dealing with the inventory problem the basic thing to be remember the developing technologies along with the inventory models. New technologies are growing due to recent research done on inventory. Seliaman et al. [1] are developing new techniques day by day to make inventory management easier.

Backorders are for products that a firm cannot currently fill because demand exceeds supply. Backordering can refer to items that are presently in production or those that have not yet started production. For handling the backorder communication is the key. By communicating the presence of backorder, the supplier gets the information about the customer’s actual demand for the product what is inbound, and when the balance items will be there. This allows both the suppliers and the customers to continue the operations uninterrupted. As the backorder may impact inventory and other holding costs.

In this article, we made an advanced model of inventory for the firms/companies who frequently deal with backorders. In this model we take the price-sensitive demand, backorder as a quadratic function with shortages in a finite planning horizon, and a case of linear backorder is discussed with lead time is zero.

2. Research Gap and Problem Identification

In this study, we aim to address the research gap related to price-dependent quadratic backorder. Despite the extensive research conducted in this field, there is still a lack of understanding regarding a very few research conducted in the finite planning horizon. Therefore, this study seeks to contribute to the existing literature by price-sensitive demand and price-dependent quadratic backorder. The research question addressed in this study is a replenishment policy for the quadratic backorder and linear backorder inventory models discussed in the finite planning horizon.

3. Literature Review

Inventory is a very interesting topic for researchers. So much work has been done from the decade and still mostly work is going to be done. Firstly, the classical EOQ formula was discovered by Harris [2] which is also known as ‘Square-Root formula’. The first book on inventory management was written by Raymond [3]. Veinott [4] studied that in the real-life situations, the demand rate is dynamic. So, they developed the first dynamic economic order quantity model which is developed by modifying Harris’s Square-Root model.

Ouyang et al. [5] provided a model by taking shortages and solving the total shortages as a combined form of lost sales and backorder. Scarf et al. [6] estimated a stochastic model of multi-period with shortages and given a policy (s, S) for an optimal solution with backorders. Veinott [4] think that finding the exact backorder cost was a hard task so they developed the model which calculates the backorder cost. They considered back-ordering as a constant function, with shortages. Zangwill [7] proposed a multi-period model along with shortages and backorders. Fogarty and Aucamp [8] gave the model with shortages and back-ordering. Aardal et al. [9] proposed a model by taking the random demand (q, r) model given by Hadley and Within. Backorders are not considered but they assure that the yearly backorders cannot cross the upper boundaries.

image003.png

Figure 1. Inventory model diagram

Table 1. Survey of existing literature

 Ref. Demand Type Shortages Backorder Type Finite Planning Horizon [2] Classical EOQ (Square-Root) - - - [4] Dynamic EOQ Yes (Dynamic) Dynamic - [35] Time-dependent Yes - - [5] - Lost sales & backorder Lost sales & backorder - [6] Stochastic Optimal with backorders Optimal - [7] Multi-period Yes Yes - [12] Normally distributed Expected total cost with backorder Expected total cost - [16] - Yes Linear - [15] Time-dependent Partially backlogging Partial - [17] - Complete backordering Complete - [18] - No & yes with complete backordering No & complete - [19] Power depending on time Yes, with complete backordering Complete Yes [23] Linear holding cost (on price) Partially backlogged Partial - [24] Time-dependent holding cost Partially backlogged Partial - [25] Time-dependent holding cost Partially backlogged Partial - [26] Stock-dependent Partially backlogging Partial - [36] Linear plus fixed - Linear & fixed - [32] Linear holding cost (on price) Partially backlogged Partial - [33] Time-dependent holding cost Completely backlogged Complete - [34] - Partially backlogged Linear & quadratic - This paper Price-sensitive Yes Complete linear and quadratic Yes
4. Assumptions

i. The total stock level is initially zero.

ii. The cost of storing stays constant.

iii. The lagging time is zero.

iv. The cost of ordering is predetermined.

v. Under a finite planning horizon, shortages are acceptable and a continuous one.

vi. Back ordering is complete and described as a quadratic function and linear.

5. Mathematical Solution of the Model

The initial inventory equation is given by,

$\frac{d{{I}_{j+1}}\left( t \right)}{dt}+\left( {{\theta }_{1}} \right){{I}_{j+1}}\left( t \right)=-D\left( t \right) \\ {{t}_{j}}<t<{{s}_{j+1}}$                     (1)

where, j=1, 2, 3, …, n1.

$\frac{d{{I}_{j+1}}\left( t \right)}{dt}=-D\left( t \right)-{{\theta }_{1}}{{I}_{j+1}}\left( t \right) \\ {{t}_{j}}<t<{{s}_{j+1}}$                      (2)

Considering the boundary condition ${{I}_{i+1}}\left( {{s}_{j}} \right)=0$.

Solution of Eq. (2) is,

${{I}_{j+1}}\left( t \right)={{e}^{-{{\theta }_{1}}*t}}\underset{t}{\overset{{{s}_{j+1}}}{\mathop \int }}\,D\left( u \right){{e}^{u}}du$                          (3)

${{I}_{j+1}}\left( t \right)=\underset{t}{\overset{{{s}_{j+1}}}{\mathop \int }}\,D\left( u \right){{e}^{{{\theta }_{1}}\left( u-t \right)}}du$                  (4)

${{I}_{j+1}}\left( t \right)=\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-t \right)}}-1 \right]D\left( t \right)$                      (5)

During the shortage phase, the instantaneously arising shortage ${{I}_{b}}\left( t \right)$ is offered by,

${{I}_{b}}\left( t \right)={{D}_{1}}\left( {{t}_{j+1}}-{{s}_{j}} \right)$                               (6)

where, ${{D}_{1}}=a-bp-c{{p}^{2}}$ is the price dependent quadratic backorder.

${{I}_{b}}\left( t \right)=a-bp-c{{p}^{2}}\left( {{t}_{j}}-{{s}_{j}} \right)$                            (7)

Considering the boundary condition, ${{I}_{b}}\left( {{s}_{j}} \right)=0$.

${{Q}_{j+1}}={{I}_{j+1}}\left( {{t}_{j}} \right)=\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right)$                          (8)

where, $D\left( t \right)=a-b*p$.

Considering the reorganization of the ordering, ${{S}_{j+1}}$ can be given as,

${{S}_{j+1}}=\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,{{I}_{b}}\left( t \right)dt=\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( a-b*p-c*{{p}^{2}} \right)\left( {{t}_{j}}-{{s}_{j}} \right)dt$                            (9)

The entire purchase amount for a limited time frame of planning,

${{Q}_{nt}}=\underset{j=1}{\overset{{{n}_{1}}}{\mathop \sum }}\,{{Q}_{j+1}}=\underset{j=1}{\overset{{{n}_{1}}}{\mathop \sum }}\,\left\{ {{I}_{j+1}}+{{S}_{j+1}} \right\}$                            (10)

${{Q}_{j+1}}=\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right) +\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( a-b*p-c*{{p}^{2}} \right)\left( {{t}_{j}}-{{s}_{j}} \right)dt~$                      (11)

The total retailer cost over a specified time horizon is given by,

Total cost = Resupply expenses + cost of retaining stocks + purchasing cost + storage cost

${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}}+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,{{I}_{j+1}}\left( t \right)dt~+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*{{Q}_{j+1}}+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,s\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,{{I}_{b}}\left( t \right)dt$                       (12)

${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}}+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,{{I}_{j+1}}\left( t \right)dt+ \underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*{{Q}_{j+1}}+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,s\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,{{I}_{b}}\left( t \right)dt$

${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}}+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-t \right)}}-1 \right]D\left( t \right)dt\\ +\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right) +\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( a-b*p-c*{{p}^{2}} \right)\left( {{t}_{j}}-{{s}_{j}} \right)dt)$

${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}} +H\underset{{{t}_{j-1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-t \right)}}-1 \right]D\left( t \right)dt \\ +\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-t \right)}}-1 \right]D\left( t \right)dt+{{W}_{h}}c *(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-{{t}_{j-1}} \right)}}-1 \right]D\left( t \right)+{{W}_{h}} \\ *(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right) +s\left( a-b*p-c*{{p}^{2}} \right){{\left( {{t}_{j-1}}-{{s}_{j-1}} \right)}^{2}} +s\left( a-b*p-c*{{p}^{2}} \right){{\left( {{t}_{j}}-{{s}_{i}} \right)}^{2}}$      (13)

To achieve the lowest possible total cost in the inventory system, the essential conditions for minimizing the total cost are as follows:

$\frac{\partial TC\left( {{t}_{j}},{{s}_{j}},~~{{n}_{1}} \right)}{\partial {{t}_{j}}}=0,~j=1,~2,~3,~\ldots ,n$                         (14)

$\frac{\partial TC\left( {{t}_{j}},{{s}_{j}},~~/n \right)}{\partial {{s}_{j}}}=0,j=1,~2,~3,\ldots ,n$                     (15)

$\frac{\partial {{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}}=\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,-H *\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right)~\\ -\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}} \right]D\left( t \right) +\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,2*s\left( a-b*p-c*{{p}^{2}}~ \right)\left( {{t}_{j}}-{{s}_{j}} \right)\\ +\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,2*s\left( a-b*p-c*{{p}^{2}}~ \right)\left( {{t}_{j}}-{{s}_{j}} \right)$              (16)

$\frac{\partial {{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{j}}}=\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H\underset{{{t}_{i-1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-{{t}_{j}} \right)}} \right]D\left( t \right)dt~ \\-\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-{{t}_{j-1}} \right)}} \right]D\left( t \right) -\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,2*s\left( a-b*p-c*{{p}^{2}} \right)\left( {{t}_{j}}-{{s}_{j}} \right)$                 (17)

The total cost's Hessian matrix must be positive definite for a fixed n in order for the total cost to be least (i.e. ${{\nabla }^{2}}TC$).

$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}^{2}}=\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H*\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}} \right]D\left( t \right) \\ +\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{\theta }_{1}}*{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}} \right]D\left( t \right) +\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,2*s*\left( a-b*p-c*{{p}^{2}} \right)$                 (18)

$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{i}}^{2}} =\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,H*{{\theta }_{1}}\left( \underset{{{t}_{j-1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-{{t}_{j}} \right)}} \right]D\left( t \right)dt+1 \right)\\ -\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,{{W}_{h}}*{{\theta }_{1}}(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}-{{t}_{j-1}} \right)}} \right]D\left( t \right)+\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,2*s\left( a-b*p-c*{{p}^{2}} \right)$                        (19)

$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{i}}\partial {{s}_{i}}}=\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,-2*s\left( a-b*p-c*{{p}^{2}} \right)$                               (20)

5.1 Total cost of supplier

${{T}_{S}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*Ss+Cs*\underset{j=0}{\overset{{{n}_{1}}-1}{\mathop \sum }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}-{{t}_{j}} \right)}}-1 \right]D\left( t \right)+\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( a-b*p-c*{{p}^{2}}~ \right)\left( {{t}_{j}}-{{s}_{j}} \right)dt~$                               (21)

5.2 Numerical illustration

A numerical example to validate our model, using specific parameter values a=1.25, b=0.2, c=18.4, r=60, e=2.7, ${{W}_{h}}$=2, H=4, p=0.01, S=2, ${{s}_{1}}=0$, ${{\theta }_{1}}=0.03$ expressed in their appropriate units. For the solution of Eq. (16) and Eq. (17), Mathematica (version 12) was the computational program that we utilized. 'MATHEMATICA VERSION 12’ provides efficiently handles the calculations and analysis required for the inventory model considering backorders.

5.3 Theorems

Theorem 1: If the following conditions are satisfied:

(i) $\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}^{2}}\ge 0$,

(ii) $\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{j}}^{2}}\ge 0,$

(iii) $\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}^{2}}-\left| \frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}~\partial {{s}_{j}}} \right|\ge 0$ and

(iv) $\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{j}}^{2}}-\left| \frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}~\partial {{s}_{j}}} \right|\ge 0$ for all j= 1, 2, ..., n

Then, ${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)$ will be positive definite. This set of conditions is sufficient to ensure that ${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)$ is at its minimum for a fixed value of n1. The theorem establishes that ${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)$ is indeed positive. Therefore, we can compute the optimal values of tjand sj for a given positive integer n1 using iterative methods and Mathematica software based on Eq. (16) and Eq. (17).

Theorem 2: When considering a convex set S ⊆ Rn, a cost function is deemed convex across S if it satisfies the condition that, for any x1 and x2 belonging to S, and for any λ within the interval [0, 1], the following inequality holds: λf(x1) + (1 − λ) f(x2) ≥ f(λx1 + (1 − λ)x2). Should this inequality always be held as a strict inequality, then the function f is denoted as a strictly convex cost function on S.

Theorem 3: Consider an open convex subset S, which is non-empty, of Rn, and a cost function f: S → R that is twice differentiable on S. In this context, f is convex on S if and only if the Hessian matrix ∇2 f(x) is positive semi-definite for all x in S.

Theorem 4: In the scenario where S is an open convex set in ${{R}^{n}}$ and f: S → R is a cost function that is twice differentiable, if the Hessian matrix ∇2 f(x) is positive definite for all x in S, then f is a strictly convex function on S.

$\nabla T_R\left(t_j, s_j, n_1\right)=\left[\begin{array}{cccccccccc}\frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1 \cdot \partial s_1} & 0 & 0 & \cdots & \cdots & 0 & 0 & 0\\ \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_2 \partial t_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1 \partial t_2} & 0 & \cdots & \cdots & 0 & 0 & 0 \\ 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \cdot \partial s_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \cdot \partial s_2} & \cdots & \cdots & 0 & 0 & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & 0 & \cdots &\cdots & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1-1} \partial s_{n_{1-1}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{1-1}}{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{1-1}} \partial t_{n_1}} \\ 0 & 0 & 0 & 0 & \cdots & \cdots & 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1} \cdot \partial s_{n_{1-1}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1}{ }^2}\end{array}\right]$

6. Sensitivity Analysis

The associated total cost for various resupply cycles, i.e., for n = 1, 2, ... are given in Table 2. From Table 3, Figures 2-5, we notice that for each resupply cycle, the most efficient number of replenishments time for the corresponding minimum total cost gets supplied in appropriate units. The optimal solutions for ${{t}_{i}}$ and ${{s}_{i+1}}$ for n = 4 are given in Tables 4 and 5, Figures 6 and 7 respectively. In Table 6 total cost for retailer, supplier and quantity is given for optimal value.

Table 2. Total cost for the retailer for different replenishment cycle

 ↓a →n 1 2 3 4 5 6 0.81675 28.8679 28.494 28.6444 29.2919 304364 32.078 0.9375 32.5612 31.5384 31.1177 31.2676 31.9882 33.2794 1.089 37.1951 35.3582 34.2208 33.7465 33.9352 34.7868 1.25 42.1195 39.4175 37.5185 36.3808 36.0042 36.3888

Table 3. The optimal solutions for ${{t}_{j}}$ (replenishment time)

 ↓a →tj t0 t1 t2 t3 t4 t5 0.81675 0 3.3374 4 0.9375 0 2.95889 3.53907 4 1.089 0 2.58054 3.13523 3.53925 4 1.25 0 2.20229 2.73147 3.13543 3.5393 4

Table 4. The optimal solutions for ${{s}_{j}}$ (time of shortage)

 ↓a →sj s0 s1 s2 s3 s4 s5 0.81675 0 3.59373 4 0.9375 0 3.19009 3.59601 4 1.089 0 2.78658 3.19216 3.59608 4 1.25 0 2.38314 2.78839 3.19226 3.59613 4

Table 5. Total cost for retailer, supplier and quantity is given for optimal value

 ↓a ${{T}_{R}}$ ${{T}_{S}}$ ${{Q}_{nt}}$ 0.81675 28.494 12.2224 9.40793 0.9375 31.1177 16.7166 8.72184 1.089 33.7465 21.2081 8.02711 1.25 36.0042 25.6457 7.15235

Table 6. Sensitivity analysis of the parameters

 Parameters % Changes Optimal Replenishment Cycle Total Order Quantity ${{Q}_{nt}}$ Total Cost of Retailer ${{T}_{R}}$ Total Cost of Supplier ${{T}_{S}}$ a $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\\end{matrix} \right.$ 6 5 5 4 4 6.53586 7.87023 7.15235 8.29353 7.36848 38.8763 37.6106 36.0042 34.3355 32.2903 30.1608 25.8611 25.6457 21.2881 21.0105 b $\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$ 3 3 3 4 4 8.2967 8.57701 8.85732 7.23527 7.45728 30.1857 30.8002 31.4147 31.9957 32.4866 3.69051 3.81515 3.9398 4.13207 4.25878 c $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\ \end{matrix} \right.$ 5 5 5 5 5 7.16731 7.16731 7.16731 7.16731 7.16731 36.0382 36.0382 36.0382 36.0382 36.0382 5.57456 5.57456 5.57456 5.57456 5.57456 $\theta$ $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\ \end{matrix} \right.$ 5 5 5 5 5 7.53094 7.34484 7.15235 6.95334 6.74774 36.8115 36.4147 36.0042 35.5798 35.1413 5.68158 5.62539 5.56728 5.50721 5.44518 Wh $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\ \end{matrix} \right.$ 5 5 5 5 5 5.87409 6.47731 7.15235 7.8991 8.71747 33.6968 34.7681 36.0042 37.4047 38.9696 5.44791 5.49681 5.56728 5.65929 5.77282 r $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\ \end{matrix} \right.$ 5 5 5 5 5 7.83037 7.50244 7.15235 6.7769 6.37212 37.1705 36.6031 36.0042 35.3707 34.6989 5.23008 5.37882 5.56728 5.80891 6.12359 U $\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} -10 \\ -20 \\\end{matrix} \\ \end{matrix} \right.$ 6 5 5 5 4 6.11979 7.48108 7.15235 6.78255 8.55565 40.4676 38.3417 36.0042 33.7416 31.2096 6.73796 5.85373 5.56728 5.26615 4.7597

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Figure 2. Convexity of total cost for retailer in 2nd replenishment cycle

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Figure 3. Convexity of total cost for retailer in 3rd replenishment cycle

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Figure 4. Convexity of total cost for retailer in 4th replenishment cycle

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Figure 5. Convexity of total cost for retailer in 5th replenishment cycle

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Figure 6. Increasing order of replenishment time tj

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Figure 7. Increasing order of replenishment time sj

7. Analyses of Sensitivity for the Illustration

We will now talk about how the ideal solution responds to variations in the values of various parameters. The comparative study is carried out by altering all of the parameter’s a, b, c, θ, Wh, r, and U by ±20% and ±10%, one at a time, while keeping the other parameters constant. The effect on total cost due to percentage changes in parameters a, b, c, θ, U, r, Wh and all parameters is shown in Figures 8-15. A detailed analysis of the table acknowledges the following perceptions:

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Figure 8. Effect on total cost of retailer and supplier due to parameter ‘a’

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Figure 9. Effect on total cost of retailer and supplier due to parameter ‘b’

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Figure 10. Effect on total cost of retailer and supplier due to parameter ‘c’

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Figure 11. Effect on total cost of retailer and supplier due to parameter ‘θ’

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Figure 12. Effect on total cost of retailer and supplier due to parameter ‘U’

The optimal replenishment cycle, n, is sensitive to varying in most parameters. It is extremely responsive to variations in the parameter ‘a’. While decreasing 'a' by 20%, the optimal replenishment cycle, n, decreases from 5 to 4, shows a 20% decrease. On the other hand, with a 20% increase in 'a', the cycle increases to 6, a 20% increase. From this, we analyze that as 'a' expands or contracts, the optimal replenishment cycle moves in conjunction. Variations in parameters impact total cost and efficiency, providing a deeper understanding of the system's behavior under different scenarios.

Similarly, changes in the parameter 'b' also show a significant impact on the replenishment cycle. An increase of 20% in 'b' retains the cycle at 3, but a decrease of 10% in 'b' moves it to 4, a 33.33% increase. This implies that as 'b' reduces, there is an impulse to have more extended cycles.

The total cost for retailer ${{T}_{R}}$ is sensitive to changes in θ and Wh. For instance, when a 20% decrease in θ shows a decrease in ${{T}_{R}}$ by approximately 3.37%. Meanwhile, a 20% increase in Wh shows an increase in ${{T}_{R}}$ by about 3.78%. These changes indicate the parameter's direct effect on the retailer's total costs.

The total cost for supplier, ${{T}_{S}}$, on the other hand, reacts differently to changes in parameters. An evident observation is with 'a'. A 20% increase in 'a' decreases the ${{T}_{S}}$ by approximately 27.72%.

The total order quantity, ${{Q}_{nt}}$, shows significant changes with parameters 'b', 'θ', 'Wh', and 'U'. For 'b', a 20% increase results in an increase of approximately 7.53% in ${{Q}_{nt}}$. A same pattern seen for 'U'; a 20% increase in 'U' shows a decrease in ${{Q}_{nt}}$ by approximately 14.42%.

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Figure 13. Effect on total cost of retailer and supplier due to parameter ‘r’

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Figure 14. Effect on total cost of retailer and supplier due to parameter ‘Wh

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Figure 15. Effect on total cost of retailer and supplier due to all parameter

Table 7. Total cost of retailer and supplier for linear back order

 Linear Back-Order Condition Replenishment Cycle (n*) ${{Q}_{nt}}$ Order Quantity Time Intervals (Years) TR Total Cost of Retailer TS Total Cost of Supplier tj sj c=0 2 9.42888 0.174335 0.980015, 28.5422 12.2287 3 8.74094 1.10189 1.76513 1.85753 2.42678 2.50215 3.00577 31.1614 16.7223 3.07021 3.52538 3.58222 4.0000 1.1019 1.76504 4 8.04422 1.85749 2.4267 2.50211 3.00571 33.9692 21.2133 3.07018 3.52535 3.5820 4.00000

In the above solution, we considered quadratic back order $\left( a-bp-c{{p}^{2}} \right).$ If we put c=0, then we form a linear backorder case for the model. Table 7 discusses the order quantity, the total cost of retailer and supplier for the linear back order.

8. Conclusion

The optimization of replenishment policies outlined in this article is invaluable for businesses striving to enhance their supply chain management efficiency. By accurately modeling parameters such as price-sensitive demand and backordering, and minimizing total costs within a finite time horizon, companies can make informed decisions that lead to optimized inventory levels, reduced stockouts, and ultimately improved customer satisfaction. This approach provides a systematic framework for strategic planning, enabling businesses to allocate resources effectively, mitigate risks, and maximize profitability in a dynamic and competitive market environment.

In this article, we tackled the optimization problem associated with a replenishment policy, focusing on various parameters that influence the cost and efficiency of the system. Specifically, we considered a scenario where demand is influenced by price, modeled as (a-bp), and assumed complete backordering. Backordering was modeled both as a quadratic function and a linear function, with shortages addressed within a finite time horizon H.

Our model's primary objective was to lower the overall expense related to the replenishment procedure. We learned a lot more about how alterations to parameters like a, b, c, p, U, H, $\theta ,~{{W}_{h}}$ affect the total cost via the results of our study. Firstly, we get optimal value at 2nd cycle then in 3rd, 4th and lastly in 5th highlighting the dynamic nature of the system.

Future research can be done taking multi-item models also discussed the model in finite planning in the future this can be extended for infinite time horizon.

Acknowledgement

We would like to extend our sincere gratitude to the editorial team and reviewers for their insightful comments and constructive feedback, which have greatly improved the quality of this manuscript. We are deeply appreciative of the contributions and collaboration of our co-authors Dr. Nitin Kumar Mishra and Dr. Ranu. Their expertise, dedication, and valuable input were instrumental in the completion of this research.

Nomenclature
 H Fixed time horizon. D The demand rate is D and D(t)= a-bp. r The amount that is carried per unit per order. Or The cost of replenishing or purchasing per order. S The shortage cost per unit time. Ij The total inventory carried out during the interval [tj, sj]. Sj The total amount of shortages in the interval [sj, tj+1]. The time at which the inventory level reaches zero in the jth replenishment cycle j=1, 2, 3, …, n. tj The jth replenishment time j=1, 2, 3, …, n. n The number of orders during the time horizon H. D1 D1=a-bp-cp2 is the price dependent quadratic backordering. Q The total optimal order quantity during the planning horizon H. Ib Instantons shortage during the shortage period. θ An inventory dependent parameter.
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