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A Parameter-Tuned Genetic Algorithm to Solve Multi-Product Economic Production Quantity Model with Defective Items, Rework, and Constrained Space

Seyed Hamid Reza Pasandideh, Ph.D., Assistant Professor Railway Faculty, Iran University of Science and Technology, Tehran, Iran

Phone: +98 (21) 77491029, Fax: +98 (21) 77451568, e-mail: SHR_pasandideh@sbu.ac.ir

Seyed Taghi Akhavan Niaki, Professor Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

Phone: +98 21 66165740, Fax: +98 21 66022702, e-mail: niaki@sharif.edu

Seyedeh Sameieh Mirhosseyni, M.Sc. Department of Industrial Management, KAR University, Qazvin Branch

Phone: +98 (282) 2232182, Fax: +98 (282) 2225883, e-mail: s.s.mir.h@gmail.com

Abstract

The economic production quantity (EPQ) model is often used in manufacturing

environments to assist firms in determining the optimal production lot-size that minimizes the

overall production-inventory costs. While there are some unrealistic assumptions in the EPQ

model that limit its real-world applications, in this research some of these assumptions such as 1)

infinite availability of warehouse space, 2) all of the produced items being perfect and 3) the

existence of one product type are relaxed. In other words, we develop a multi-product EPQ

model in which there are some imperfect items of different product types being produced such

that reworks are allowed and that there is a warehouse space-limitation. Under these conditions,

we formulate the problem as a non-linear integer-programming model and propose a genetic

algorithm to solve it. At the end, a numerical example is presented to identify the optimal values

of the genetic algorithm parameters and to illustrate the applications of the proposed

methodology to more realistic real-world problems.

Keywords: EPQ, multi-product; imperfect and scrap items; constrained space; genetic algorithm

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1. Introduction and literature review

The economic production quantity (EPQ) model can be considered as an extension to the

well known economic order quantity (EOQ) model that was introduced by Harris (1913). It is a

technique to find the optimum production quantity by considering costs of procurement,

inventory holding, and shortages.

In real-life manufacturing systems, there is usually more than one type of products, the

demand for each product is a random variable, and that the generation of defective items and

random breakdowns of production equipment are inevitable. As a result, the first few

assumptions of the EPQ model (single product, deterministic demand, producing only perfect

items, etc.) may not be realized in many real-life problems.

Many researchers have developed different EOQ and EPQ models assuming the

existence of scrap items. For instance, Hayek and Salameh (2001) assumed that all of the

produced defective items are repairable and derived an optimal operating policy of the EPQ

model. By considering rework time in their model, the basic assumptions of their work were not

only to allow backorders but also to permit all of the defective items to be reworkable to become

perfect. In the research by Rosenblatt and Lee (1986), which has been proposed an EPQ model

for a production system that contains defective products, the main assumption was that the

production system produces 100% non-defective items from the starting point of the production

time until a time point that was considered a random variable. At this time, the system becomes

out of control and starts to produce defective items until the end of the production period.

Furthermore, they assumed that the distribution of the time lag until the system becomes out of

control is exponential. Kim and Hong (1999) extended the Rosenblatt and Lee’s model (1986)

with the assumption of an arbitrary distribution of the time lag. Salameh and Jaber (2000)

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developed an EPQ model in which the shortages were not allowed and a uniformly distributed

fraction of the ordered lot contained imperfect quality items. Chiu et al. (2007) presented a

procedure to determine the optimal run time for an EPQ model with scrap, rework and stochastic

machine breakdowns.

Hou (2007) presented an EPQ model with imperfect production processes, in which the

set up cost and process quality are functions of capital expenditure. This model illustrates the

relationship among production run length, set up reduction, and process quality improvement in

an imperfect production system. He showed that investment in set up reduction would lead to the

reduction in optimal production run length and would reduce lot size, whereas investment in

process quality improvement would result in an increase in optimal production run length and lot

size. At the end, he pointed out that it was very important to investigate the optimal allocation of

investment between both options.

Lin (1999) introduced an integrated EPQ model subject to an imperfect production process

and constraint on the raw materials. His basic model assumes that at the beginning of each

production run, the production facility is in an in-control state. Then, after a period of time, the

facility shifts to an out-of-control state. The elapsed time to the shift is a random variable having

an exponential distribution with a given mean. Moreover, there is a constraint on the availability

of the storage space for raw materials. Then he extended the model to incorporate two cases that

have a dynamic deterioration in the production process. In addition, he studied the model for the

situation where elapsed time to the shift and the percentage of defective items are function of the

production setup cost.

In several instances of practice, producing new or recovering defective products take

place on a common facility. Consequently, it is necessary to coordinate the production and

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rework activities with respect to the timing of operations and with regard to appropriate lot sizes

for both processes. Buscher and Lindner (2007) presented a lot size model, which addresses all

of these aspects. In addition, they cited that it was very important to assign completed units at

one stage to partial lots - called batches - for shipment to the next operation.

Liao et al. (2009) studied the maintenance and production programs of an EPQ model for

an imperfect process involving a deteriorating production system with increasing hazard rate.

The imperfect repair restores the system to an operating state, but leaves it failed until perfect

preventive maintenance (PM) is performed. They introduced two types of PM, namely imperfect

and perfect PM. The probability that the perfect PM is performed depends on the number of

imperfect maintenance operations performed since the last renewal cycle. In addition, they

suggested that if the PM rate is estimated based on the actual data, analysts can use the learning

curves to project the PM costs in the integrated EPQ model.

One of the most important aspects in extensions of the EOQ and EPQ models is to

fuzzify their parameters. For instance, Lee and Yao (1998) in their research fuzzified both the

demand and the production quantity to solve the problem of the economic production quantity

per cycle.

Many researchers assumed that there is a large enough storage space to hold products.

However, in reality, there is usually a limitation on available warehouse space for raw materials

or finished goods. In addition, the cost of warehouse holding sometimes outweighs the benefits

of having no limitation on space enormously and the manufacturers prefer to have limited space.

Hence, the storage space limitation will surely affects the quantity of the lot-sizes and needs to

be considered in the model. In this paper, a multi-product EPQ model is considered in which

there is limited warehouse space. In addition, the rate of imperfect and scrap items is known and

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reworking makes the imperfect products perfect. To solve this problem, we first define the

problem and the model in section 2 and 3, respectively. To solve the model, we present a genetic

algorithm in section 4. To demonstrate the application of the proposed methodology, in section

5, we present a numerical example in which the parameters of the proposed GA are fine-tuned.

Finally, the conclusion and some recommendations for future research come in section 6.

2. Problem definition

The proposed model of this research is an applied one that is developed based on the real

constraints and environments of production companies. Consider a manufacturing company that

receives raw materials from a supplier to produce n products. All of the produced items are

inspected to be classified as perfect, imperfect (defective but repairable) and scrap (defective and

not repairable) products. Suppose the required time of the inspection is included within the

production time such that it can separately be assumed zero. This assumption is not far from

reality, because in many instances the inspection task and producing an item occur

simultaneously.

All of the imperfect products are reworked to be perfect and the scrap products are sold

at reduced price. The work in process inventory (WIP) consists of three types of materials around

the manufacturing machines: 1) raw material, 2) perfect products and 3) imperfect products.

Furthermore, the warehouse space of the company for all perfect products is limited, shortage

and delay are not allowed, and that all parameters, such as the demand rate, the rate of imperfect

and scrap items production, the set up cost, etc. are all known and deterministic. The objective is

to determine the optimal production quantities of the products that minimize the total costs while

satisfying the constraint.

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Three main specifications of the proposed model of this research that have led to its

novelty are 1) the allowance of several products, 2) rework and imperfect product are allowed,

and 3) the warehouse space to store raw materials and finished goods is limited. By allowing

these conditions simultaneously, the created model is different from the other models in the EPQ

literature.

3. Problem modeling

In order to model the problem at hand, the classical EPQ model will be extended to

contain the perfect, the imperfect and the scrap items along with the warehouse capacity. To do

this, we first define the parameters in section 3.1. Then, the pictorial representation of the

inventory problem will be given by an inventory graph in section 3.2. In section 3.3, different

costs of the system will be derived. Finally, the model of the problem will be presented in section

3.4.

3.1. Parameters and notations

For products 1,2,...,i n , we define the parameters of the model as follow:

n Number of products

iQ Order quantity of the ith product

iP Production rate of the ith product

iD Demand rate of the ith product

iA Set up cost per cycle of the ith product

ih Holding cost rate of the ith product

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iM Raw material cost per unit of the ith product

iS Set up time of the ith product

im Machining time per unit of the ith product

iR Production cost rate per unit time of the ith product

ic Average production cost per unit of the ith product

iv Average value added per unit of the ith product

iw Average investment per unit of WIP of the ith product

iI Average amount of warehouse inventory of the ith product

1ip Imperfect production percentage of the ith product

2ip Scrap production percentage of the ith product

1is Perfect production cost of the ith product

2is Scrap production cost of the ith product

iT Cycle time of the ith product

iTP Total time per cycle to produce the ith product

it Average production time per unit of the ith product

if Required space per perfect unit of the ith product

F Total available warehouse space for all products

iPTC Total procurement cost of the ith product

iOTC Total set up cost of the ith product

iITC Total inspection cost of the ith product

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iWIPTC Total holding cost for WIP of the ith product

iHTC Total holding cost for perfect products of the ith product

TC Total annual cost of all products

3.2. The inventory graph

In order to calculate all inventory costs, it is necessary to survey the work in process and

warehouse inventory. For the problem at hand, the graph of raw material quantity versus time is

demonstrated in Fig1.a. In addition, the graphs of the perfect and scrap WIP inventory versus

time are illustrated in Figures 1.b and 1.c, respectively. In this problem, the rate of demand is

constant and hence the graph of the final product quantity in the warehouse is similar to the EOQ

model and is given in Fig 1.d. We note that in Fig 1.d, the amount of produced perfect products

in the warehouse in each cycle is reduced based on the deterministic rate of demand.

Insert Figure (1) about here

3.3. Costs calculations

Since the shortage and delay are not permitted, the total inventory costs of all products

per year (TC ), is the sum of total procurement cost ( PTC ), total set up cost ( OTC ), total

inspection cost ( ITC ), total holding cost for WIP inventory ( WIPTC ) and the total holding cost

for warehouse inventory ( HTC ) for all products. In other words, we have:

1

i i i i i

n

P O I WIP Hi

TC TC TC TC TC TC

(1)

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In any cycle, since the set up time, the production time and the reworking time are equal

to iS , i im Q and 1ii im p Q , respectively, the total time to produce product i, ( iTP ), is given in

equation (2).

1 11i ii i i i i i i i iTP S m Q m p Q S m Q p (2)

Hence, the average production time for each unit of product i is:

11i

i ii i

i i

TP St m p

Q Q (3)

Based on iR which is the rate of production cost per unit time, iv and ic are obtained as:

11i

ii i i i i

i

Sv R t R m p

Q

(4)

11i

ii i i i i i

i

Sc M v M R m p

Q

(5)

Since delays are not allowed, the supply and the demand quantities are equal and we have:

2

2

11 i

i

i

i i i ii

p Qp Q D T T

D

(6)

As 1is and 2i

s represent the price of the perfect and the scrap items, respectively, the average

revenue in unit time is obtained as:

2 1 2 2 21 2

2

1 ; 1, 2,...,

1i i i i i

i i

i

i i

i i ii

p Q s p Q s pTR D s D s i n

T p

(7)

Note that for the problem at hand the revenue in unit time does not depend on the lot size.

Now, based on equations (2) to (6), the inventory costs of equation (1) are calculated as

follows.

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Since the annual rate of demand for each product is known, the total procurement cost for

product i per unit time is obtained as:

2

= ; 1, 2,...,1i

i

i i i iP

i

m Q m DTC i n

T p

(8)

For each product, the setup process accrues only once and hence the set up cost per unit

time of the ith product can be obtained as:

2

= ; 1, 2,...,1i

i

i i iO

i i

A A DTC i n

T Q p

(9)

Assuming 100% inspection and that all of the imperfect products transform to perfect

ones after reworks, the inspection of each product occurs once and its associated cost per unit

time is obtained as

2

; 1, 2,...,1i

i

i i i iI

i

I Q I DTC i n

T p

(10)

In order to calculate the holding cost of WIP inventory of the ith product, since

iw denotes the average investment per unit of WIP inventory (including raw materials, perfect

and imperfect items) and ih represents the holding cost rate of the ith product, then

iWIP i iTC h w (11)

The average raw material inventory of each product is the total amount of raw materials

(the surface under its corresponding inventory graph) divided by the cycle time. Accordingly, the

average investment value of the raw material is obtained by the product of the average raw

material inventory and the price per unit of the raw material. The average investment value of the

perfect and imperfect products can be calculated similarly. Hence, the average investment value

per unit of the WIP inventory of product i is given in equation (12).

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2 2

1 1

2

1 1 11 12 2 2

2

1 2 12 1

i i

i i

i

i i i i i ii i

i i i i i ii i i i

i i ii i i i i i

i

Q TP Q p TP Q p TP Q TPw M c c M c

T T T T

D R SS m p Q M R m p

Qp

(12)

Hence, based on equations (11) and (12), the average holding cost of the WIP inventory of

product i becomes

1 1

2

1 2 1 ; 1,2,...2 1i i i

i

i i i iWIP i i i i i i

i

h D R STC S m p Q M R m p i n

Qp

(13)

In order to calculate the holding cost of the warehouse inventory, we first need to

estimate the average warehouse inventory. Regarding Figure 4, we have

2

2

11 12 1

2

i

i

i i

i ii

Q p TI Q p

T

(14)

Hence, using equation (5) and (14) the holding cost of the warehouse inventory for product i

becomes

1 2

11 1

2i i i

iH i i i i i i i i

i

STC h c I h M R m p Q p

Q

(15)

Finally, the total annual inventory cost of all products described in equation (1) is given

in equation (16).

1

i i i i i

n

P O I WIP Hi

TC TC TC TC TC TC

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2 22

1 11 2

1 2

1 11

1 2 12 1

11 1

2

i ii

i i

i

i i

i i i i i i

i

ni i i i

i i i i i ii i

ii i i i i

i

m D A D I D

p pQ p

h D R SS m p Q M R m p

Qp

Sh M R m p Q p

Q

(16)

3.4. Problem formulation

As described earlier, the goal is to determine the economic production quantities of the

products such that the total annual inventory cost obtained in equation (16) is minimized within

the warehouse space limitation. Since the space limitation can be modeled as

21

1i

n

i ii

p Q f F

(17)

The mathematical programming model of the problem at hand becomes

2 22

1 11 2

1 2

1 11

1 2 12 1

11 1

2

i ii

i i

i

i i

i i i i i i

i

ni i i i

i i i i i ii i

ii i i i i

i

m D A D I D

p pQ p

h D R SMin TC S m p Q M R m p

Qp

Sh M R m p Q p

Q

s.t.:

21

1

0 and integer ; 1, 2,...

i

n

i ii

i

p Q f F

Q i n

(18)

In the next section, an efficient algorithm is proposed to solve this model.

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4. A solution algorithm

In most EOQ or EPQ models that have been developed so far, researchers have tried to

consider some constraints such as defective items, shortages, backorders, and so on such that the

objective function of the model becomes concave and the model can easily be solved by some

mathematical approaches like the Lagrangian or the derivative methods. However, since the

objective function of the nonlinear integer programming model in (18) is a complex and

sophisticated one, reaching an analytical solution (if any) is difficult and time-consuming (Gen &

Cheng 1997). As a result, in this section a meta-heuristic stochastic search algorithm is used to

solve the model.

Many researchers have successfully used meta-heuristic methods to solve complicated

optimization problems in different fields of scientific and engineering disciplines. Some of these

meta-heuristic algorithms are simulating annealing (Aarts and Korst (1989), Taleizadeh et al.

(2008)), threshold accepting (Dueck and Scheuer (1990)), Tabu search (Joo and Bong (1996)),

genetic algorithms (Pasandideh & Niaki (2006), Najafi & Niaki (2006), Taleizadeh et al. (2008b,

2009a, 2009b, 2009c), neural networks (Abbasi & Niaki (2007)) Gaidock et al. (2002)), ant

colony optimization (Dorigo and Stutzle (2004)), fuzzy simulation (Taleizadeh et al. (2009a),

evolutionary algorithm (Laumanns et al. (2002), Taleizadeh et al. (2009b), and harmony search

(Lee and Geem (2004), Geem et al. (2001)). Among these meta-heuristic algorithms, the genetic

algorithm has shown to be an efficient one to solve the nonlinear integer programming model of

the problem at hand (Gen & Cheng 1997).

The usual form of Genetic Algorithm (GA) was described by Goldberg (1989). Since

then many researchers have applied and expanded this concept in different fields of study.

Genetic algorithm was inspired by the concept of survival of the fittest. In genetic algorithms, the

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optimal solution is the winner of the genetic game and any potential solution is assumed to be a

creature that is determined by different parameters. These parameters are considered as genes of

chromosomes that could be assumed to be binary strings. In this algorithm, the better

chromosome is the one with higher fitness value. In practical applications of genetic algorithms,

populations of chromosomes are created randomly. The size of these populations is different in

each problem. Some hints about choosing the proper population size exist in different reports

(Man et al. 1997).

In the next subsections, we describe the proposed GA to solve the model at hand.

4.1. Chromosomes

In a GA, a chromosome is a string or trail of genes, which is considered as the coded

figure of a possible solution (appropriate or none-appropriate). Chromosome representation is a

very important part of the GA method description. While in some researches a chromosome is

complied in binary code, a decimal (real)-mode code is used in others. The success of the coding

format depends on the other routines of GA, especially the crossover and the mutation operations

(Gen & Cheng 1997).

In this paper, the chromosomes are strings of the quantities of the products ( )jQ and are

given in real-mode code, and hence the crossover and the mutation operators of this research is

based on the real-mode code of the chromosome that is described later. Figure (2) shows a

typical chromosome structure, in which the genes are quantity of the products.

Insert Figure (2) about here

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An infeasible chromosome is defined as the one that does not satisfy the constraints of the model

given in equation (18).

4.2. Population

A group of chromosomes is called population. One of the characteristics of a GA is that

instead of focusing on a single point of the search space (or one chromosome) it works on a

population of chromosomes. Each population or generation of chromosomes has the same size

which is well-known as the population size and is denoted by N. In this research, the initial

population is randomly generated regarding the population sizes that vary between 20 and 60.

4.3. Crossovers

Crossover is the main genetic operator. In a crossover operation, it is necessary to mate

pairs of chromosomes at a time to create offspring. Crossover operates on the parents

chromosomes with the probability of cP . If no crossover occurs, the offspring's chromosomes

will be the very same as their parents.

One simple way to achieve crossover is to randomly create a binary chromosome

corresponding to the chromosome at hand. Then the genes of the chromosome at hand that

correspond to zeros in the binary chromosome are not changed. However, those that correspond

to ones are changed. Figure (3) demonstrates the crossover operation for four products.

Insert Figure (3) about here

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In this research, we use single point crossover with different values of the cP parameter

ranging between 0.45 and 0.85. We note that an infeasible chromosome that does not satisfy the

constraints of the models (18) does not move to the new population.

4.4. Mutation

Mutation is the second operation in a GA method for exploring new solutions and it

operates on each of the chromosomes resulted from the crossover operation. Mutation is a

background operator, which produces random change in chromosomes and may result in a

chromosome with higher fitness value. In mutation, we replace a gene with a randomly selected

number within the boundaries of the parameter (Gen & Chen 1997). We create a random number

RN between (0,1) for each gene. If RN is less than a predetermined mutation probability mP , then

the mutation occur in the gene. Otherwise, the mutation operation is not performed in that gene.

Usual value of mP is 0.1 (sometimes 0.2) per chromosome or 0.1 (sometimes 0.05) over the

numbers of genes in a chromosome. Based on the later approach, in this research different values

between 0.005 and 0.05 are chosen as different values of mP . We note that an infeasible

chromosome that does not satisfy the constraints of the models in (18) does not move to the new

population. Figure (4) shows an example of the mutation operator for four products in which mP

is chosen to be 0.05.

Insert Figure (4) about here

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4.5. Objective Function Evaluation

After producing the new chromosomes by crossover and mutation operations, we need to

evaluate them. Whether a solution (represented by a chromosome) is appropriate or not depends

on the objective function evaluation. In a minimization problem, the more appropriate the

solution is the less the amount of the objective function (fitness value) will be. The chromosomes

that are the fittest will take part in offspring generation with more probability. For the model at

hand, the fitness value is the total inventory cost given in equation (18). For constrained

optimization problems, the main issue is to control the feasibility of the chromosomes. To do this

and to avoid infeasible chromosomes, the penalty policy given in Gen & Chen (1997) is

employed. Since we are faced with a minimization problem, a positive value is assigned to the

penalty. This penalty is a squared function of violation of right hand side of the space constraint

of the model. Thus, high penalties are given to more infeasible chromosomes. For a feasible

chromosome, the penalty is set to zero. In this case, the fitness value of a chromosome is

evaluated as the weighted sum of its objective function.

4.6. Chromosome selection

After producing the offspring with crossover and mutation operators and then evaluating

their fitness value, the next population of size N is made out of the chromosomes with the highest

fitness. This selection strategy is called the deterministic mechanism.

4.7. Stopping criteria

The last step in a GA method is to check if the algorithm has found a solution that is good

enough to meet the user’s expectations. Stopping criteria is a set of conditions such that when

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satisfied a good solution is obtained. Different criteria used in literature are as follows: 1)

Stopping of the algorithm after a specific numbers of generations, 2) reaching a maximum

number of evaluations, 3) no improvement in the objective function, and 4) reaching a specific

value of the objective function.

In this research, a combination of the first and the third criterion is used to stop the

algorithm.

In short, the steps involved in the G.A algorithm used in this research are:

1. Setting the parameters cP , mP and N

2. Initializing the population randomly

3. Evaluating the objective function for all chromosomes

4. Selecting individual for mating pool

5. Applying the crossover operation for each pair of chromosomes with probability cP

6. Applying mutation operation for the genes of the chromosome with probability mP

7. Replacing the current population by the resulting new population

8. Evaluating the objective function

9. If stopping criteria is met, then stop. Otherwise, go to step 4

The proposed GA is coded using the embedded algorithm in MATLAB. In order to

demonstrate the application of the proposed GA and to evaluate its performance, in the next

section we bring a numerical example.

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5. A numerical example

Consider a four-product inventory control problem with annual rate of production iP ,

annual demand rate of iD , annual inventory holding cost per unit of ih , fixed ordering cost per

cycle of iA , raw material purchasing cost per unit of iM , production cost per unit per unit time

of iR , annual setup time of iS , machining time per unit of im , imperfect production percentage

of 1ip , scrap production percentage of 2i

p , and average amount of inventory iI . The required

warehouse space per perfect unit is if and F is the total available warehouse space. The

corresponding numerical data are given in Table (1). The total available warehouse space is

3000. The goal is to find the integer order quantities of the products ( ; 1,2,3,4iQ i as the

decision variables) such that the total inventory cost is minimized while the warehouse space

constraint is satisfied.

Insert Table (1) about here

Since the parameters of many meta-heuristic algorithms like GA play important roles in

the quality of the solution and that without tuning the parameters the algorithm may not work

properly, they need to be adjusted such that a better solution is reached. As a result, in this

research, a regression analysis using the SAS software is used to identify the significant

parameters of the proposed GA and to find a proper relationship between the response (the

quality of the solution defined as the total cost) and the parameters. At the end, the LINGO

software will be used to optimize the relationship function and to find the optimal values of the

significant GA parameters.

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To find the optimal values of the GA parameters, using MATLAB computer software,

the algorithm is employed 110 times, each time changing its parameters in their corresponding

ranges and obtaining the response value. The crossover and mutation operations rates vary in the

range of 0.45-0.85 and 0.005-0.05, respectively. Furthermore, different integer population sizes

between 20 and 60 are considered in this experiment. Table (2) shows the results of the

experimentation.

Insert Table (2) about here

In the next step, all possible regression procedure (Neter et al. 1999) was used to find the

significant GA parameters. Table (3) shows the results of different criteria for different

regression functions. These results show that the model with all GA parameters is the best.

Insert Table (3) about here

In order to study the interaction effects between the GA parameters, consider the

regression equation including all main and interaction effects in (19).

0 1 2 3 12 13 23 123( ) c m c m c m c mE Y N P P NP NP P P NP P (19)

Given the data in Table (2), the Backward Elimination Procedure of the SAS software

was then employed to study the regression model in (19). The final analysis of variance table of

this procedure is given in Table (4). Furthermore, the regression parameter estimates are given in

Table (5). Moreover, the summary of the steps in backward elimination procedure is given in

Table (6).

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Insert Table (4) about here

Insert Table (5) about here

Insert Table (6) about here

Based on the results of Table (5), the estimated regression function is:

175496 2334.53408 11335422 276243 17017396 396226m m c m c mY N P NP P P NP P (20)

The Lingo software solves the estimated regression function (the objective function) that

needs to be minimized along with the constraints (the space constraint and integer chromosome)

within the GA parameter ranges. The optimum values of the GA parameter are 0.85, 0.05 and 60

for crossover rate, mutation rate and pop-size respectively. Table (7) shows the optimum results.

Insert Table (7) about here

Finally, the proposed GA with the optimal parameters given in Table (7) employed to the

problem at hand. The graph of the fitness value in terms of the generation numbers is given in

Figure (5). The results of Figure (5) show that the minimum cost is 8584.8 and the algorithm

converges after 5 generations. At this cost, the optimal decision variables are given in Figure (6).

Note that the meta-heuristic GA provides a near optimal solution and not necessarily the optimal

one.

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Insert Figure (5) about here

Insert Figure (6) about here

In order to demonstrate the proposed GA is an effective procedure to solve the multi-

product economic production quantity model, example (1) is solved as a nonlinear integer model

by Lingo software as well. Figure (7) shows the optimal values of decision variables along with

the optimal cost of 8489.4. These values are obtained at the 978th iteration of the software.

However, we note that the optimum solution was obtained in the fifth generation of the proposed

GA. This means that while the proposed GA is an effective tool to provide a near optimum

solution, it is a faster approach.

6. Conclusions and recommendations for future research

One inevitable aspect of manufacturing systems is producing defective products. In this

paper, we developed a multi-product EPQ model in which defective items and reworking are

considered. In addition, the warehouse space is limited for all products. In this condition, we

formulated the problem as a nonlinear integer programming and solved it using a genetic

algorithm. At the end, a numerical example was presented to demonstrate not only the

application of proposed methodology, but also to evaluate the effectiveness of the GA algorithm

in solving the multi-product economic order quantity model of this research. In this example, the

optimum values of the GA parameters were obtained using regression analysis along with

LINGO software.

Some recommendations for future research follow:

22  

a) Some gradient-based algorithms such as sequential quadratic programming or other

heuristic search techniques such as ant colony optimization, simulated annealing, etc.

may be used to solve the problem at hand and compare the results with the ones of the

proposed genetic algorithm.

b) Some other limitations such as delay, shortage, budget limitation, etc. can be augmented

to the model.

c) The model can be extended to involve stochastic or fuzzy natures of some parameters

such as the demand rate.

7. Acknowledgement

The authors are thankful for the constructive comments of the anonymous reviewers that

improved the presentation of the paper.

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27  

Q Fig. a Fig. c

2p Q

time

time

21 p Q 21 p Q

Fig. b Fig. d

time time

TP TP

T T

Figure (1): Inventory graphs of raw material (Fig. a), perfect products (Fig. b), scrap items (Fig.

c) and the product in the warehouse (Fig. d)

28  

1Q 2Q 3Q . . . . nQ

Figure (2): The structure of a chromosome

29  

[32 26 45 18] [26 26 45 24]

Binary Chromosome [1 0 0 1]

[26 35 32 24] [32 35 32 18]

Figure (3): An example of a crossover operation

30  

[26 35 32 40] [28 35 22 21]

Randomly Generated Chromosome [0.25 0.63 0.48 0.015]

Figure (4): A graphical representation of the mutation operator

31  

Figure (5): The graph of the convergence path

32  

1 6Q 2 72Q 3 85Q 4 52Q

Figure (6): The optimal values of the decision variables by the proposed GA

33  

1 6Q 2 70Q 3 88Q 4 51Q

Figure (7): The optimal values of the decision variables by LINGO

34  

Table (1): Numerical data of the example

Product iA iM iD

iS im 1 ip 2i

p iR ih iI if

1 12 8 32 0.0009 0.01 0.14 0.05 12 0.16 15 13

2 13 9 25 0.001 0.04 0.25 0.09 14 0.25 12 16

3 14 5 42 0.002 0.098 0.15 0.05 12 0.42 14 10

4 15 9 38 0.004 0.03 0.28 0.11 11 0.25 12 8

35  

Table (2): The experimental results No. N cP mP Fitness ( )Y No. N cP mP Fitness ( )Y

1 60 0.85 0.05 13515 56 40 0.75 0.03 103394 2 60 0.85 0.05 13480 57 40 0.75 0.03 103410 3 60 0.85 0.05 13570 58 40 0.75 0.03 103412 4 60 0.85 0.05 13560 59 40 0.75 0.03 103385 5 60 0.85 0.05 13615 60 40 0.72 0.03 100860 6 60 0.85 0.05 13642 61 40 0.72 0.03 100896 7 60 0.85 0.05 13689 62 35 0.71 0.03 117160 8 60 0.85 0.05 13602 63 35 0.70 0.03 115950 9 60 0.85 0.05 13408 64 35 0.70 0.03 115925 10 60 0.85 0.05 13712 65 35 0.70 0.03 115987 11 60 0.85 0.05 13600 66 35 0.70 0.03 116120 12 60 0.85 0.05 13589 67 35 0.69 0.03 114740 13 60 0.84 0.05 14596 68 35 0.69 0.03 114350 14 60 0.84 0.05 14582 69 35 0.68 0.03 113540 15 60 0.84 0.05 14608 70 35 0.66 0.03 111120 16 60 0.84 0.05 14615 71 30 0.66 0.03 126400 17 60 0.83 0.05 15677 72 30 0.66 0.01 109420 18 60 0.83 0.05 15656 73 30 0.65 0.01 108890 19 55 0.83 0.05 15683 74 30 0.65 0.01 108842 20 55 0.83 0.05 13695 75 30 0.65 0.01 108910 21 55 0.83 0.05 15642 76 30 0.65 0.01 108925 22 55 0.82 0.05 44881 77 30 0.65 0.01 108823 23 55 0.82 0.05 44842 78 30 0.65 0.01 108875 24 55 0.81 0.05 45343 79 30 0.63 0.01 107840 25 55 0.81 0.05 45392 80 30 0.63 0.01 107820 26 55 0.81 0.05 45416 81 30 0.60 0.01 106260 27 50 0.81 0.05 72848 82 30 0.60 0.01 106242 28 50 0.80 0.05 72691 83 30 0.60 0.009 105720 29 50 0.80 0.05 72743 84 30 0.59 0.009 105250 30 50 0.80 0.05 72586 85 30 0.58 0.009 104780 31 50 0.80 0.05 72653 86 30 0.57 0.009 104300 32 50 0.80 0.05 72721 87 30 0.57 0.008 103930 33 50 0.80 0.05 72740 88 30 0.55 0.008 103080 34 50 0.80 0.05 72680 89 30 0.60 0.008 105190 35 50 0.80 0.05 72702 90 30 0.59 0.008 104770 36 50 0.79 0.05 72534 91 30 0.58 0.008 104350 37 50 0.79 0.05 72521 92 30 0.57 0.008 103930 38 50 0.79 0.05 72508 93 25 0.55 0.008 114040 39 50 0.79 0.05 72592 94 25 0.54 0.008 113520 40 50 0.79 0.05 72468 95 25 0.54 0.007 113310 41 50 0.79 0.05 72521 96 25 0.53 0.007 112850 42 50 0.79 0.05 72542 97 25 0.52 0.007 112400 43 45 0.79 0.05 98801 98 25 0.52 0.006 112310

36  

44 45 0.78 0.05 98026 99 25 0.51 0.006 111920 45 45 0.78 0.05 98001 100 25 0.50 0.006 111530 46 45 0.78 0.05 98090 101 20 0.50 0.006 122090 47 45 0.78 0.05 98121 102 20 0.49 0.006 121630 48 45 0.77 0.05 97250 103 20 0.49 0.005 121800 49 45 0.77 0.03 85675 104 20 0.48 0.005 121410 50 45 0.76 0.03 85210 105 20 0.48 0.005 121387 51 45 0.75 0.03 84744 106 20 0.47 0.005 121030 52 45 0.75 0.03 84736 107 20 0.46 0.005 120640 53 40 0.75 0.03 103370 108 20 0.45 0.005 120250 54 40 0.75 0.03 103356 109 20 0.45 0.005 120195 55 40 0.75 0.03 103389 110 20 0.45 0.005 120189

37  

Table (3): All possible regression results

No. of Variables

No. of Regression

Parameters ( )p

Model Variables

2pR pSSR pSSE pMSE 2

pR pC

1 2 N 0.7899 1.274425*10^11 33889112377 313788078 0.7880 2

1 2 cP 0.5990 96642323837 64689323326 598975216 0.5953 2

1 2 mP 0.5621 90691515695 70640131468 654075291 0.5581 2

2 3 N, cP 0.8690 1.40202*10^11 21130603338 197482274 0.8666 3

2 3 , mN P 0.8496 1.370745*10^11 24257117137 226702029 0.8468 3

2 3 ,c mP P 0.5990 96642323837 64689323326 598975216 0.5953 1.9956

3 4 , ,c mN P P 0.8877 1.432105*10^11 18121106939 170953839 0.8845 4

38  

Table (4): The analysis of variance table Source DF SS MS F-Value Pr > F Model 5 1.487032E11 29740646676 244.93 <0.0001 Error 104 12628413785 121427056

Corrected Total 109 1.613316E11

39  

Table (5): The parameter estimates Variable

Parameter Estimate

Standard Error Type II SS F-Value Pr > F

Intercept 175496 11963 26130530408 215.20 <0.0001

N -2334.53408 390.53098 4339152528 35.73 <0.0001

mP -11335422 3518445 1260344785 10.38 0.0017

mNP 276243 86195 1247185779 10.27 0.0018

c mP P 17017396 4395554 1820012503 14.99 0.0002

c mNP P -396226 94735 2124143120 17.49 <0.0001

40  

Table (6): The summary of backward elimination method

Step Variable Removed

Number of

Variables in

Partial R-Square

Model R-Square pC F-Value Pr > F

1 cP 6 0.0000 0.9217 6.0419 0.04 0.8383

2 cNP 5 0.0000 0.9217 4.0495 0.01 0.9304

41  

Table (7): The Lingo optimum solution

*N *

cP *mP Cost

60 0.85 0.05 10244.89