Optimal production cycle time for inventory model with linear time dependent exponential distributed deterioration

Abstract

We study an inventory system with linear time varying exponential distributed deterioration in which production and demand rates are constant. The mathematical model is developed to obtain the total cost per unit time of an inventory system. The inventory controlling systems in terms of first order differential equations are solved numerically. The optimal cycle time is derived and the results are applied to numerical problems. The effect of changes in the model parameters on decision variables and the average total cost of an inventory system are studied through numerical examples.

Keywords

Inventory-production model exponential distribution linear time dependent deterioration constant demand performance measure

1 Introduction

Items deterioration is a matter of prime importance in inventory theory. The problem of inventory models with deteriorating items has received a great attention in the recent years, as shown by surveys of [6 , 11]. In general, almost all items deteriorate over time. The impact of product deterioration should not be neglected in the decision process of production lot size. Deterioration can be classified as age-dependent on-going deterioration (e.g., blood, fish and strawberries) and age-independent on-going deterioration (e.g., alcohol and gasoline, radioactive chemicals and grain products). Most researchers assumed a constant rate of deterioration in deteriorating inventory. But, the time-varying deteriorating item is more realistic than the constant deteriorating item.

An exponentially deteriorating inventory model was proposed initially by [5]. Covert and Philip [3] used a variable deterioration rate of two parameter Weibull distribution to formulate an economic order quantity model. Choi and Hwang [2] developed a model with deteriorating items for the production rate determination to minimize the total cost function over a finite planning period. Raafat [10] extended the model of [9] where the finished product is subject to a constant rate of decay following the works of [1, 13]. Yang and Wee [19] considered a multi-lot-size production-inventory system with constant production and demand rates for deteriorating items. Tadj et al. [17] formulated the problem of optimal control of an inventory system with ameliorating and deteriorating items. El-Gohary and El-Syed [4] formulated the optimal control model of a multi-items inventory system with different types of deteriorations. This type of inventory control problems have also been studied by [8 , 18]. Most of the past works on inventory models did not consider simultaneously the inventory productionsystem with linear time dependent exponential distributed deteriorating items.

The problem of an inventory system with an exponential distributed deteriorating item, in which production and demand rates are constant, is considered here. The probability density function of an exponential distribution is given by $f (t) = θ exp {- θ t}, t \geq 0,$ where the θ > 0 rate or inverse scale parameter. The probability distribution function is $F (t) = 1 - exp {- θ t}, t \geq 0 .$

The deterioration rate θ (t) as a function of time on-hand inventory is given by $θ (t) = \frac{f (t)}{1 - F (t)} = θ, t \geq 0, θ > 0 .$

The novelty we take into consideration is that a deterioration rate is linear time varying followed by an exponential distribution. We are concerned with an inventory system for a single product subject to a one-parameter exponential deterioration item to determine a production cycle time in this study. The product is provided with price discount for its deteriorating rate. The objective of this paper is to develop a mathematical model for obtaining an optimal purchase quantity for a linear time dependent deterioration rate associated with exponential distribution during the cycle time. To minimize the total cost per unit time of an inventory-production system is also of interest. The optimal cycle time of the model is derived here. The effect of changes in the model parameters on decision variables and total cost of an inventory-production system is studied through numerical examples.

This paper is organized as follows. In Section 2 basic assumptions and notations are employed for the development of the inventory model. In Section 3 we develop a mathematical model for an optimal purchase quantity and the solution of this model in the inventory-production system. Illustrative numerical examples and final conclusions of the results are presented in the subsequent sections.

2 Basic model notations and assumptions

The model is based on the following notations:

p: units of the production rate per unit time.

d: actual demand of the product per unit time.

θ: a deterioration rate of exponential distribution.

θt: linear time dependent exponential distributed deterioration rate.

A: set up cost per order.

h > 0: inventory holding cost coefficient per unit time.

I (t): inventory level of the product.

k: production cost per unit time.

r: price discount per unit cost.

T: optimal cycle time.

T₁: production period.

T₂: time during which there is no production of the product i.e., T₁ = T - T₂ .

I₁ (t): inventory level for product during the production period, i.e., 0 ≤ t ≤ T₁ .

I₂ (t): inventory level for product during the period when there is no production, i.e., T₁ ≤ t ≤ T₂ .

I (0): initial inventory.

The following are the assumptions applied in the proposed model:

A finite planning horizon is assumed.

The production unit of the product is available and it can meet the demand.

The inventory system deals with a single item.

The demand rate for the product is known and constant.

Shortage is not allowed.

Once the production is terminated, the product starts deteriorating and the price discount is considered.

The linear time dependent deterioration of the product in inventory follows the exponentialdistribution.

3 Development of the model and solutions

The state level of inventory I (t) amplifies due to a production rate and depletes due to a demand rate and a deterioration rate θ of a linear time varying exponential distribution. The dynamics of the state equation of an inventory level I (t) of the product at time t over period [0, T₁] is governed by the differential equation $\begin{matrix} {dI}_{1} (t) & = & [p - d] dt, 0 \leq t \leq T_{1} \end{matrix}$ (3.1) $\begin{matrix} {dI}_{2} (t) & = & [- θ {tI}_{2} (t) - d] dt, 0 \leq t \leq T_{2} \end{matrix}$ (3.2)with boundary conditions I₁ (0) =0 and I₂ (T₂) =0.

The solution of differential Equation (3.1) using the boundary equation I₁ (0) =0 is $I_{1} (t) = (p - d) t, for all t \in [0, T_{1}],$ (3.3)

In order to solve the solution of differential Equation (3.2), we estimate the integrating factor of Equation (3.2) is v (t) = exp [∫θtdt] = exp [θt² /2] and the corresponding solution is v (t) I₂ (t) = ∫v (t) (- d) dt = - d∫ exp [θt² /2] dt . However, using the integrating factor and Taylor expansion until the second order and ignoring the higher orders of e^θt² /2, we have, $\begin{matrix} exp [θ t^{2} / 2] I (t) = - d \int e^{θ t^{2} / 2} dt \\ = - d [\int (1 + θ t^{2} / 2) dt] = - d [t + \frac{θ t^{3}}{6}] + c . \end{matrix}$

Using the boundary condition I₂ (T₂) = 0, t = T₂, so we have, $\begin{matrix} 0 = - d [T_{2} + \frac{θ T_{2}^{3}}{6}] + c, c = d [T_{2} + \frac{θ T_{2}^{3}}{6}] . \end{matrix}$

Substituting the value of c, we obtain $\begin{matrix} exp [θ t^{2} / 2] I_{2} (t) = - d [t + \frac{θ t^{3}}{6}] + d [T_{2} + \frac{θ T_{2}^{3}}{6}], \end{matrix}$ from which we then have the solution of (3.2); i.e.: $\begin{matrix} I_{2} (t) = d exp [- θ t^{2} / 2] [T_{2} + \frac{θ T_{2}^{3}}{6} - t - \frac{θ t^{3}}{6}], \\ for all 0 \leq t \leq T_{2} \end{matrix}$ (3.4)

Thus, the order quantity is given by $Q = I_{2} (0) = d [T_{2} + \frac{θ T_{2}^{3}}{6}] .$ (3.5)

To remove the difficulty of exponential terms, taking Tailor’s series expansion of the exponential function e^θT, and neglecting those terms of degree greater than or equal to 2 in θ, we have the inventory holding cost in the system during production cost per unit time; i.e.: $\begin{matrix} IHC = \frac{h}{T} [\int_{0}^{T_{1}} I_{1} (t) dt + \int_{0}^{T_{1}} I_{2} (t) dt] \\ = \frac{h}{T} [\int_{0}^{T_{1}} (p - d) tdt + \int_{0}^{T_{2}} {dexp}^{θ t^{2} / 2} \\ (T_{2} + \frac{θ T_{2}^{3}}{6} - t - \frac{θ t^{3}}{6}) dt] \\ = \frac{h}{2 T} [(p - d) T_{1}^{2} + ({dT}_{2}^{2} + \frac{d θ T_{2}^{4}}{6})] \end{matrix}$ (3.6)

The number of units that deteriorated; D (T) during one cycle time is the difference between the maximum inventory (order quantity) and the number of units used to meet the demand. So, from Equation (3.5) the deterioration cost per unit time is given by $\begin{matrix} DC = \frac{k}{T} [I_{2} (0) - \int_{0}^{T_{2}} ddt] \\ = \frac{k}{T} [d (T_{2} + \frac{θ T_{2}^{3}}{6}) - {dT}_{2}] = \frac{kd θ T_{2}^{3}}{6 T} . \end{matrix}$ (3.7)

Price discount is offered as a fraction of production cost for the units in the period [0,T₂] $PD = \frac{kr}{T} \int_{0}^{T_{2}} ddt = \frac{{krdT}_{2}}{T}$ (3.8)

The production cost per unit time is given by $PC = \frac{{pkT}_{1}}{T}$ (3.9)

The set up cost per unit time is given by $SC = \frac{A}{T} .$ (3.10)

The average total cost per unit time is given by $\begin{matrix} K (T) & = & [IHC + DC + PD + PC + SC] \\ = & \frac{h}{2 T} [(p - d) T_{1}^{2} + ({dT}_{2}^{2} + \frac{d θ T_{2}^{4}}{6})] \\ + \frac{k}{T} (\frac{d θ T_{2}^{3}}{6} + {rdT}_{2} + {pT}_{1}) + \frac{A}{T} \end{matrix}$ (3.11)

To minimize the total cost per unit time K (T) to express in terms of T in Equation (3.11) so that there is only one variable T in the equation. At the moment when the production run is terminated during a cycle: $I_{1} (T_{1}) = I_{2} (0) = (p - d) T_{1} = d [T_{2} + \frac{θ T_{2}^{3}}{6}] .$

Neglecting higher terms and approximation $T_{2} = \frac{p - d}{d} T_{1} .$ (3.12)

Again, $T = T_{1} + T_{2} = T_{1} + \frac{p - d}{d} T_{1},$ from which $T = \frac{p}{d} T_{1}, and T_{1} = \frac{d}{p} T .$ (3.13)

Now, by using Equations (3.12) and (3.13) to express T₂ and T₁ in terms of T in Equation (3.11), we have $\begin{matrix} K (T) = \frac{h (p - d) dT}{2 p^{2}} \\ [d + (p - d) (1 + \frac{θ (p - d)^{2} T^{2}}{6 p^{2}})] \\ + \frac{kd (p - d)}{p} {\frac{θ (p - d)^{2} T^{2}}{6 p^{2}} + r} \\ + kd + \frac{A}{T} \end{matrix}$ (3.14)

As K (T) is a convex function of T, hence, there exists a unique value of T (say T^*) that minimizes the total cost per unit time K (T). The necessary condition for total cost K (T) to be minimum is $\frac{\partial K (T)}{\partial T} = 0$ and $\frac{\partial^{2} K (T)}{\partial T^{2}} > 0 for all T > 0 .$ So, T^* can be found by solving the equation $\frac{\partial K}{\partial T} = 0$ , i.e., T^* satisfies the following equation: $\begin{matrix} - \frac{A}{T^{2}} + \frac{h}{2 p^{2}} (p - d) d^{2} \\ + \frac{hd {(p - d)}^{2}}{2 p^{2}} (1 + \frac{θ {(p - d)}^{2} T^{2}}{2 p^{2}}) \\ + \frac{kd θ {(p - d)}^{3} T}{3 p^{3}} = 0, \end{matrix}$ (3.15)and $\begin{matrix} \frac{\partial^{2} K (T)}{\partial T^{2}} = \frac{2 A}{T^{3}} + \frac{hd θ {(p - d)}^{4} T}{2 p^{4}} \\ + \frac{kd θ {(p - d)}^{3} T}{3 p^{3}} > 0, for all T > 0 \end{matrix}$ i.e., the second derivative is found to be positive.

When the optimal cycle time, T^* is obtained, the optimal order quantity, Q^* can be obtained as follows: $Q^{*} = d [T^{*} + \frac{θ T^{* 3}}{6}] .$ (3.16)

4 Numerical solutions

Consider an inventory system with A = $2000/setup, p = 100 units/unit time, d = 30 units/unit time, k = $40/unit time, h = 3 units/unit time, θ= 0.08 and r = 2 units/unit time. In order to get the value of the optimal cycle time, T^* satisfying the Equation (3.15), we used the Regula Falsi numerical method. This method is easy to program to find a numerical estimate of an equation. However, it can be very slow and needs an initial interval around the solution. The T^* value based on these parameters is 4.6612 unit time. Using the Equation (3.13) we estimate T₁= 1.3984, we then substitute the value of T₁ in (3.12) to obtain T₂= 3.2628. Accordingly, using the Equations (3.6), (3.7), (3.8), (3.9), (3.10) and (3.14) we obtain IHC = $161.4165, DC = $119.2322, PD = $1679.9794, PC = $1200.0343, SC = $429.0741, K(T) = $3589.7278 and Q^*= 180.3452.

Figures 1 to 7 illustrate the relationships between the inventory level, inventory holding cost, production run time, production cycle time, deterioration rate and total cost. Briefly, Fig. 1 shows that the system’s inventory level increases during its production run time while its inventory level decreases during the time having no production. Figure 2 shows that the average total cost increases when the deterioration rate increases. Figures 3 and 4 show that the average total cost decreases when the production run time and cycle time increase. Figure 5 shows that when the inventory holding cost increases then the average total cost decreases. Figure 6 shows that when the deterioration rate increases then the average total cost decreases. Lastly, Fig. 7 shows that the deterioration cost is incurred at the start of the production cycle time and it increases until the optimal.

This study presents the direct application of an inventory model to the business enterprises that considers the fact that the storage item is deteriorated during storage periods and in which the deterioration depends upon the time. The model is solved analytically by minimizing the total inventory cost and finally the proposed model is verified by the numerical and graphical analysis. The obtained results point toward the validity and stability of the model.

5 Conclusion

In this paper, an inventory system with a linear time dependent exponential distributed deterioration item has been studied and the mathematical model is developed for the optimal purchase quantity. An optimal cycle time for the model is derived. The average total cost per unit time is derived and is found be a relatively simple expression. However, this study minimizes the total cost for a linear time dependent exponential distributed deterioration rate. This proposed model can further be enriched by taking more realistic assumptions such as a finite replenishment rate, a probabilistic demand rate, etc.

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