An EPQ Model with Unit Production Cost and Set-Up Cost as Functions of Production Rate

Extensive research has been devoted to economic production quantity (EPQ) problem. However, no attention has been paid to problems where unit production and set-up costs must be considered as functions of production rate. In this paper, we address the problem of determining the optimal production quantity and rate of production in which unit production and set-up costs are assumed to be continuous functions of production rate. Based on the traditional economic production quantity (EPQ) formula, the cost function associated with this model is proved to be nonconvex and a procedure is proposed to solve this problem. Finally, utility of the model is presented using some numerical examples and the results are analyzed.


Introduction
The economic production quantity (EPQ) model has been widely used in practice because of its simplicity.However, there are some drawbacks in the assumption of the original EPQ model and many researchers have tried to improve it with different viewpoints.Recently, the classical EPQ model has been generalized in many directions.Some authors extended the EPQ model by incorporating the effect of learning in setups and process quality.Also, set-up time reduction on production run length and varying parameters have received significant attention.The relationship between set-up cost and production run length is also influenced by the learning and forgetting effects.The effect of learning and forgetting in setups and in product quality is investigated by Jaber and Bonney [1].Porteus studied the effect of process deterioration on the optimal production cycle time [2].
Darwish generalized the EPQ model by considering a relationship between the set-up cost and the production run length [3].Jaber investigated the lot sizing problem for reduction in setups with reworks and interruptions to restore the process to an "in-control" state [4].Unlike the model presented by Khouja [5], he considered that the setup cost and defect rate decrease as the number of restoration activities increases.Afshar-Nadjafi and Abbasi considered an EPQ model with depreciation cost and process quality cost as continuous functions of time [6].Freimer et al. studied the effect of imperfect yield on EPQ decisions.They considered set-up cost reductions and process quality improvements as types of investments in the production processes [7].
Furthermore, the classical EPQ model has been investigated in many other ways; for example, the effect of varying production rate on the EPQ model was investigated by Khouja [8].Huang introduced the EPQ model under conditions of permissible delay in payments [9].Salameh and Jaber developed the EPQ model for items of imperfect quality [10].Jaber et al. applied first and second laws of thermodynamics on inventory management problem.They showed that their approach yields higher profit than that of the classical EPQ model [11].
Recently, Hou considered an EPQ model with imperfect production processes, in which the set-up cost and process quality are functions of capital expenditure [12].Tsou presented a modified inventory model which accounts for imperfect items and Taguchi's cost of poor quality [13].
The assumption of the fixed unit production and set-up costs is one of the classical EOQ shortcomings.To the author's knowledge, none of the above EPQ models considered Modelling and Simulation in Engineering the unit production and set-up costs as continuous functions of the production rate.
In this paper, the classical EPQ model is extended by considering unit production cost and set-up cost as continuous functions of production rate.We use a simple method to solve the extended economic production quantity model of minimizing the total annual cost.Also, numerical examples are used to show the utility of the proposed models.The paper is organized as follows.The model assumptions and notation are presented in Section 2.Then, the model is developed in Section 3. In Section 4 we explain a brief summary of the results given in this paper based on a numerical example.Finally, Section 5 contains the conclusions.

Assumptions and Notation
In this section, we derive a mathematical statement for the EPQ model with unit production cost and set-up cost as continuous functions of production rate.The basic EPQ model is that of determining a production quantity of an item, subject to the following conditions related to the production facility and marketplace [14].
(i) Demand rate, is continuous, known, and constant.
(ii) Production rate is greater than or equal to demand rate.
(iii) All demands must be met.
(iv) Holding costs are determined by the value of the item.
(v) Set-up time is assumed to be zero.
(vi) There are no quantity constraints.
(vii) No shortages are allowed.
Most of the assumptions in our mathematical model are the same as those in the conventional EPQ.Besides, we impose the following additional assumptions.
(i) Unit production cost of product is a decreasingly continuous function of production rate.
(ii) Set-up cost is an increasingly continuous function of production rate.
The classic EPQ model assumes that unit production cost and set-up cost are fixed.However, in real production environment, this assumption does not accurately reflect the reality, because it can often be observed that the unit production cost and set-up cost depend on the production rate.Processes with low production rates require less set-up cost than that of high rates.This is because the effort and automation needed to perform a set-up activity are related to the condition of the production process, that is, for high production rates, the production process is more likely to be subjected to higher level of equipment resulting in a higher set-up cost.In mass production processes by increasing the investment in hightech equipment, the production rate increases but also the set-up cost will increase accordingly.For example, the set-up cost of a CNC is much more than in simple turning machines.Also, processes with high production rates result in low unit production cost.This is because the human resource and raw materials are used efficiently with comparison to processes with low production rates.These assumptions add complexity of the model where a closed form solution was not possible and the convexity of the cost function was not validated.
In order to state the problem mathematically, let  be production quantity (real positive decision variable),  annual demand rate of product,  annual production rate of product (real positive decision variable),  unit production cost of product,

Model Development
In this section we develop EPQ model with unit production cost and set-up cost as continuous functions of production rate.The set-up cost in the proposed model is assumed to be an increasingly continuous function of production rate as follows (modified from that of Jaber and Bonney [1]): where the factor  is the shape factor of the set-up cost, the parameter  0 is a positive constant that can be interpreted as set-up cost associated with the classical EPQ model ( = 0), and  max is maximum production rate which serves as an upper limit on the set-up cost.Also,  max =  0   max is defined as maximum set-up cost (related to maximum production rate,  max ).The factor  can be estimated by the curve-fitting approach using historical data on set-up cost of the process.
The unit production cost in the proposed models is a decreasingly continuous function of production rate as follows: where the factor  is the shape factor of unit production cost, the parameter  0 is a positive constant that can be interpreted as unit production cost associated with the classical EPQ model ( = 0), and  max is maximum production rate which serves as an upper limit on the set-up cost.Also,  min =  0  − max is defined as minimum unit production cost (related to maximum production rate,  max ).The factor  can be estimated by the curve-fitting approach using historical data on unit production cost of the process.The behavior of () and () is shown in Figure 1.This figure shows that the set-up cost increases with the production rate.However, after the production rate reaches maximum rate,  max , the process set-up cost is fixed at  max .Inversely, unit production cost decreases with the production rate.However, after the production rate reaches maximum rate,  max , the unit production cost fixed at  min .Although parameters  and  can be any real numbers in general, logically 0 ≤  ≤ 1 and 0 ≤  ≤ 1 are acceptable.
The model presented in this section assumes that all demands are satisfied from inventory; that is, no stock-out situation occurs.The objective is to find economic production quantity  * and economic production rate  * , in order to minimize the annual total cost ATC.The behavior of inventory level in EPQ model is illustrated in Figure 2. It shows that when the inventory level vanishes, production is started at a rate .Since  exceeds , the inventory position will increase at a rate  − , satisfying demand and building inventory, until  units are produced.At that point in the cycle, the inventory level will be a maximum.

ATC is computed as follows
where the first term is the production cost, the second term is the set-up cost, and the third term is the holding cost.The elements of Hessian matrix are Evaluation of  1 and  2 shows that Then ATC is a nonconvex function.Therefore, taking the partial derivatives of ATC with respect to the  and  and solving equations (/)ATC = 0 and (/)ATC = 0 do not guarantee the necessary conditions for  and  to be optimal.For a given amount of production rate , (3) gives an expression of ATC as a function of only  which is called reduced ATC.Although ATC is a nonconvex function, it is obvious from expression (5) that reduced ATC is a convex function.Therefore, taking the first derivative of reduced ATC with respect to  and setting (/) reduced ATC = 0 yield the only positive solution at This problem poses a difficult computational task due to the nonconvexity involved.To obtain the economic production Algorithm 1 starts with an initial solution (production rate and production quantity) for the problem and by initializing the control parameters.This first solution is considered as the current and best solution.In the inner cycle of the procedure, repeated, while  +  ≤  max , a neighborhood solution of the current solution is obtained by adding to the current solution a fixed amount, , where  > 0 is the steplength parameter.The generated solution replaces the current one.This procedure continues until  is equal to maximum production rate,  max .Finally, the last best solution is reported as the optimal solution.

Numerical Example and Discussion
To illustrate the usefulness of the model developed in Section 3, let us consider the inventory situation where a stock is replenished with  units.The parameters needed for analyzing the above inventory situation are given as follows.
maximum production rate: 500 unit/year, demand rate,  = 220 units/year, holding cost rate,  = 0.2 annually, maximum production rate,  max = 500 units/year,  0 = 75$/unit,  0 = 100$/cycle,  = 0.09, Using the above mentioned procedure gives  * = 130.614,  * = 500, and ATC * = 10058.55,whereas with classical EPQ model (with production rate 500), we have  * = 72.375 and ATC * = 17107.95.It is clear that significant losses are realized when the classical EPQ model is used instead of the proposed model.Annual total cost function against production rate and production quantity is sketched in Figure 3. Now, we demonstrate the utility of the model and study the effect of the shape parameter of the set-up cost and unit production cost on the optimal solution.In order to assess the loss of using the classical EPQ model, its performance is compared with that of the proposed model.Let ATC * EPQ denote the optimal total expected cost using the classical EPQ model ( =  = 0).Now define the percent loss due to using the classical EPQ model instead of the proposed model as in [3] as follows: Table 1 gives the optimal solutions for selected values of  ranging from 0 to 0.9 and  fixed at 0.1.It is apparent from Table 1 that the effect of the  on economic production quantity is not monotonously increasing or decreasing.On the contrary, lot size has a convex behavior as  increases.
Another important observation in this example is that the economic production rate is located in extreme points ( + 1 = 221 or  max = 500).The results also indicate that the loss due to using the classical EPQ model increases with .This is because of the fact that for high values , the unit production cost in the classical EPQ deviates significantly from the actual situation.
In Table 2, the optimal solutions for selected values of  ranging from 0 to 0.9 and  fixed at 0.09 are examined.The results show that the economic production quantity increases as  increases; lot size is inflated when  approaches unity from below.The results also indicate that the loss due to using the classical EPQ model decreases slightly with .
Table 3 shows the simultaneous effect of values  and  on optimal solutions.The results show that the economic production quantity has a convex behavior as  and  increase; lot size is inflated when  and  approach unity from below.Inversely, the economic production rate has a concave behavior as  and  increase, and also the economic production rate is located in extreme points.The results also indicate that the loss due to using the classical EPQ model increases with  and .This is because of the fact that for high values  and , the unit production cost and set-up cost in the classical EPQ deviate significantly from the actual situation.

Conclusions
In this paper, economic production quantity (EPQ) model has been developed considering varying both the unit production cost and set-up cost.We have considered the unit production cost as a continuous decreasing function of production rate and the set-up cost as a continuous increasing function of production rate.The problem is described with a mathematical model, and then a simple procedure is proposed to solve it.From the numerical results, we could clearly see that loss due to using the classical EPQ model is significant.Also, the results showed that the shape parameters,  and , which are a property of the system, have marginal and simultaneous effect on optimal policies, significantly.For example, lot size is inflated when  and  approach unity from below.The results of this study can help managers make optimal decisions on equipment selection (with optimal production rate).Also, the optimal production quantity based on the optimal value of production rate can be determined.

Table 1 :
Optimal solutions for different values of .

Table 2 :
Optimal solutions for different values of .

Table 3 :
Optimal solutions for different values of  and .