An inventory model for delayed deteriorating items with power demand considering shortages and lost sales

Abstract

This article considers an EOQ model for a delayed deteriorating item in which the demand varies with time and follows a power pattern. Shortages are allowed with partially backlogged and lost sales. We develop a mathematical model for the problem.The proposed model aims at minimizing the total inventory cost which depends on the length of time with positive and negative inventory level. Numerical examples with the effect of various changes in some possible parameters combination of the model are given to illustrate the effectiveness of the model and to gain some managerial decision.

Keywords

Deterioration delayed power demand lost sales shortages

1. Introduction

Deterioration is one of the terms that cannot be overlooked in inventory management.In fact,it is a known fact that almost all items deteriorates over time.Many times, the rate at which the items deteriorate is so insignificant that there is little need to consider the deterioration in resolving the inventory models.However,in the real life,there are many items that are subject to a noticeable rate of deterioration. Some items begin to deteriorate immediately throughout the planning horizon such as alcohol and gasoline and some items like dry fruits and vegetables have a shelf life and begin to deteriorate after a time lag. Hence the impart of item deterioration cannot be ignored in the inventory decision-making.

Deterioration can therefore be defined as damage, decay, obsolescence and lost of value in a product as time progresses. Many researchers over decades have studied inventory models for deteriorating items. In Whitin [39], the author studied fashionable items which deteriorates over a certain period of time. Later, Ghare & Schrader [26] developed a model for an inventory system with constant demand and exponential deterioration rate. Also, Mishra [29] proposed a production lot-size inventory model for items with both time-varying and constant deterioration rate. Whereas Aggarwal [32] studied an order-level inventory model by modifying and correcting the average inventory holding cost proposed earlier by Shah and Jaiswal [44]. A detailed review of deteriorating items were carried out by Dave [40], Raafat [11], Goyal and Giri [31], Bakker et al. [23], Janssen et al. [18]. Recently, Tiwari et al. [36] developed a deteriorating supply chain model for items with optimal pricing and lot-sizing.The demand is assumed to be dependent on selling price and displayed of stock level under restricted storage capacity.Also in this direction, Bhunia et al. [2] presented a single deteriorating inventory model for items with variable demand rate that depends on stock level and marketing decisions. Later Sunil et al. [30] proposed a deteriorating inventory model for items with expiration date.The model considered joint pricing and partial two-level credit facility with partially backlogging shortages.The total profit was maximized by determining optimal selling price, replenishment cycle time and inventory time to get to zero.

Other deteriorating inventory models that incorporate other factors like delay in payments, preservation technology and so forth, includes among others: Jaggi et al. [8] who presented a two-warehouse deteriorating inventory model for items with imperfect quality and one level of trade credit policy. Later, a deteriorating inventory model for items with controllable deteriorating rate in which demand is dependent on price and stock under preservation technology with shortages was proposed by Umakanta et al. [42]. Shaikh et al. [1] also investigated a fuzzy deteriorating inventory model for items with selling price and advertisement dependent demand under permissible delay in payment and partially backlogging shortages. More also, a retailers replenishment and price policies deteriorating inventory model for items under price dependent demand, trade credit facility and preservation technology investment was presented by Mishra et al. [41].

In EOQ inventory models, it is commonly assumed that the demand of good is constant, unfortunately this may not be feasible in some situation. It will be more preferable to considered that the demand varies with time.The study of inventory model with time-varying demand is important because it allow to appropriately modeling the behavior and evolution of the inventory according to Sicilia et al. [24] who developed an inventory model for deteriorating items with partial backlogging and time-varying demand process. Several research works have been carried out in this direction.Thus, Pal and Datta [38] investigated an order level inventory system that has power demand pattern with variable rate of deterioration. Hariga [19] proposed a deterministic inventory model for deteriorating items with shortages and linearly time-varying demand. Also, Benkheroof and Mahmoud [16] developed an inventory model with increasing time-varying demand for constant deteriorating items over a finite planning horizon with shortages allowed. Later, Giri et al. [4] proposed an EOQ model for items that deteriorate with time varying demand and cost. Whereas Chen [27] developed an inventory model in which he considered the impact of time-varying demand and production rate. In addition, Dye et al. [9] studied an inventory model for deteriorating items with time-varying demand and shortages partially backlogged. Also, Rajeswari and Vanjikkodi [25] analyzed an inventory model with constant deterioration that follows power demand pattern with shortages partially backlogged. Rajoria et al. [43] developed a deterministic inventory model for deteriorating items with power demand under inflation with partial backlogging. Also, Mishra and Singh [33] proposed an EOQ model with quadratic deterioration rate and power demand. Later, Singh et al. [37] analyzed an optimal inventory model for deteriorating items with time-proportional and constant deterioration rate, considering demand rate as both constant and linear function of time and no shortage. On the other hand, Sanni and Chigbu [34] analyzed an optimal replenishment policy for items with Weibull distribution deterioration rate and demand depending on stock level with partial backlogging. Benkherouf et al. [17] proposed an inventory model for finite horizon problem considering product substitution and demand varying with time. Afterward, Chen et al. [5] investigated an optimal and replenishment inventory model for deteriorating items in which the demand rate depend on stock displayed, time-varying and price.

Many researchers assumed that the deterioration starts as soon as the retailer receives the commodity. However, this is not always true since some commodities will have a period of time to retain the freshness of their original quality. During this period, no deterioration takes place. Wu [14] was the first to introduce and analyzed an optimal replenishment model for non instantaneous deteriorating items under stock dependent demand with partial backlogging. Later, Uthayakumar and Geetha [28] presented an inventory model for non-instantaneous deteriorating items with partial backlogging under inflation and time discounting. Also, Musa and Sanni [3] analyzed an inventory model for delayed deteriorating items under permissible delay in payment.In addition, Palaniel and Uthayakumar [21] investigated an EOQ model for non-instantaneous items that has Weibull distribution deterioration rate with power demand. On the other hand, Sharmila and Uthayakumar [10] considered a model for non-instantaneous deteriorating items with power demand and permissible delay in payment with shortages. Whereas Jaggi et al. [6] investigated an inventory model for non-instantaneous item that deteriorates under inflation with partial backlogged. Furthermore, Sunil et al. [35] considered a two-warehouse inventory model for non-instantaneous deteriorating items under inflation and permissible delay in payment. Shortages are permitted and partially backlogged. A year later Jaggi et al. [7] provided a two-warehouse inventory model for non-instantaneous deteriorating items considering different dispatching policy with partially backlogged shortages. Also, Soni and Suther [13] developed a non-instantaneous deteriorating inventory model for items with price and promotional efforts that has stochastic dependent demand. Deterioration rate is proportional to time and shortages are permitted and partially backlogged. In like manner, Li et al. [12] studied a non-instantaneous inventory model for deteriorating item incorporating pricing, replenishment and preservation technology investment with partially backlogged shortage.

Most of the works mentioned above to the knowledge of the authors did not considered delayed deteriorating items simultaneously with lost sale that are partially backlogged.Sales are lost when customers are not willing to come back when their demand are not met as at the time they placed an order. In the developmental stages of many businesses, they experiences a situation where their demand increases with time.It is therefore important to consider power demand pattern that will be suitable for such an inventory system.

In this works, an inventory model for delayed deteriorating items is considered simultaneously with lost sales that are partially backlogged.Demand follows a power pattern to take cares of how demands occurs during the cycle. The outline of this paper is given as follows. Section 2 deals with the notation and assumptions used for the model whereas in Section 3, the mathematical model for the inventory system is developed. In Section 4, we provide a solution procedure for the model. Numerical examples and sensitivity analysis is carried out in Section 5. We end the paper by providing conclusion as well as possible future work that could be carried out.

2. Assumptions and notation

The following assumptions were used:

The inventory system involves a single item.

Deterioration takes place after the life span of the items.

There are no replenishment or repair taking place for any deteriorating items.

The replenishment takes place at infinite rate with zero lead time.

The demand rate, B (t), at any time t is $B (t) = \frac{d t^{\frac{1}{n} - 1}}{{nT}^{\frac{1}{n} - 1}}$ , where d is the average demand whereas n is the pattern index with 0< n < ∞.

Shortages are allowed with the backlogging rate depending on the length of the waiting time for the next replenishment. The negative inventory of the backlogging rate is given by $G (t) = \frac{1}{1 + γ (T - t)}$ , where γ is the backlogging parameter and the waiting period is given by (T - t). The remaining fraction 1 - G (t) is consider as lost sales.

Notation used in the model are given as follows.

A is the ordering cost.

α is the deteriorating rate where (0 < α < 1).

K₁ is the holding cost per unit per year.

K₂ is the deteriorating cost per unit per year.

K₃ is the shortage cost for backlogged items per unit per year.

K₄ is the cost of lost sale per unit per year.

T is the cycle length.

t_d is the length of time when the items experience no deterioration.

t₁ is the length of time when the inventory has no shortage.

Q is the quantity order during a cycle of length T.

S is the inventory level during [0, T].

P is the back-ordered unit during the stock out period.

I₁ (t) is the level of positive inventory at time t, where 0 ≤ t ≤ t_d, when no deterioration occurs.

I₂ (t) is the level of positive inventory at time t, where t_d ≤ t ≤ t₁, when deterioration occurs.

I₃ (t) is the level of negative inventory at time t, where t_d ≤ t ≤ T.

ψ (t₁, T) is the total cost per unit per time.

3. Mathematical formulation

The inventory system for the model is given in Fig. 1. Initially, a lot size of Q unit enters the system at the beginning of each cycle, where Q = P + S. The deterioration takes place after time t_d and reaches zero inventory level at time t₁. The shortage occurs in the interval [t₁, T] and there are partially backlogged and lost sales at the end of cycle time.

Fig.1

Graphical Representation of Inventory Model

The inventory system in Fig. 1 can be described by the following differential equations given by:

$\frac{d I_{1} (t)}{dt} = - \frac{{dt}^{\frac{1}{n} - 1}}{{nT}^{\frac{1}{n} - 1}}, 0 \leq t \leq t_{d}$ (1) $\frac{d I_{2} (t)}{dt} + α I_{2} (t) = - \frac{{dt}^{\frac{1}{n} - 1}}{{nT}^{\frac{1}{n} - 1}}, t_{d} \leq t \leq t_{1}$ (2) $\frac{d I_{3} (t)}{dt} = \frac{- B (t)}{1 + γ (T - t)}, t_{1} \leq t \leq T$ (3) with the boundary conditions given by I₁ (0) = S, I₂ (t₁) = 0 and I₃ (t₁) =0. Upon solving the differential Equations in (1), (2) and (3), we then obtain $I_{1} (t) = S - \frac{d}{T^{\frac{1}{n} - 1}} [t^{\frac{1}{n}}] 0 \leq t \leq t_{d} .$ (4) From equation (2), we obtain $\begin{matrix} I_{2} (t) e^{α t} & = - \int \frac{d t^{\frac{1}{n} - 1}}{n T^{\frac{1}{n} - 1}} e^{α t} d t \\ = - \frac{d}{n T^{\frac{1}{n} - 1}} \int^{} e^{α t} t^{\frac{1}{n} - 1} d t . \end{matrix}$ We only consider the first three terms of the exponential function by neglecting the higher terms since they tend to become very small. It follows that $\begin{matrix} I_{2} (t) e^{α t} & = - \frac{d}{T^{\frac{1}{n} - 1}} [\int (1 + α t + \frac{α^{2} t^{2}}{2!}) t^{\frac{1}{n} - 1} dt] \\ = - \frac{d}{T^{\frac{1}{n} - 1}} [\int t^{\frac{1}{n} - 1} dt + α \int t^{\frac{1}{n}} dt + \\ \frac{α^{2}}{2} \int t^{\frac{1}{n} + 1} dt] \\ = - \frac{d}{T^{\frac{1}{n} - 1}} [t^{\frac{1}{n}} + \frac{α t^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t^{\frac{1}{n} + 2}}{2 (2 n + 1)}] + C . \end{matrix}$ Upon setting the boundary condition I₂ (t₁) =0, we then obtain $\begin{matrix} C & = \frac{d}{T^{\frac{1}{n} - 1}} [t_{1}^{\frac{1}{n}} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \\ I_{2} (t) & = \frac{d e^{- α t}}{T^{\frac{1}{n} - 1}} [(t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}}) + \frac{α}{n + 1} (t_{1}^{\frac{1}{n} + 1} - t^{\frac{1}{n} + 1}) \\ + \frac{α^{2}}{2 (2 n + 2)} (t_{1}^{\frac{1}{n} + 2} - t^{\frac{1}{n} + 2})] . \end{matrix}$ Expanding further and considering the first three terms, we then obtain $\begin{matrix} I_{2} (t) & = \frac{d}{T^{\frac{1}{n} - 1}} [t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}} + \frac{n α t^{\frac{1}{n} + 1}}{n + 1} - α t t_{1}^{\frac{1}{n}} \\ + \frac{α^{2} t^{2} t_{1}^{\frac{1}{n}}}{2} - \frac{α^{2} t^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{α^{2} t^{\frac{1}{n} + 2}}{n + 1} - \frac{α^{2} t t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} \\ - \frac{α^{2} t^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \end{matrix}$ (5) From Fig. 1, we have I₁ (t_d) = I₂ (t_d) at time t_d which gives $\begin{matrix} S & = \frac{d}{T^{\frac{1}{n} - 1}} [t_{1}^{\frac{1}{n}} + \frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \end{matrix}$ (6) Substituting equation (6) into equation (4) yields $\begin{matrix} I_{1} (t) & = \frac{d}{T^{\frac{1}{n} - 1}} [t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}} + \frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} \\ + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \end{matrix}$ (7)

During the shortage interval [t₁, T], the demand at time t is partially backlogged. Thus, the solution to equation (3) with boundary condition I₃ (t₁) =0 is given by $\begin{matrix} I_{3} (t) & = \frac{d}{T^{\frac{1}{n} - 1}} [(t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}}) (1 - γ T) \\ + \frac{γ}{n + 1} (t_{1}^{\frac{1}{n} + 1} - t^{\frac{1}{n} + 1})] \end{matrix}$ (8)

The maximum back ordered inventory, P, is obtained when t = T. Then, from equation (8) we obtain $\begin{array}{l} P = - \frac{d}{T^{\frac{1}{n} - 1}} [(t_{1}^{\frac{1}{n}} - T^{\frac{1}{n}}) (1 - γ T) \\ + \frac{γ}{n + 1} (t_{1}^{\frac{1}{n} + 1} - T^{\frac{1}{n} + 1})] \end{array}$ (9)

Finally, from Fig. 1, we have $\begin{matrix} Q & = \frac{d}{T^{\frac{1}{n} - 1}} [t_{1}^{\frac{1}{n}} - α t_{d} t^{\frac{1}{n}} + \frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1}] - \frac{d}{T^{\frac{1}{n} - 1}} [(t_{1}^{\frac{1}{n}} \\ - T^{\frac{1}{n}}) (1 - γ T) + \frac{γ}{n + 1} (t_{1}^{\frac{1}{n} + 1} - T^{\frac{1}{n} + 1})] \end{matrix}$ (10)

Total relevant inventory cost per cycle consists of the following cost components:

The ordering cost is A.

The inventory holding cost is given by $\begin{matrix} HC & = K_{1} [\int_{0}^{t_{d}} I_{1} (t) dt + \int_{t_{d}}^{t_{1}} I_{2} (t) dt] \\ = \frac{K_{1} d}{T^{\frac{1}{n} - 1}} [\int_{0}^{t_{d}} (t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}} + \frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} \\ - α t_{d} t_{1}^{\frac{1}{n}} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} \\ + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}) dt] \\ + \frac{K_{1} d}{T^{\frac{1}{n} - 1}} [\int_{t_{d}}^{t_{1}} (t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}} + \frac{n α t^{\frac{1}{n} + 1}}{n + 1} \\ - α t t_{1}^{\frac{1}{n}} + \frac{α^{2} t^{2} t_{1}^{\frac{1}{n}}}{2} - \frac{α^{2} t^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{α^{2} t^{\frac{1}{n} + 2}}{n + 1} - \frac{α^{2} t t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} \\ - \frac{α^{2} t^{\frac{1}{n} + 2}}{2 (2 n + 1)}) dt] \end{matrix}$ Upon simplification, we obtain $\begin{matrix} HC & = \frac{K_{1} d}{T^{\frac{1}{n} - 1}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n}}}{3} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ + \frac{t_{1}^{\frac{1}{n} + 1} (2 - α^{2} t_{d}^{2})}{2 (n + 1)} + \frac{n α t_{d}^{\frac{1}{n} + 2}}{2 n + 1} + \\ + \frac{α t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} + \frac{α^{2} t_{1}^{\frac{1}{n} + 3}}{6 (3 n + 1)} \\ - \frac{α^{2} n^{2} t_{d}^{\frac{1}{n} + 3}}{(n + 1) (3 n + 1)}] \end{matrix}$ (11)

The deterioration cost is $\begin{matrix} DC & = K_{2} [I_{2} (t_{d}) - \int_{t_{d}}^{t_{1}} B (t) dt] \\ DC & = \frac{d}{T^{\frac{1}{n} - 1}} [\frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \end{matrix}$ (12)

Shortage cost per cycle as a result of backlog is given by $\begin{matrix} SC & = K_{3} [\int_{t_{1}}^{T} - I_{3} (t) dt] \\ = \frac{- K_{3} d}{T^{\frac{1}{n} - 1}} [\int_{t_{1}}^{T} ((t_{1}^{\frac{1}{n}} - t^{\frac{1}{n}}) (1 - γ T) \\ + \frac{γ}{n + 1} (t_{1}^{\frac{1}{n} + 1} - t^{\frac{1}{n} + 1})) dt] \\ = \frac{- K_{3} d}{T^{\frac{1}{n} - 1}} [T t_{1}^{\frac{1}{n}} (1 - γ T) - \frac{t_{1}^{\frac{1}{n} + 1}}{n + 1} (1 - 2 γ T) \\ - \frac{γ t_{1}^{\frac{1}{n} + 2}}{2 n + 1} + \frac{n T^{\frac{1}{n} + 1}}{n + 1} + \frac{2 n^{2} γ T^{\frac{1}{n} + 2}}{(2 n + 1) (n + 1)}] \end{matrix}$ (13)

The lost sale cost during interval [0. T] is given by $\begin{array}{l} L C = K_{4} [\int_{t_{1}}^{T} {1 - \frac{B (t) d t}{1 + γ (T - t)}}] \\ L C = \frac{K_{4} d}{T^{\frac{1}{n} - 1}} [\frac{n γ T^{\frac{1}{n} + 1}}{n + 1} + \frac{γ t_{1}^{\frac{1}{n} + 1}}{n + 1} - γ T t_{1}^{\frac{1}{n}}] \end{array}$ (14)

Finally, the total relevant inventory cost per unit time is given by $ψ (t_{1}, T) = \frac{1}{T}$ (ordering cost + holding cost + deteriorating cost + shortage cost and lost sale cost) or simplifies as below $\begin{matrix} ψ (t_{1}, T) & = \frac{A}{T} + \frac{K_{1} d}{T^{\frac{1}{n}}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n}}}{3} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ + \frac{t_{1}^{\frac{1}{n} + 1} (2 - α^{2} t_{d}^{2})}{2 (n + 1)} + \frac{n α t_{d}^{\frac{1}{n} + 2}}{2 n + 1} \\ + \frac{α t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} + \frac{α^{2} t_{1}^{\frac{1}{n} + 3}}{6 (3 n + 1)} \\ - \frac{α^{2} n^{2} t_{d}^{\frac{1}{n} + 3}}{(n + 1) (3 n + 1)}] + \frac{K_{2} d}{T^{\frac{1}{n}}} \\ [\frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] \\ + \frac{- K_{3} d}{T^{\frac{1}{n}}} [T t_{1}^{\frac{1}{n}} (1 - γ T) - \frac{t_{1}^{\frac{1}{n} + 1}}{n + 1} (1 - \\ 2 γ T) - \frac{γ t_{1}^{\frac{1}{n} + 2}}{2 n + 1} + \frac{n T^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{2 n^{2} γ T^{\frac{1}{n} + 2}}{(2 n + 1) (n + 1)}] + \frac{K_{4} d}{T^{\frac{1}{n}}} [\frac{n γ T^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{γ t_{1}^{\frac{1}{n} + 1}}{n + 1} - γ T t_{1}^{\frac{1}{n}}] \end{matrix}$ (15) Therefore, we are interested in finding the values of t₁:and:T that minimizes the function ψ (t₁, T) given in Equation (15) in the feasible region F (t₁, T) :0 ≤ t₁ ≤ T, T > 0.

4. Solution procedure

We consider the partial derivatives of ψ (t₁, T) with respect to the decision variable t₁ and T such that $\frac{\partial ψ (t_{1}, T)}{\partial t_{1}} = 0 and \frac{\partial ψ (t_{1}, T)}{\partial T} = 0,$ (16)

provided that $[\frac{\partial^{2} ψ (t_{1}, T)}{\partial t_{1}^{2}}] [\frac{\partial^{2} ψ (t_{1}, T)}{\partial T^{2}}] - {[\frac{\partial^{2} ψ (t_{1}, T)}{\partial t \partial T}]}^{2} > 0$ (17)

From equation (16) and (17), we obtain $\begin{matrix} \frac{\partial ψ (t_{1}, T)}{\partial t_{1}} & = \frac{K_{1} d}{T^{\frac{1}{n}}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n} - 1}}{3 n} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n} - 1}}{2 n} \\ + \frac{t_{1}^{\frac{1}{n}} (2 - α^{2} t_{d}^{2})}{2 n} + \frac{α t_{1}^{\frac{1}{n} + 1}}{2 n} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{6 n}] \\ + \frac{K_{2} d}{T^{\frac{1}{n}}} [\frac{- α t_{d} t_{1}^{\frac{1}{n} - 1}}{n} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n} - 1}}{2 n} + \frac{α t_{1}^{\frac{1}{n}}}{n} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n}}}{n} + \frac{α^{2} t_{1}^{\frac{1}{n} + 1}}{2 n}] - \frac{K_{3} d}{T^{\frac{1}{n}}} [\frac{T t_{1}^{\frac{1}{n} - 1}}{n} \\ (1 - γ T) - \frac{t_{1}^{\frac{1}{n}}}{n} (1 - 2 γ T) - \frac{γ t_{1}^{\frac{1}{n} + 1}}{n}] \\ + \frac{K_{4} d}{T^{\frac{1}{n}}} [\frac{γ t_{1}^{\frac{1}{n}}}{n} - \frac{γ T t_{1}^{\frac{1}{n} - 1}}{n}] = 0 \end{matrix}$ (18) and $\begin{matrix} \frac{\partial ψ (t_{1}, T)}{\partial T} & = - \frac{A}{T^{2}} - \frac{K_{1} d}{{nT}^{\frac{1}{n} + 1}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n}}}{3} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ + \frac{t_{1}^{\frac{1}{n} + 1} (2 - α^{2} t_{d}^{2})}{2 (n + 1)} + \frac{n α t_{d}^{\frac{1}{n} + 2}}{2 n + 1} \\ + \frac{α t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} + \frac{α^{2} t_{1}^{\frac{1}{n} + 3}}{6 (3 n + 1)} \\ [1 pt] & - \frac{α^{2} n^{2} t_{d}^{\frac{1}{n} + 3}}{(n + 1) (3 n + 1)}] - \frac{K_{2} d}{n T^{\frac{1}{n} + 1}} \\ [\frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] + \frac{(1 - n) K_{3} d t_{1}^{\frac{1}{n}}}{n T^{\frac{1}{n}}} \\ + \frac{(2 - \frac{1}{n}) K_{3} d γ t_{1}^{\frac{1}{n}}}{T^{\frac{1}{n} - 1}} - \frac{K_{3} d t_{1}^{\frac{1}{n} + 1}}{n (n + 1) T^{\frac{1}{n} + 1}} \\ + \frac{K_{3} dn}{n + 1} + \frac{(1 - n) 2 K_{3} d γ t_{1}^{\frac{1}{n} + 1}}{(n^{2} + n) T^{\frac{1}{n}}} \\ - \frac{K_{3} d γ t_{1}^{\frac{1}{n} + 2}}{(2 n^{2} + n) T^{\frac{1}{n} + 1}} - \frac{4 n^{2} γ T K_{3} d}{(2 n^{2} + 3 n + 1)} \\ + \frac{K_{4} dn γ}{n + 1} - \frac{K_{4} d γ t_{1}^{\frac{1}{n} + 1}}{(n^{2} + n) T^{\frac{1}{n} + 1}} \\ + \frac{(1 - n) K_{4} d γ t_{1}^{\frac{1}{n}}}{n T^{\frac{1}{n}}} = 0 \end{matrix}$ (19) The second partial derivatives of the cost function ψ(t₁, T) and the Hessian at the point (t₁, T) is presented in the appendix. To obtain an optimal value that minimizes the total cost per unit time, we solve equations (18) and (19) simultaneously. We can observe that the equations are nonlinear. Here, we use excel solver to find the optimum solution and check for convexity using equation (17). We can observe from our model that, if t_d = 0 and γ = 0, it then becomes an inventory model with instantaneous deterioration and complete backlogging where the total relevant cost per unit time is given by $\begin{array}{l} ψ (t_{1}, T) & = \frac{A}{T} + \frac{K_{1} d}{T^{\frac{1}{n}}} [\frac{t_{1}^{\frac{1}{n} + 1} (2 - α^{2} t_{d}^{2})}{2 (n + 1)} \\ + \frac{α t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} + \frac{α^{2} t_{1}^{\frac{1}{n} + 3}}{6 (3 n + 1)}] \\ + \frac{K_{2} d}{T^{\frac{1}{n}}} [\frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1}] \\ - \frac{K_{3} d}{T^{\frac{1}{n}}} [T t_{1}^{\frac{1}{n}} - \frac{t_{1}^{\frac{1}{n} + 1}}{n + 1} - \frac{n T^{\frac{1}{n} + 1}}{n + 1}] \end{array}$ (20)

This is similar to the one obtained in Sicilia et al. [24] although in our case we did not consider the purchasing cost.

5. Numerical examples

We illustrate the proposed model with some numerical examples as given below.

Example 1 as found in [24] A = 50, d = 100, K₁ = 2, K₂ = 4, K₃ = 12, K₄ = 10, n = 0.5, α = 0.1, t_d = 0.4, γ = 0.2 in appropriate units. The following results were obtained where t₁ = 0.3847 years and T = 0.4396 years. From (16), ψ (t₁, T) =161.7829, Q = 43.9107 units, S = 33.6628 units, P = 10.24791 units. From (18), the Hessian is given by H (t₁, T) =11040994 and is positive which implies that (t₁, T) is the minimum point.

Example 2. Upon repeating the same Example 1 with n = 3, the following results were obtained where T = 0.9690 years, t₁ = 0.8306 years, S = 92.4478 units, ψ (t₁, T) =97.9974, P = 4.7828 units, Q = 97.2306 units. From (18), the Hessian is H (t₁, T) =1259413.774 (positive) which implies that (t₁, T) is the minimum point.

5.1. Sensitivity analysis

Table 1 shows the effect of changes of some model parameters on the decision variables based on the first example. A careful study of the results obtained in the table within the specified range of values of the selected parameters indicates the following observations:

Table 1
Sensitivity Analysis of the Parameters in the Inventory model

P *	V *	C *	Change in
P *	V *	C *	T	t ₁	ψ (t₁, T)	S	P	Q
d	120	+20	0.4025	0.3532	177.2935	37.2167	11.0510	48.2677
110	+10	0.4196	0.3677	169.7146	35.4536	10.6553	46.1090
100	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
90	-10	0.4636	0.4050	153.435	31.8399	9.8269	41.6668
80	-20	0.4928	0.4297	144.5877	29.9802	9.3904	39.3706
A	60	+20	0.4824	0.4209	177.1182	36.7254	11.4455	28.1706
55	+10	0.4612	0.4030	169.6344	35.2080	10.8524	46.0601
50	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
45	-10	0.4175	0.3660	153.5158	32.0857	9.6299	41.7156
40	-20	0.3948	0.3467	144.7732	30.4713	8.9959	39.4672
K ₁	2.4	+20	0.4134	0.3538	173.4363	30.2922	11.0032	41.2954
2.2	+10	0.4255	0.3681	167.8253	31.8641	10.6309	42.4950
2.0	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
1.8	-10	0.4566	0.4040	155.2507	35.7506	9.8546	45.6053
1.6	-20	0.4771	0.4270	148.1566	38.2175	9.4520	47.6694
K ₂	4.8	+20	0.4399	0.3850	161.7425	33.6965	10.2432	43.9397
4.4	+10	0.4398	0.3849	161.7625	33.6798	10.2455	43.9254
4.0	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
3.6	-10	0.4395	0.3845	161.8038	33.6455	10.2503	43.8958
3.2	-20	0.4393	0.3844	161.8251	33.6277	10.2528	43.8805
K ₃	14.4	+2 0	0.4301	0.3834	164.5753	34.1857	8.7852	42.9709
13.2	+10	0.4345	0.3840	163.2752	33.9434	9.4604	43.4038
12	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
10.8	-10	0.4458	0.3855	160.0530	33.3343	11.1781	44.5124
9.6	-20	0.4532	0.3864	158.0237	32.9443	12.2937	45.2380
K ₄	12	+20	0.4378	0.3845	162.3090	33.7621	9.9683	43.7304
11	+10	0.4387	0.3846	162.0492	33.7131	10.1062	43.8193
10	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
9	-10	0.4406	0.3848	161.5098	33.6112	10.3937	44.0049
8	-20	0.4416	0.3849	161.2297	33.5581	10.5438	44.1020
n	0.6	+20	0.4791	0.4181	154.5227	38.1816	9.6680	47.8496
0.55	+10	0.4594	0.4014	157.9807	35.9456	9.9342	45.8798
0.50	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
0.45	-10	0.4201	0.3681	165.9415	31.3378	10.6306	41.9684
0.40	-20	0.4011	0.3521	170.4411	28.9706	11.1148	40.0853
t _d	0.48	+20	0.4516	0.3972	160.7676	35.0046	10.1675	45.1721
0.44	+10	0.4451	0.3905	161.2168	34.2805	10.1934	44.4739
0.4	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
0.36	-10	0.4353	0.3798	162.456	33.1484	10.3298	43.4783
0.32	-20	0.4320	0.3759	163.2244	32.7324	10.4373	43.1696
α	0.12	+20	0.4400	0.3851	161.7348	33.7033	10.2433	43.9456
0.11	+10	0.4398	0.3849	161.7586	33.6833	10.2451	43.9284
0.10	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
0.09	-10	0.4395	0.3845	161.8079	33.6418	10.2508	43.8926
0.08	-20	0.4393	0.3843	161.8335	33.6203	10.2538	43.8740
γ	0.24	+20	0.4379	0.3845	162.2799	33.7565	9.9755	43.7321
0.22	+10	0.4388	0.3846	162.0341	33.71022	10.1100	43.8202
0.20	0	0.4396	0.3847	161.7829	33.6628	10.2479	43.9107
0.18	-10	0.4405	0.3848	161.5262	33.6143	10.3894	44.0037
0.16	-20	0.4414	0.3849	161.2636	33.5646	10.5346	44.0992

Note: P*=Parameters, V*=Values, C*=% Changes

As the demand rate d increases, the quantity order Q also increases leading to the increase in the total cost ψ (t₁, T). In addition, T and t₁ decreases. The economic implication of this is that as the demand rate gets higher, the stock takes less amount of time to complete and so T and t₁ decreases. An increase in the demand rate leads to an increase in order quantity and the inventory total cost.

As the ordering cost A increases, T, t₁, Q, ψ (t₁, T) also increases. The economic implication of this is that, it is advisable to order higher quantity when the ordering cost is high in order to prevent too many order and cost.

T, Q, t₁ decreases with the increase in the holding cost, K₁. It is observed that there is an increase in the inventory total cost as the holding cost increases. The economic implication of this is that as the holding cost increases, it is better to reduce the cycle length and order quantity to keep the cost of inventory as low aspossible.

As the deteriorating cost, K₂, increases, there is an increase in T, t₁, Q which leads to slight decrease in the inventory total cost. This implies that when the deteriorating cost is higher, there is a need to order more quantity and increase the cycle period so as to take the opportunity of reduced deteriorating cost.

As K₁ and K₂ increases, there is a decrease in T, t₁ and Q which leads to an increase in the inventory total cost ψ (t₁, T).

It is observed that as the backlogging rate γ increases, the parameters T, t₁ and Q reduces while there is a slight increase in the total inventory cost.

An increase in the parameters n and t_d leads to an increase in T, t₁ and Q while the total inventory cost reduces.

6. Conclusion

In this article, we present an EOQ inventory model for delayed deteriorating items with power demand upon considering shortages and lost sales. It extends the similar model carried out by Sicilia et al. [24]. We incorporate delayed deterioration and lost sale. The import of demand rate, constant rate of deterioration and partial backlogging rate of order quantity and total inventory cost per unit time are reported. Numerical examples were presented to illustrate the application of the model developed. From the above observations, the effect of demand rate, deteriorating rate and backlogging rate on optimal replenishment policy can not be easily neglected. The proposed model can be used in inventory control of some delayed deteriorating items such as food items,vegetables, milks, fish,meat etc.

This model in the future can be extended to economic production model considering variable deterioration with power demand pattern and shortages allow. The other area of interest may be to consider the effect of other factors likeadvertisement, inflation, preservation technology, product substitution etc.on the demand of the product.

Conflicts of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Footnotes

Acknowledgments

The authors are thankful to the associate editor and the three anonymous reviewers for their valuable, constructive and detailed suggestions.This research work is supported by the University of Malaya for providing enabling environment and also with grant number FRGS (FP015 - 2015A).

Appendix A

The second partial derivatives of the cost function ψ (t₁, T) is given by (21) $\begin{matrix} \frac{\partial^{2} ψ (t_{1}, T)}{\partial t_{1}^{2}} & = \frac{K_{1} d}{T^{\frac{1}{n}}} [\frac{(1 - n) α^{2} t_{d}^{3} t_{1}^{\frac{1}{n} - 2}}{3 n^{2}} \\ - \frac{(1 - n) α t_{d}^{2} t_{1}^{\frac{1}{n} - 2}}{2 n^{2}} + \frac{t_{1}^{\frac{1}{n} - 1} (2 - α^{2} t_{d}^{2})}{2 n^{2}} \\ + \frac{(1 + n) α t_{1}^{\frac{1}{n}}}{2 n^{2}} + \frac{(2 n + 1) α^{2} t_{1}^{\frac{1}{n} + 1}}{6 n^{2}}] \\ + \frac{K_{2} d}{T^{\frac{1}{n}}} [\frac{(n - 1) α t_{d} t_{1}^{\frac{1}{n} - 2}}{n^{2}} \\ + \frac{(1 - n) α^{2} t_{d}^{2} t_{1}^{\frac{1}{n} - 2}}{2 n^{2}} + \frac{α t_{1}^{\frac{1}{n} - 1}}{n^{2}} \\ - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} - 1}}{n^{2}} + \frac{(n + 1) α^{2} t_{1}^{\frac{1}{n}}}{2 n^{2}}] \\ - \frac{K_{3} d}{T^{\frac{1}{n}}} [\frac{(1 - n) (1 - γ T) {Tt}_{1}^{\frac{1}{n} - 2}}{n^{2}} \\ - \frac{(1 - 2 γ T) t_{1}^{\frac{1}{n} - 1}}{n^{2}} - \frac{(n + 1) γ t_{1}^{\frac{1}{n}}}{n^{2}}] \\ + \frac{K_{4} d}{T^{\frac{1}{n}}} [\frac{γ t_{1}^{\frac{1}{n} - 1}}{n^{2}} - \frac{(1 - n) γ T t_{1}^{\frac{1}{n} - 2}}{n^{2}}] \end{matrix}$ (22) $\begin{matrix} \frac{\partial^{2} ψ (t_{1}, T)}{\partial T^{2}} & = \frac{2 A}{T^{3}} + \frac{K_{1} d (n + 1)}{n^{2} T^{\frac{1}{n} + 2}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n}}}{3} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} \\ + \frac{t_{1}^{\frac{1}{n} + 1} (2 - α^{2} t_{d}^{2})}{2 (n + 1)} + \frac{n α t_{d}^{\frac{1}{n} + 2}}{2 n + 1} + \frac{α t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} \\ + \frac{α^{2} t_{1}^{\frac{1}{n} + 3}}{6 (3 n + 1)} - \frac{α^{2} n^{2} t_{d}^{\frac{1}{n} + 3}}{(n + 1) (3 n + 1)}] \\ + \frac{K_{2} d (n + 1)}{n^{2} T^{\frac{1}{n} + 2}} [\frac{n α t_{d}^{\frac{1}{n} + 1}}{n + 1} - α t_{d} t_{1}^{\frac{1}{n}} \\ + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n}}}{2} - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2} + \frac{α t_{1}^{\frac{1}{n} + 1}}{n + 1} \\ + \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{n + 1} - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n} + 1}}{n + 1} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{2 (2 n + 1)} \\ - \frac{α^{2} t_{d}^{\frac{1}{n} + 2}}{2 (2 n + 1)}] - \frac{(1 - n) K_{3} d t_{1}^{\frac{1}{n}}}{n^{2} T^{\frac{1}{n} + 1}} \\ + \frac{(2 n^{2} - 3 n + 1) γ K_{3} d t_{1}^{\frac{1}{n}}}{n^{2} T^{\frac{1}{n}}} + \frac{K_{3} d t_{1}^{\frac{1}{n} + 1}}{n^{2} T^{\frac{1}{n} + 2}} \\ + \frac{(n - 1) 2 γ K_{3} d t_{1}^{\frac{1}{n} + 1}}{n^{2} (n + 1) T^{\frac{1}{n} + 1}} + \frac{(n + 1) γ K_{3} d t_{1}^{\frac{1}{n} + 2}}{n^{2} (2 n + 1) T^{\frac{1}{n} + 2}} \\ - \frac{4 n^{2} γ K_{3} d}{(2 n^{2} + 3 n + 1)} + \frac{K_{4} d γ t_{1}^{\frac{1}{n} + 1}}{n^{2} T^{\frac{1}{n} + 2}} \end{matrix}$ (23) $\begin{matrix} \frac{\partial^{2} ψ (t_{1}, T)}{\partial t \partial T} & = \frac{- K_{1} d}{{nT}^{\frac{1}{n} + 1}} [\frac{α^{2} t_{d}^{3} t_{1}^{\frac{1}{n} - 1}}{3 n} - \frac{α t_{d}^{2} t_{1}^{\frac{1}{n} - 1}}{2 n} \\ + \frac{t_{1}^{\frac{1}{n}} (2 - α^{2} t_{d}^{2})}{2 n} + \frac{α t_{1}^{\frac{1}{n} + 1}}{2 n} + \frac{α^{2} t_{1}^{\frac{1}{n} + 2}}{6 n}] \\ - \frac{K_{2} d}{{nT}^{\frac{1}{n} + 1}} [\frac{- α t_{d} t_{1}^{\frac{1}{n}} - 1}{n} + \frac{α^{2} t_{d}^{2} t_{1}^{\frac{1}{n} - 1}}{2 n} \\ + \frac{α t_{1}^{\frac{1}{n}}}{n} - \frac{α^{2} t_{d} t_{1}^{\frac{1}{n}}}{n} + \frac{α^{2} t_{1}^{\frac{1}{n} + 1}}{2 n}] \\ - \frac{(n - 1) K_{3} d t_{1}^{\frac{1}{n} - 1}}{n^{2} T^{\frac{1}{n}}} + \frac{(2 n - 1) K_{3} γ d t_{1}^{\frac{1}{n} - 1}}{n^{2} T^{\frac{1}{n} - 1}} \\ - \frac{K_{3} d t_{1}^{\frac{1}{n}}}{n^{2} T^{\frac{1}{n} + 1}} - \frac{γ K_{3} d t_{1}^{\frac{1}{n} + 1}}{n^{2} T^{\frac{1}{n} + 1}} \\ - \frac{(n - 1) 2 γ K_{3} d t_{1}^{\frac{1}{n}}}{n^{2} T^{\frac{1}{n}}} - \frac{K_{4} d γ t_{1}^{\frac{1}{n}}}{n^{2} T^{\frac{1}{n} + 1}} \\ - \frac{(n - 1) K_{4} γ d t_{1}^{\frac{1}{n} - 1}}{n^{2} T^{\frac{1}{n}}} \end{matrix}$

References

A.A.

Shaikh ,

A.K.

Bhunia ,

L.E.

Cárdenas-Barrón ,

Sahoo and

Tiwari , A fuzzy inventory model for a deteriorating item with variable demand, permissible delay in payments and partial backlogging with shortage follows inventory (SFI) policy, International Journal of Fuzzy Systems 20(5) (2018), 1606–1623.

A.K.

Bhunia ,

A.A.

Shaikh ,

Dhaka ,

Pareek and

L.E.

Cardenas-Barron , An application of genetic algorithm and PSO in an inventory model for a single deteriorating item with variable demand dependent on marketing strategy and displayed stock level, Scientia Iranica 25(3) (2018), 1641–1655.

Musa and

Sani , Inventory ordering policies of delayed deteriorating items under permissible delay in payments, International Journal of Production Economics 136(1) (2012), 75–83.

B.C.

Giri ,

Goswami and

K.S.

Chaudhuri , An EOQ model for deteriorating items with time varying demand and costs, Journal of the Operational Research Society 47(11) (1996), 1398–1405.

Liuxin ,

Xian ,

M.F.

Keblis and

Gen , Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent demand, Computers & Industrial Engineering (2018).

Jaggi ,

Tiwari and

Goel , Replenishment policy for non-instantaneous deteriorating items in a two storage facilities under inflationary conditions, International Journal of Industrial Engineering Computations 7(3) (2016), 489–506.

C.K.

Jaggi ,

S.K.

Goel and

Tiwari , Two-warehouse inventory model for non-instantaneous deteriorating items under different dispatch policies, Investigación Operacional 38(4) (2018), 343–365.

C.K.

Jaggi ,

L.E.

Cárdenas-Barrón ,

Tiwari and

A.A.

Shafi , Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments, Scientia Iranica Transaction E, Industrial Engineering 24(1) (2017), 390.

C.Y.

Dye ,

H.J.

Chang and

J.T.

Teng , A deteriorating inventory model with time-varying demand and shortagedependent partial backlogging, European Journal of Operational Research 172(2) (2006), 417–429.

10.

Sharmila and

Uthayakumar , Non-instantaneous deteriorating items with power demand rate and shortages under permissible delay in payments, Communications in Applied Analysis 20(4) (2016).

11.

Raafat , Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society 42(1) (1991), 27–37.

12.

Li ,

He ,

Zhou and

Wu , Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items, Omega 84 (2018), 1–13.

13.

H.N.

Soni and

D.N.

Suthar , Pricing and inventory decisions for non-instantaneous deteriorating items with price and promotional effort stochastic demand, Journal of Control and Decision (2018), 1–25.

14.

K.S.

Wu , An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging, Production Planning & Control 12(8) (2001), 787–793.

15.

K.S.

Wu ,

L.Y.

Ouyang and

C.T.

Yang , An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging, International Journal of Production Economics 101(2) (2006), 369–384.

16.

Benkherouf and

M.G.

Mahmoud , On an inventory model for deteriorating items with increasing time-varying demand and shortages, Journal of the Operational Research Society 47(1) (1996), 188–200.

17.

Benkherouf ,

Skouri and

Konstantaras , Inventory decisions for a finite horizon problem with product substitution options and time varying demand, Applied Mathematical Modelling 51 (2017), 669–685.

18.

Janssen ,

Claus and

Sauer , Literature review of deteriorating inventory models by key topics from 2012 to 2015, International Journal of Production Economics 182 (2016), 86–112.

19.

Hariga , An EOQ model for deteriorating items with shortages and time-varying demand, Journal of the Operational research Society 46(3) (1995), 398–404.

20.

Omar and

Yeo , A model for a production?repair system under a time-varying demand process, International Journal of Production Economics 119(1) (2009), 17–23.

21.

Palanivel and

Uthayakumar , An EOQ model for non-instantaneous deteriorating items with power demand, time dependent holding cost, partial backlogging and permissible delay in payments, World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 8(8) (2014), 1127–1137.

22.

Palanivel and

Uthayakumar , Two-warehouse inventory model for non-instantaneous deteriorating items with partial backlogging and inflation over a finite time horizon, Opsearch 53(2) (2016), 278–302.

23.

Bakker ,

Riezebos and

R.H.

Teunter , Review of inventory systems with deterioration since 2001, European Journal of Operational Research 221(2) (2012), 275–284.

24.

Sicilia ,

González-De-la-Rosa ,

Febles-Acosta and

Alcaide-López-de-Pablo , An inventory model for deteriorating items with shortages and time-varying demand, International Journal of Production Economics 155 (2014), 155–162.

25.

Rajeswari and

Vanjikkodi , Deterioration inventory model with power demand and partial backlogging, International Journal of Mathematical Archive (IJMA) ISSN 2229-5046 2(9) (2011).

26.

P.M.

Ghare and

G.F.

Schrader , A model for exponentially decaying inventory, Journal of industrial Engineering 14(5) (1963), 238–243.

27.

P.S.

Chen , The impact of time-varying demand and production rates on determining inventory policy, Mathematical Methods of Operations Research 54(3) (2001), 395–405.

28.

Uthayakumar and

K.V.

Geetha , A replenishment policy for non-instantaneous deteriorating inventory system with partial backlogging, Tamsui Oxford Journal of Mathematical Sciences 25(3) (2009), 313–332.

29.

R.B.

Misra , Optimum production lot size model for a system with deteriorating inventory, The International Journal of Production Research 13(5) (1975), 495–505.

30.

Tiwari ,

L.E.

Cárdenas-Barrón ,

Goh and

A.A.

Shaikh , Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two level partial trade credits in supply chain, International Journal of Production Economics 200 (2018), 16–36.

31.

S.K.

Goyal and

B.C.

Giri , Recent trends in modeling of deteriorating inventory, European Journal of Operational Research 134(1) (2001), 1–16.

32.

S.P.

Aggarwal , Note on: An order level lot size inventory model for deteriorating items by YK Shah, AIIE Transactions 11(4) (1979), 344–346.

33.

S.S.

Mishra and

P.K.

Singh , Partial backlogging EOQ model for queued customers with power demand and quadratic deterioration: Computational approach, American Journal of Operational Research 3(2) (2013), 13–27.

34.

S.S.

Sanni and

P.E.

Chigbu , Optimal replenishment policy for items with three-parameter weibull distribution deterioration, stock-level dependent Demand and partial backlogging, Yugoslav Journal of Operations Research 27(2) (2017).

35.

Tiwari ,

L.E.

Cárdenas-Barrón ,

Khanna and

C.K.

Jaggi , Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics 176 (2016), 154–169.

36.

Tiwari ,

C.K.

Jaggi ,

Gupta and

L.E.

Cárdenas-Barrón , Optimal pricing and lot-sizing policy for supply chain system with deteriorating items under limited storage capacity, International Journal of Production Economics 200 (2018), 278–290.

37.

Singh ,

P.J.

Mishra and

Pattanayak , An optimal policy for deteriorating items with time-proportional deterioration rate and constant and time-dependent linear demand rate, Journal of Industrial Engineering International 13(4) (2017), 455–463.

38.

T.K.

Datta and

A.K.

Pal , Order level inventory system with power demand pattern for items with variable rate of deterioration, Indian Journal of Pure and Applied Mathematics 19(11) (1988), 1043–1053.

39.

T.M.

Whitin , Theory of inventory management, Princeton University Press, 1957.

40.

Dave , Survey of literature on continuously deteriorating inventory models? A rejoinder, Journal of the Operational Research Society 42(8) (1991), 725–725.

41.

Mishra ,

Tijerina-Aguilera ,

Tiwari and

L.E.

Cárdenas-Barrón , Retailer's joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments, Mathematical Problems in Engineering (2018).

42.

Mishra ,

L.E.

Cárdenas-Barrón ,

Tiwari ,

A.A.

Shaikh and

Treviño-Garza , An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of operations research 254(1-2) (2017), 165–190.

43.

Y.K.

Rajoria ,

Saini and

S.R.

Singh , EOQ model for decaying items with power demand, partial backlogging and inflation, International Journal of Applied Engineering Research 10(9) (2015), 22861–22873.

44.

Y.K.

Shah and

M.C.

Jaiswal , An order-level inventory model for a system with constant rate of deterioration, Opsearch 14(3) (1977), 174–184.

An inventory model for delayed deteriorating items with power demand considering shortages and lost sales

Abstract

Keywords

1. Introduction

2. Assumptions and notation

3. Mathematical formulation

5.1. Sensitivity analysis

Table 1 Sensitivity Analysis of the Parameters in the Inventory model

Conflicts of interest

Footnotes

Acknowledgments

Appendix A

References

Table 1
Sensitivity Analysis of the Parameters in the Inventory model