© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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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 pricesensitive 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.
pricesensitive demand, shortage, linear backorder, quadratic backorder, supply chain, finite planning horizon
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 pricesensitive 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.
In this study, we aim to address the research gap related to pricedependent 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 pricesensitive demand and pricedependent 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.
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 ‘SquareRoot formula’. The first book on inventory management was written by Raymond [3]. Veinott [4] studied that in the reallife situations, the demand rate is dynamic. So, they developed the first dynamic economic order quantity model which is developed by modifying Harris’s SquareRoot 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 multiperiod 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 backordering as a constant function, with shortages. Zangwill [7] proposed a multiperiod model along with shortages and backorders. Fogarty and Aucamp [8] gave the model with shortages and backordering. 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.
Various other models discussed in the literature are compared and contrasted to showcase the advancements made in inventory management research. Çetinkaya and Parlar [10] established a generalized model by taking two different types of backorder costs. Sarkar et al. [11] concerned with optimal inventory replenishment for a degrading item with timequadratic demand and timedependent partial backlogging. The analytical model yields optimum solutions, which are demonstrated numerically. Liao and Shyu [12] gave a model of predefined lot size and demand is assumed to be regularly distributed, with lead time as the variable, the estimated total cost with the backorder is minimized. Pan et al. [13] established an inventory model by taking the lead time & backorder discounts are negotiable in the way that the supplier may take into account the future & present loss & profit. The buyer may be ready to obtain the item as quickly as it can be obtained to ensure production may restart. Bayındır et al. [14] established an EPQ model taking general stock dependent backordering. SanJosé et al. [15] proposed an EOQ model for a single item with partially backlogging, shortages timedependent, partial backordering, the demand rate is backlogged at any instant is a constant fraction with shortages & obtained an optimal policy & less inventory cost. Pan and Hsiao [16] extended the work of literature [5]. Taken an integrated inventory system with shortages and backorder as well as lead time are negotiable. A provider may provide waiting consumers with a backorder cost reduction in the first of two models they described, which had normally distributed demand, and widely dispersed demand in the second. Sazvar et al. [17] established an inventory model for deteriorating goods by taking shortages and complete backordering. Ghasemi and Afshar Nadjafi [18] proposed two models taking holding cost as increasing continuous functions. The first model with no shortages & the second model is with shortages and complete backordering. Kumar et al. [19] proposed an economic policy by taking demand as power depending on time, with shortages and complete backordering. Mishra and Ranu [20] discussed the importance of supplierretailer coordination in managing deteriorating inventory with decreasing demand, addressing a research gap in supply chain literature. It presents a numerical solution and conducts a sensitivity analysis to illustrate the concept further.
Backordering was studied over the decade and still the work is going on. Back ordering is a major problem for the business, organization that’s why researchers readily study backorder taking different types of backordering like linear, nonlinear, exponential, negative exponential, constant function and quadratic function, etc.) Grubbström and Erdem [21] applied algebraic approach to develop the equations for both the EOQ (Economic Order Quantity) and the Economic Production Quantity (EPQ), while taking into account a single backordering cost that is only linear with respect to time. CárdenasBarrón [22] developed an algebraic method to prove the mathematical equations for EOQ and EPQ with a single cost of backordering, only linear (depending on time). Taleizadeh et al. [23] proposed an EOQ model by taking linear holding cost (depend on price), partially backlogged & backorder is a linear function. Taleizadeh et al. [24] proposed two EOQ models (a) by taking holding cost linear dependent on time, partially backlogged, backorders are linear function, lost sale cost as fixed and partially delayed payments. Taleizadeh et al. [25] by taking holding cost linear depends on time, partially backlogged, backorders are linear functions, lost sale cost is fixed & partially prepayments. Yang [26] established an EOQ model by taking nonlinear stock dependent holding cost, partially backlogging, backorders are linear, a lost sale is fixed, the demand rate is stock dependent. By taking different types of backordering singly researchers were not satisfied with the output, so they started taking two types of backorders together, like linear plus fixed, linear, and quadratic Some of the literature surveys are as follows, Unwin [27] firstly took the linear plus fixed backorder and solved by calculus and solved the system of equations &they get the firstorder condition. Sphicas [28] extended the study of literature [21] by taking two parameters combine i.e., Linear & fixed backorder cost for the EOQ & EPQ models. They discussed two conditions first is when fixed backorder cost is high then we can't get any optimal backorder &the second case if the backorder cost is very lesser than there should be optimally some of the backorders. The result reveals that linear backorder cost plays no role. Chung and CárdenasBarrón [29] given the complete solution procedure for the EOQ/EPQ models, and backorder cost is taken as fixed & linear. Most of the models are failed to give an argument & surety of the optimal situation but Chung and Lin [30] given every aspect of the approved solution procedure, we ensure the most effective possible solution. They discussed two cases in their paper for the existence of optimal solution & if the conditions are not satisfied then how to identify the condition by which optimal solution is sure. And derives four theorems & two lemmas for an optimal solution. Mishra and Namwad [31] discussed an inventory model that addresses items with minimal lead time and deterioration, utilizing cubic demand and deterioration functions. It emphasizes the advantages of employing cubic functions for practical applicability, numerical validation, and graphical representation. Additionally, it includes a numerical example and a comprehensive sensitivity analysis. Wee et al. [32] proposed an EOQ model by taking linear holding cost (depend on price), partially backlogged & backorder is linear & fixed function. Sphicas [33] proposed an EOQ model holding cost is linear & dependent on time, completely backlogged, and backorders are fixed &linear. Hu et al. [34] proposed a model of backordering as linear & quadratic function, partially backlogged. Figure 1 is the inventory model diagram. In Table 1, a literature survey is carried out.
Figure 1. Inventory model diagram
Table 1. Survey of existing literature
Ref. 
Demand Type 
Shortages 
Backorder Type 
Finite Planning Horizon 
[2] 
Classical EOQ (SquareRoot) 
 
 
 
[4] 
Dynamic EOQ 
Yes (Dynamic) 
Dynamic 
 
[35] 
Timedependent 
Yes 
 
 
[5] 
 
Lost sales & backorder 
Lost sales & backorder 
 
[6] 
Stochastic 
Optimal with backorders 
Optimal 
 
[7] 
Multiperiod 
Yes 
Yes 
 
[12] 
Normally distributed 
Expected total cost with backorder 
Expected total cost 
 
[16] 
 
Yes 
Linear 
 
[15] 
Timedependent 
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] 
Timedependent holding cost 
Partially backlogged 
Partial 
 
[25] 
Timedependent holding cost 
Partially backlogged 
Partial 
 
[26] 
Stockdependent 
Partially backlogging 
Partial 
 
[36] 
Linear plus fixed 
 
Linear & fixed 
 
[32] 
Linear holding cost (on price) 
Partially backlogged 
Partial 
 
[33] 
Timedependent holding cost 
Completely backlogged 
Complete 
 
[34] 
 
Partially backlogged 
Linear & quadratic 
 
This paper 
Pricesensitive 
Yes 
Complete linear and quadratic 
Yes 
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.
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, …, n_{1}.
$\frac{d{{I}_{j+1}}\left( t \right)}{dt}=D\left( t \right){{\theta }_{1}}{{I}_{j+1}}\left( t \right) \\ {{t}_{j}}<t<{{s}_{j+1}}$ (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)
${{I}_{j+1}}\left( t \right)=\underset{t}{\overset{{{s}_{j+1}}}{\mathop \int }}\,D\left( u \right){{e}^{{{\theta }_{1}}\left( ut \right)}}du$ (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, ${{D}_{1}}=abpc{{p}^{2}}$ is the price dependent quadratic backorder.
${{I}_{b}}\left( t \right)=abpc{{p}^{2}}\left( {{t}_{j}}{{s}_{j}} \right)$ (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, $D\left( t \right)=ab*p$_{.}
Considering the reorganization of the ordering, ${{S}_{j+1}}$ can be given as,
${{S}_{j+1}}=\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,{{I}_{b}}\left( t \right)dt=\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( ab*pc*{{p}^{2}} \right)\left( {{t}_{j}}{{s}_{j}} \right)dt$ (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)
${{Q}_{j+1}}=\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}}1 \right]D\left( t \right) +\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( ab*pc*{{p}^{2}} \right)\left( {{t}_{j}}{{s}_{j}} \right)dt~$ (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 $
${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}}+\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,H\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}t \right)}}1 \right]D\left( t \right)dt\\ +\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,{{W}_{h}}*(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}}1 \right]D\left( t \right) +\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( ab*pc*{{p}^{2}} \right)\left( {{t}_{j}}{{s}_{j}} \right)dt)$
${{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*{{O}_{r}} +H\underset{{{t}_{j1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}t \right)}}1 \right]D\left( t \right)dt \\ +\underset{{{t}_{j}}}{\overset{{{s}_{j+1}}}{\mathop \int }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}t \right)}}1 \right]D\left( t \right)dt+{{W}_{h}}c *(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}{{t}_{j1}} \right)}}1 \right]D\left( t \right)+{{W}_{h}} \\ *(\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}}1 \right]D\left( t \right) +s\left( ab*pc*{{p}^{2}} \right){{\left( {{t}_{j1}}{{s}_{j1}} \right)}^{2}} +s\left( ab*pc*{{p}^{2}} \right){{\left( {{t}_{j}}{{s}_{i}} \right)}^{2}}$ (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)
$\frac{\partial {{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}}=\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,H *\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}}1 \right]D\left( t \right)~\\ \underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}} \right]D\left( t \right) +\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s\left( ab*pc*{{p}^{2}}~ \right)\left( {{t}_{j}}{{s}_{j}} \right)\\ +\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s\left( ab*pc*{{p}^{2}}~ \right)\left( {{t}_{j}}{{s}_{j}} \right)$ (16)
$\frac{\partial {{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{j}}}=\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,H\underset{{{t}_{i1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}{{t}_{j}} \right)}} \right]D\left( t \right)dt~ \\\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}{{t}_{j1}} \right)}} \right]D\left( t \right) \underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s\left( ab*pc*{{p}^{2}} \right)\left( {{t}_{j}}{{s}_{j}} \right)$ (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. ${{\nabla }^{2}}TC$).
$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{j}}^{2}}=\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,H*\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}} \right]D\left( t \right) \\ +\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,{{\theta }_{1}}*{{W}_{h}}*(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}} \right]D\left( t \right) +\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s*\left( ab*pc*{{p}^{2}} \right)$ (18)
$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{s}_{i}}^{2}} =\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,H*{{\theta }_{1}}\left( \underset{{{t}_{j1}}}{\overset{{{s}_{j}}}{\mathop \int }}\,\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}{{t}_{j}} \right)}} \right]D\left( t \right)dt+1 \right)\\ \underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,{{W}_{h}}*{{\theta }_{1}}(\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j}}{{t}_{j1}} \right)}} \right]D\left( t \right)+\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s\left( ab*pc*{{p}^{2}} \right)$ (19)
$\frac{{{\partial }^{2}}{{T}_{R}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)}{\partial {{t}_{i}}\partial {{s}_{i}}}=\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,2*s\left( ab*pc*{{p}^{2}} \right)$ (20)
5.1 Total cost of supplier
${{T}_{S}}\left( {{t}_{j}},{{s}_{j}},{{n}_{1}} \right)={{n}_{1}}*Ss+Cs*\underset{j=0}{\overset{{{n}_{1}}1}{\mathop \sum }}\,\frac{1}{{{\theta }_{1}}}\left[ {{e}^{{{\theta }_{1}}\left( {{s}_{j+1}}{{t}_{j}} \right)}}1 \right]D\left( t \right)+\underset{{{s}_{j}}}{\overset{{{t}_{j}}}{\mathop \int }}\,\left( ab*pc*{{p}^{2}}~ \right)\left( {{t}_{j}}{{s}_{j}} \right)dt~$ (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 n_{1}. 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 t_{j}and s_{j} for a given positive integer n_{1} using iterative methods and Mathematica software based on Eq. (16) and Eq. (17).
Theorem 2: When considering a convex set S ⊆ R^{n}, a cost function is deemed convex across S if it satisfies the condition that, for any x_{1} and x_{2} belonging to S, and for any λ within the interval [0, 1], the following inequality holds: λf(x_{1}) + (1 − λ) f(x_{2}) ≥ f(λx_{1} + (1 − λ)x_{2}). 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 nonempty, of R^{n}, 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 semidefinite 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.
$\nabla T_R\left(t_j, s_j, n_1\right)=\left[\begin{array}{cccccccccc}\frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1 \cdot \partial s_1} & 0 & 0 & \cdots & \cdots & 0 & 0 & 0\\ \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_2 \partial t_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1 \partial t_2} & 0 & \cdots & \cdots & 0 & 0 & 0 \\ 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \cdot \partial s_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \cdot \partial s_2} & \cdots & \cdots & 0 & 0 & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & 0 & \cdots &\cdots & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_11} \partial s_{n_{11}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{11}}{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{11}} \partial t_{n_1}} \\ 0 & 0 & 0 & 0 & \cdots & \cdots & 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1} \cdot \partial s_{n_{11}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1}{ }^2}\end{array}\right]$
The associated total cost for various resupply cycles, i.e., for n = 1, 2, ... are given in Table 2. From Table 3, Figures 25, 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 
→t_{j} 
t_{0} 
t_{1} 
t_{2} 
t_{3} 
t_{4} 
t_{5} 
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 
→s_{j} 
s_{0} 
s_{1} 
s_{2} 
s_{3} 
s_{4} 
s_{5} 
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 
W_{h} 
$\left\{ \begin{matrix} +20 \\ +10 \\ 0 \\ \begin{matrix} 10 \\ 20 \\\end{matrix} \\ \end{matrix} \right.$ 
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 
Figure 2. Convexity of total cost for retailer in 2^{nd }replenishment cycle
Figure 3. Convexity of total cost for retailer in 3^{rd }replenishment cycle
Figure 4. Convexity of total cost for retailer in 4^{th }replenishment cycle
Figure 5. Convexity of total cost for retailer in 5^{th }replenishment cycle
Figure 6. Increasing order of replenishment time t_{j}
Figure 7. Increasing order of replenishment time s_{j}
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, W_{h}, 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, W_{h} and all parameters is shown in Figures 815. A detailed analysis of the table acknowledges the following perceptions:
Figure 8. Effect on total cost of retailer and supplier due to parameter ‘a’
Figure 9. Effect on total cost of retailer and supplier due to parameter ‘b’
Figure 10. Effect on total cost of retailer and supplier due to parameter ‘c’
Figure 11. Effect on total cost of retailer and supplier due to parameter ‘θ’
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%.
Figure 13. Effect on total cost of retailer and supplier due to parameter ‘r’
Figure 14. Effect on total cost of retailer and supplier due to parameter ‘W_{h}’
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 BackOrder Condition 
Replenishment Cycle (n^{*}) 
${{Q}_{nt}}$ Order Quantity 
Time Intervals (Years) 
T_{R} Total Cost of Retailer 
T_{S} Total Cost of Supplier 

t_{j} 
s_{j} 

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 $\left( abpc{{p}^{2}} \right).$ 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.
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 pricesensitive 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 (abp), 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 multiitem models also discussed the model in finite planning in the future this can be extended for infinite time horizon.
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 coauthors Dr. Nitin Kumar Mishra and Dr. Ranu. Their expertise, dedication, and valuable input were instrumental in the completion of this research.
H 
Fixed time horizon. 
D 
The demand rate is D and D(t)= abp. 
r 
The amount that is carried per unit per order. 
O_{r} 
The cost of replenishing or purchasing per order. 
S 
The shortage cost per unit time. 
I_{j} 
The total inventory carried out during the interval [t_{j}, s_{j}]. 
S_{j} 
The total amount of shortages in the interval [s_{j}, t_{j+1}]. 
The time at which the inventory level reaches zero in the j^{th }replenishment cycle j=1, 2, 3, …, n. 

t_{j} 
The j^{th} replenishment time j=1, 2, 3, …, n. 
n 
The number of orders during the time horizon H. 
D_{1} 
D_{1}=abpcp^{2} is the price dependent quadratic backordering. 
Q 
The total optimal order quantity during the planning horizon H. 
I_{b} 
Instantons shortage during the shortage period. 
θ 
An inventory dependent parameter. 
[1] Seliaman, M., CárdenasBarrón, L., Rushd, S. (2020). An algebraic decision support model for inventory coordination in the generalized nstage nonserial supply chain with fixed and linear backorders costs. Symmetry, 12(12): 1998. https://doi.org/10.3390/sym12121998
[2] Harris, F.W. (1990). How many parts to make at once. Operations Research, 38(6): 947950. https://doi.org/10.1287/opre.38.6.947
[3] Raymond, F.E. (1931). Quantity and Economy in Manufacture. McGrawHill.
[4] Veinott, A.F. (1964). Review of Hadley and whitin. Journal of the American Statistical Association, 59(305): 283285. https://doi.org/10.2307/2282878
[5] Ouyang, L.Y., Yeh, N.C., Wu, K.S. (1996). Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47: 829832. https://doi.org/10.1057/jors.1996.102
[6] Scarf, H., Arrow, K., Karlin, S., Suppes, P. (1960). The optimality of (S, s) policies in the dynamic inventory problem. In Optimal Pricing, Inflation, and the Cost of Price Adjustment, The MIT Press.
[7] Zangwill, W.I. (1969). A backlogging model and a multiechelon model of a dynamic economic LoT size production system—A network approach. Management Science, 15(9): 506527. https://doi.org/10.1287/mnsc.15.9.506
[8] Fogarty, D.W., Aucamp, D.C. (1985). Implied backorder costs. IIE Transactions, 17(1): 105107. https://doi.org/10.1080/07408178508975279
[9] Aardal, K., Jonsson, Ö., Jönsson, H. (1989). Optimal inventory policies with servicelevel constraints. Journal of the Operational Research Society, 40: 6573. https://doi.org/10.1057/jors.1989.6
[10] Çetinkaya, S., Parlar, M. (1998). Nonlinear programming analysis to estimate implicit inventory backorder costs. Journal of Optimization Theory and Applications, 97: 7192. https://doi.org/10.1023/a:1022623016607
[11] Sarkar, T., Ghosh, S.K., Chaudhuri, K. (2012). An optimal inventory replenishment policy for a deteriorating item with timequadratic demand and timedependent partial backlogging with shortages in all cycles. Applied Mathematics and Computation, 218(18): 91479155. https://doi.org/10.1016/j.amc.2012.02.072
[12] Liao, C.J., Shyu, C.H. (1991). An analytical determination of lead time with normal demand. International Journal of Operations & Production Management, 11(9): 7278. https://doi.org/10.1108/eum0000000001287
[13] Pan, J.C.H., Lo, M.C., Hsiao, Y.C. (2004). Optimal reorder point inventory models with variable lead time and backorder discount considerations. European Journal of Operational Research, 158(2): 488505. https://doi.org/10.1016/s03772217(03)003667
[14] Bayındır, Z.P., Birbil, Ş.İ., Frenk, J.B.G. (2007). A deterministic inventory/production model with general inventory cost rate function and piecewise linear concave production costs. European Journal of Operational Research, 179(1): 114123. https://doi.org/10.1016/j.ejor.2006.03.026
[15] SanJosé, L.A., Sicilia, J., GarcíaLaguna, J. (2014). Optimal lot size for a production–inventory system with partial backlogging and mixture of dispatching policies. International Journal of Production Economics, 155: 194203. https://doi.org/10.1016/j.ijpe.2013.08.017
[16] Pan, J.C.H., Hsiao, Y.C. (2005). Integrated inventory models with controllable lead time and backorder discount considerations. International Journal of Production Economics, 93: 387397. https://doi.org/10.1016/j.ijpe.2004.06.035
[17] Sazvar, Z., Baboli, A., Akbari Jokar, M.R. (2013). A replenishment policy for perishable products with nonlinear holding cost under stochastic supply lead time. The International Journal of Advanced Manufacturing Technology, 64: 10871098. https://doi.org/10.1007/s0017001240422
[18] Ghasemi, N., Afshar Nadjafi, B. (2013). EOQ models with varying holding cost. Journal of Industrial Mathematics, 2013: 743921. https://doi.org/10.1155/2013/743921
[19] Kumar, S., Kumar, M., Sahni, M. (2021). Multiproduct economic inventory policy with time varying power demand, shortages and complete backordering. Universal Journal of Accounting and Finance, 9: 98104. https://doi.org/10.13189/ujaf.2021.090110
[20] Mishra, N.K., Ranu, R. (2023). Single supplierretailer inventory model for deteriorating items with linear decreasing demand. AIP Conference Proceedings, 2800(1): 020291. https://doi.org/10.1063/5.0162939
[21] Grubbström, R.W., Erdem, A. (1999). The EOQ with backlogging derived without derivatives. International Journal of Production Economics, 59(13): 529530. https://doi.org/10.1016/s09255273(98)000152
[22] CárdenasBarrón, L.E. (2001). The economic production quantity (EPQ) with shortage derived algebraically. International Journal of Production Economics, 70(3): 289292. https://doi.org/10.1016/s09255273(00)000682
[23] Taleizadeh, A.A., Pentico, D.W., Aryanezhad, M., Ghoreyshi, S.M. (2012). An economic order quantity model with partial backordering and a special sale price. European Journal of Operational Research, 221(3): 571583. https://doi.org/10.1016/j.ejor.2012.03.032
[24] Taleizadeh, A.A., Pentico, D.W., Jabalameli, M.S., Aryanezhad, M. (2013). An EOQ model with partial delayed payment and partial backordering. Omega, 41(2): 354368. https://doi.org/10.1016/j.omega.2012.03.008
[25] Taleizadeh, A.A., Pentico, D.W., Jabalameli, M.S., Aryanezhad, M. (2013). An economic order quantity model with multiple partial prepayments and partial backordering. Mathematical and Computer Modelling, 57(34): 311323. https://doi.org/10.1016/j.mcm.2012.07.002
[26] Yang, C. T. (2014). An inventory model with both stockdependent demand rate and stockdependent holding cost rate. International Journal of Production Economics, 155: 214221. https://doi.org/10.1016/j.ijpe.2014.01.016
[27] Unwin, A.R. (2017). Forecasting and time series analysis. Journal of the Operational Research Society, 29(6): 618. https://doi.org/10.2307/3009830
[28] Sphicas, G.P. (2006). EOQ and EPQ with linear and fixed backorder costs: Two cases identified and models analyzed without calculus. International Journal of Production Economics, 100(1): 5964. https://doi.org/10.1016/j.ijpe.2004.10.013
[29] Chung, K.J., CárdenasBarrón, L.E. (2012). The complete solution procedure for the EOQ and EPQ inventory models with linear and fixed backorder costs. Mathematical and Computer Modelling, 55(1112): 21512156. https://doi.org/10.1016/j.mcm.2011.12.051
[30] Chung, K.J., Lin, S.D. (2009). The convexities of the total annual cost functions of the inventory models with linear and fixed backorder costs. Journal of Information and Optimization Sciences, 30(5): 961967. https://doi.org/10.1080/02522667.2009.10699921
[31] Mishra, N.K., Namwad, R.S. (2023). An economic ordering policy for interest earned on sales till the permissible period without paying interest for the items kept in advance stock. AIP Conference Proceedings, 2800(1): 020301. https://doi.org/10.1063/5.0163138
[32] Wee, H.M., Huang, Y.D., Wang, W.T., Cheng, Y.L. (2014). An EPQ model with partial backorders considering two backordering costs. Applied Mathematics and Computation, 232: 898907. https://doi.org/10.1016/j.amc.2014.01.106
[33] Sphicas, G.P. (2014). Generalized EOQ formula using a new parameter: Coefficient of backorder attractiveness. International Journal of Production Economics, 155: 143147. https://doi.org/10.1016/j.ijpe.2013.09.014
[34] Hu, W.T., Kim, S.L., Banerjee, A. (2009). An inventory model with partial backordering and unit backorder cost linearly increasing with the waiting time. European Journal of Operational Research, 197(2): 581587. https://doi.org/10.1016/j.ejor.2008.06.041
[35] Silver, E.A. (1973). A heuristic for selecting lot size quantities for the case of a deterministic timevarying demand rate and discrete opportunities for replenishment. Production and Inventory Management: Journal of the American Production and Inventory Control Society,14: 6474.
[36] Montgomery, D.C., Johnson, L.A., Gardiner, J.S. (1990). Forecasting and Time Series Analysis. McGrawHill, New York.