An application of tournament genetic algorithm in a marketing oriented economic production lot-size model for deteriorating items

ARTICLE IN PRESS Int. J. Production Economics 119 (2009) 112–121

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

An application of tournament genetic algorithm in a marketing oriented economic production lot-size model for deteriorating items A.K. Bhunia a,, S. Kundu a, T. Sannigrahi a, S.K. Goyal b a b

Department of Mathematics, The University of Burdwan, Burdwan 713104, West Bengal, India Department of Decision Sciences and MIS, John Molson School of Business, Concordia University, Montreal, Quebec, Canada H3G 1M8

a r t i c l e i n f o

abstract

Article history: Received 8 November 2006 Accepted 14 January 2009 Available online 12 February 2009

The goal of this research is to discuss an application of tournament genetic algorithm (TGA) for solving an economic production lot-size (EPL) model. In this model, the production rate of the item is finite and it is assumed to be a decision variable. The demand rate is also a deterministic function of selling price and the marketing cost. The selling price per unit item is determined by a mark-up over the unit production cost. The deterioration rate at any instant is a linear increasing/decreasing function of time. Partial backlogging shortages are allowed with a variable rate dependent on the length of the waiting time up to the starting of next production. This model is formulated as a highly non-linear constrained optimization problem. To solve this problem, a TGA with steady-state selection, whole arithmetic crossover and nonuniform mutation has been developed and applied. The model has been illustrated with three numerical examples. Finally, sensitivity analyses have been shown graphically to study the variations of the average profit with respect to the different parameters. & 2009 Elsevier B.V. All rights reserved.

Keywords: Genetic algorithm Production Deterioration Partial backlogging Variable demand

1. Introduction In the classical production lot-size model, both production rate and unit production cost are assumed to be known constants and independent of each other. These assumptions are somewhat unrealistic. In practice, there are several situations where the unit production cost is dependent on several factors, like raw materials, labour charges, production rate, etc. Several researchers have developed production lot-size models for single or multiproducts taking constant or variable production rate (either as a function of demand and/or on-hand inventory). In this connection, the recent works of Gowswami and Chaudhuri (1992), Bhunia and Maiti (1998) and Abad (2000, 2003) are worth mentioning. In their models, the

 Corresponding author. Tel.: +91 342 2559881, +91 9933059881 (mobile); fax: +91 342 2634200. E-mail addresses: [email protected], [email protected] (A.K. Bhunia).

0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.01.010

unit production cost is taken as known constant. Khouja (1995) developed an economic production lot-size (EPL) model considering the production rate as a decision variable and the unit production cost as a function of production rate. Again, in the inventory control theory, the deterioration effect on items plays an important role. In addition to the demand, the stock level decreases continuously due to deterioration. This effect is observed for some commonly used physical goods like paddy, wheat, or any other foodgrains, vegetables, fruits, perfumes, etc. Therefore, while determining the optimal inventory levels of those types of goods, the loss due to deterioration effect cannot be ignored. Considering this effect, many researchers developed production lot-size models. In this area, one may refer to the works of Goyal and Gunesekaran (1997), Bhunia and Maiti (1998) and Abad (2000, 2003) among others. As an inventory problem is a decision-making problem which can be formulated as constrained/unconstrained

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non-linear optimization problem, there arises a question: how it can be solved? Generally, most of the optimization problems of different inventory systems are non-convex or non-concave problems in which both local and global optimal solutions may exist. In order to solve these problems, generally, traditional direct and gradient-based optimization methods are used. However, these methods have some limitations. Among these limitations, one is that the traditional methods (direct and gradient based) very often stuck to the local optima. To overcome these limitations, during last four decades there has been growing interest in algorithms which are based on the principle of natural evolution and genetics. A common term, accepted recently, refers to such techniques which are called as evolutionary computational methods. The well-known algorithm in this class includes genetic algorithm (GA). According to the well-known text books of Goldberg (1989), Michalewicz (1996) and Sakawa (2002), GA is a powerful computerized stochastic search and optimization algorithm based on the mechanics of natural evolution and natural genetics. Generally, this algorithm is used to solve the non-convex/non-concave optimization problems. Till now, only a very few researchers have applied it to solve the problem in the field of inventory control. Khouja et al. (1988), Sarkar and Newton (2002), Mondal and Maiti (2003), Pal et al. (2005), Mahapatra et al. (2005) and Maiti et al. (2006) developed different types of inventory problems and solved by simple GA. Recently, very few works have been done in the hierarchical GA. One may refer to the works of Garai and Chaudhuri (2003), Lai and Chang (2004) and Martikainen and Ovaska (2006). In Garai and Chaudhuri (2003), the entire search region is partitioned into a number of subregions. Then applying simple GA in each subregion, the solution of the optimization problem is obtained whereas Lai and Chang (2004) developed a clusteringbased approach using hierarchical GA to solve the automatic image segmentation problem. Recently, Martikainen and Ovaska (2006) proposed a hierarchical two-population genetic algorithm (2PGA). The 2PGA scheme constitutes of two differently sized population containing individuals of similar fitness values. Again, tournamenting has been used as a selection operator in the traditional GA. Recently, several research works have been done with tournament selection (Deb, 2000; Miettinen et al., 2003; Alvarenga and Mateus, 2004). Combining this idea and the concept of hierarchical GA, very recently, Kundu and Bhunia (2007) have developed a new methodology (called tournament genetic algorithm (TGA)). In this GA, tournamenting has been used to produce the better population from the existing two populations in the knockout fashion of well-known games (like cricket, football, etc.). In this paper, we have developed an EPL model for single deteriorating items considering (i) finite production rate, (ii) demand rate as a deterministic function of selling price and the marketing cost, (iii) linearly time dependent deterioration rate, (iv) selling price of an item determined by a mark-up over the unit production cost and (v) partially backlogged shortages. The backlogging rate is

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dependent on the waiting time up to the starting of next production. This model is formulated as a constrained optimization problem. Then the problem has been solved by TGA. Finally, the results are illustrated with the help of three numerical examples. Also, the effects of changes of different parameters are studied graphically on the average profit. 2. Assumptions and notations The following assumptions and notations are used in this paper. (i) The demand rate D(A,p) is a deterministic function of selling price p and marketing cost A per unit item, i.e., D(A,p) ¼ An(akp) where a, k, nX0. (ii) y(t) be the deterioration rate of the on-hand inventory at time t, where 0pyðtÞ51. (iii) The deteriorated units are neither repaired nor replaced during the entire cycle. (iv) Lead time is negligible. (v) The production rate is R (decision variable). (vi) The unit production cost f(R) is given by f(R) ¼ Cr+A+L/Ra+k1Rg; a, g, k140, where Cr, L are the raw material and labour cost, respectively. (vii) The inventory carrying cost Ch per unit per unit time, backordering cost Cb per unit per unit time, penalty cost Cp per unit and the set-up cost C0 per cycle are known and constant. (viii) Selling price p per unit item is determined by a mark-up over the unit production cost f(R), i.e., p ¼ mf(R), m41 where m is the mark-up rate. (ix) Shortages, if any, are partially backlogged. During the shortage period, the backlogging rate is dependent on the length of the waiting time up to the starting of production. To consider this situation, the backlogging rate is defined as [1+d(t3t)]1 where d being the parameter of backlogging rate.

3. Mathematical formulation It is assumed that the amount of stock is zero initially. Just after t ¼ 0, production starts and the items are produced at a rate R units per unit time. Customer demands are met directly and the excess amount is stored. As a result, the stock attains a level S at t ¼ t1 and then the production is stopped. After t ¼ t1, the inventory level gradually decreases to meet up the customer’s demand and partly for deterioration up to t ¼ t2. By this process, the inventory level reaches to zero at t ¼ t2, thereafter, shortages are allowed to occur during the interval (t2, t3) and all the demand are backlogged with backorder rate [1+d{t3tq(t)/R}]1 during the period (t2, t3). At time t ¼ t3, the shortages accumulate to the level S1. At this instant of time, fresh production and supply to customers, start to clear the entire backlogged quantity and to meet up the current demand. At the end of the time t ¼ T, the inventory level is zero. The entire cycle is repeated after the time period T.

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Under the above assumptions, the differential equations governing the instantaneous state of inventory level q(t) during the interval 0ptpT are as follows:

During the period (t2, T), there is no deterioration of items, it occurs only during the period (0, t2). Hence, the total number of deteriorated units over (0, t2) is given by

q0 ðtÞ þ yðtÞqðtÞ ¼ R  DðA; pÞ;

D0 ¼ Rt1  DðA; pÞt 2

(16)

Now, the inventory carrying cost is Z t 1  Z t2 Ch qðtÞ dt þ qðtÞ dt

(17)

q0 ðtÞ þ yðtÞqðtÞ ¼ DðA; pÞ;

0ptpt 1

(1)

t 1 ptpt 2

(2) t 2 ptpt 3

q0 ðtÞ ¼ DðA; pÞ=½1 þ dðt 3  t  qðtÞ=RÞ;

t 3 ptpT

q0 ðtÞ ¼ R  DðA; pÞ=½1  dqðtÞ=R;

(3) (4)

with qðtÞ ¼ 0 at t ¼ 0; t 2 and T

(5)

(6)

Furthermore; qðtÞ ¼ S at t ¼ t 1 and qðtÞ ¼ S1 at t ¼ t 3

(7)

The solutions of the differential Eqs. (1)–(2) with the conditions q(t) ¼ 0 at t ¼ 0 and t ¼ t2 are as follows:  Z t  qðtÞ ¼ fR  DðA; pÞg exp  yðuÞ du 0 Z t  Z t exp yðuÞ du dt; 0ptpt1 (8)  0 t

Z yðuÞ du

t2

Z exp

t

0

t

 yðuÞ du dt

0

t 1 ptpt 2

(9)

However, the solutions of the differential Eqs. (3)–(4) with the conditions q(t) ¼ 0 at t ¼ t2 and t ¼ T are given by

t3 

Z

1

t2

þ dqðtÞ ¼ Rf1 þ dðt 3  tÞg  DðA; pÞ;

t 2 ptpt 3

(10)

qðtÞ  DðA; pÞ log jfR  DðA; pÞ  dqðtÞg =fR  DðA; pÞgj=d ¼ Rðt  TÞ;

t 3 ptpT

(11)

Now using q(t) ¼ S at t ¼ t1, we have  Z t1  S ¼ fR  DðA; pÞg exp  yðuÞ du 0 Z t   Z t1  Z t1 exp yðuÞ du dt ¼ DðA; pÞ exp  yðuÞ du  0 0 0 Z t  Z t2 exp yðuÞ du dt (12)  0

Thus t1 and t2 are related by the equation Z t 1 Z t   Z t 2 Z t   R exp yðuÞ du dt ¼DðA; pÞ exp yðuÞ du dt 0

0

0

(13) Now, from (7) and (10), we have t 2 ¼ ½Rð1 þ dt 3 Þ  DðA; pÞ  fR  DðA; pÞ þ dS1 g expfdS1 =DðA; pÞg=ðdRÞ

(14)

Again, from (7) and (11), we have t 3 ¼ T  ½S1 þ DðA; pÞ log jfR  DðA; pÞ þ dS1 g =fR  DðA; pÞgj=d=R

 1 DðA; pÞ dt 1 þ dft 3  t  qðtÞ=Rg

(18)

Hence, the net profit (X) for the entire system is given by X ¼ /sales revenueS/production costS/carrying costS/backordering and penalty costS/set up costS i.e., X ¼ pft 2 DðA; pÞ þ RðT  t 3 Þg  Rf ðRÞðt 1 þ T  t 3 Þ Z t 1  Z T Z t2  Ch qðtÞdt þ qðtÞ dt þ C b qðtÞ dt  Cp

Z

0 t3 

t2

t1

 1 DðA; pÞ dt  C 0 1 1 þ dft 3  t  qðtÞ=Rg (19)

Therefore, the profit function p(m, A, R, t1, T, S1) (average profit per unit time for entire cycle) of inventory system is given by

pðm; A; R; t1 ; T; S1 Þ ¼ X=T

(20)

Hence, our problem is

½Rf1 þ dðt 3  t 2 Þg  DðA; pÞ expfdqðtÞ=DðA; pÞg

0

þ Cp

t2

 Z ¼ DðA; pÞ exp 

t1

t1

The sum of backordering and penalty costs is Z T Cb ½qðtÞ dt t2

Also, q(t) is continuous at t ¼ t1 and t ¼ t3.

0

0

(15)

9 Maximize pðm; A; R; t 1 ; T; S1 Þ > > > > > subject to the constraint > > > > = given by Eq: ð13Þ and R4DðA; pÞ; T4t 3 ; t 1 ot 2 ; m41 and A; R; t 1 ; T; S1 40

> > > > > > > > > ;

(21)

4. Solution procedure The above problem (21) is a constrained optimization problem for general form of y(t). It can only be solved for the specified form of y(t). But, a closed form solution cannot be obtained for m, A, R, t1, T and S1. Only a suitable numerical method is to be used to find an optimal solution. Now, we wish to determine the values of m, A, R, t1, T and S1 along with average profit for a scheduling period assuming the deterioration rate y(t) as linearly increasing function of time t, i.e., y(t) ¼ h+gt, 0ph, g51, t40. During the early stage of the inventory, intensity of deterioration is very low because t is small. However, the intensity increases with time t, but y(t) remains bounded for tb1 since g51. Substituting the expression for y(t) in (21), we obtain a non-linear constrained optimization problem of m, A, R, t1, T and S1. When the deterioration rate, y(t) is linearly time dependent decreasing function

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i.e., y(t) ¼ hgt, 0ph, g51, t40, the problem (21) can be solved similarly as for y(t) ¼ h+gt. Since the objective function in (21) is in the ratio form, i.e., p(m, A, R, t1, T, S1) ¼ X/T where X is the function of m, A, R, t1, T, S1 and the constraint (13) is highly nonlinear, the problem (21) is a non-concave maximization problem. Now, for simplicity, using the Taylor’s series expansion for the exponential function, one can easily express t1 in terms of t2, and then the problem (21) reduces to

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while t3 and T are related by the same Eq. (15) and the average profit reduces to

pðm; A; R; T; S1 Þ ¼ ½ pft 2 DðA; pÞ þ RðT  t 3 Þg  Rf ðRÞðt 1 þ T  t 3 Þ Z T  C h fR  DðA; pÞgt 22 =2R þ C b qðtÞ dt Z

t3 

t2

1 þ Cp 1 1 þ dft 3  t  qðtÞ=Rg t2   DðA; pÞ dt  C 0 =T

 (28)

)

Maximize pðm; A; R; T; S1 Þ

(22)

R4DðA; pÞ; T4t 3 ; m41 and A; R; T; S1 40

Clearly, the profit function pðm; A; R; t 1 ; T; S1 Þ is not a concave or pseudo-concave function for R4DðA; pÞ; T4t 3, m41 and A; R; S1 40. So, to obtain the global or close-toglobal optimal solution, gradient-based methods cannot be applied.

6. Implementation of TGA Now, we shall develop a TGA for solving the maximization problem (22) and the problems for special cases. This TGA combines the advanced GA with the tournamenting.

5. Special cases 6.1. Tournamenting Now, let us investigate some particular cases.

(i) When d ¼ 0 which means that the model reduces to the model with fully backlogged shortages. In this case, t1 and t2 are related by same Eq. (13) while t2, t3 and T are related by the equation Rðt 3  t 2 Þ ¼ fR  DðA; pÞgðT  t 2 Þ

(23)

and the average profit reduces to

pðm; A; R; T; S1 Þ ¼ ½ pfct2 DðA; pÞ þ RðT  t 3 Þg  Rf ðRÞðt1 þ T  t 3 Þ Z t1 Z  Ch qðtÞ dt þ 0

t2

qðtÞ dt



t

1  þ C b DðA; pÞðT  t 2 Þðt 3  t 2 Þ=2  C 0 =T

(24)

(ii) When shortages do not occur, this means that the problem reduces to without shortage problem. In this case, the problem reduces to ) Maximize pðm; A; R; t 2 Þ (25) and R4DðA; pÞ; A; R; t 2 40; m41 where

pðm; A; R; t2 Þ

 ¼ pt 2 DðA; pÞ  Rf ðRÞt 1 Z t1 Z  Ch qðtÞdt þ 0

t2

qðtÞdt



 C0

 t2

(26)

t1

(iii) If there is no deterioration effect, i.e., y(t) ¼ 0. In this case, t1 and t2 are related by the equation Rt1 ¼ t 2 DðA; pÞ

(27)

Normally, GA is applied to a number of times with different initial populations for solving numerical optimization problems to find out the best individual among all the runs. For smaller search space, this methodology gives the better results. However, for larger search space and non-convex/non-concave problem, there arises a difficulty to achieve the goal. To overcome this difficulty, we have proposed TGA in which GA is applied through binary tournamenting to upgrade the population successively in the different round of the tournament. In this binary tournamenting process, two different populations (through initialization or from any round of tournament) are considered and improved these populations by implementing the advanced GA and taken 50% of the individuals from two improved populations. There may arise different options for consideration of 50% of individuals from two improved populations P01 and P02 corresponding to two populations P1 and P2 in order to obtain the initial population P12 for the next round. Different options are as follows: Option-1: The union of the best 50% of the improved populations P01 and P 02 . Option-2: Selection of best 50% from the union of two populations P01 and P 02 . Option-3: Formation of a new population by selecting a better individual on comparing two randomly chosen individuals from each population P 01 and P02 at a time. Option-4: Selection of population, which contains the best individual among two populations. Option-5: Selection of population, which has higher average fitness value. In our experiment, all the options have been considered. The structure of Option-1 of TGA (with four teams) is

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given below: procedure TGA of option-1: begin r’1; while (ro4) do initialize P(r); [P(r) represents the r-th population] apply GA on P(r); initialize P(r+1); apply GA on P(r+1); find the union of the best 50% of P(r) and P(r+1) and obtain the population P0 (r+1); apply GA on P0 (r+1); r’r+2; end while find the union of the best 50% of P0 (2) and P0 (4) and obtain the population P00 (4); apply GA on P00 (4); print the best found result; end

6.2. Genetic algorithm The different steps of advanced GA are described as follows: Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

Generate the initial population P ¼fV 1 ; V 2 ; . . . ; V N g, where Vi is the i-th individual. Evaluate the fitness F(Vi) for each individual V i ð1pipNÞ. Apply the crossover operator to generate child chromosomes Oc ¼ fV Nþ1 ; . . . ; V NþM g. Apply the mutation operator to generate child chromosomes Om ¼ fV NþMþ1 ; . . . ; V NþMþR g. Evaluate the fitness F(Vi) for each individual V i ð1pipM þ N þ RÞ. Select N individuals according to their fitness values. Formally, the new solution set is as follows:

Pnew ¼ fV i jV i 2 P [ Oc [ Om ; FðV i Þ4FðV j Þ 8V j eP new g Step 7: Step 8: Step 9:

ði ¼ 1; 2; . . . ; NÞ

If the termination condition is not satisfied, go to Step 3, otherwise, go to Step 8. Sort the population according to their fitness value. Stop.

To implement the above GA in the different round of tournamenting, the following basic components are considered:

 Genetic parameters.  Chromosome representation and initialization of po   

pulation. Fitness function. Selection operators. Genetic operators (crossover and mutation). Termination criterion.

GA is dependent on different parameters like population size (N), maximum number of generations (MGen), crossover rate (pc) and mutation rate (pm). About the population size, there is no hard and fast rule how large it would

be. If it is too large, it may invite some difficulties during the computation. Again, according to natural genetics, it is obvious that the crossover rate is always greater than that of mutation. Generally, crossover rate ranges from 0.6 to 0.9 whereas mutation rate from 0.05 to 0.20. To design an appropriate chromosome, representation of solution is an important task in the utilization of GA. There are different types of representations among which binary and real coding representations are popular. In binary coding representation each variable is represented as binary substrings with desired precision. In this case, the string length of an individual will be large and GA would perform poorly. In real coding representation, each chromosome vector is encoded as a vector of floating point number of same length as the solution vector. This type of representation is easier to handle and is capable of representing quite large domains (or case of unknown domains). A vector Vj ¼ (Vj1,Vj2,y,Vjn), j ¼ 1,2,y,N, is used to represent a chromosome, where n be the number of variables (genes) of the optimization problem. After selection of appropriate chromosome representation, the next step is to initialize the chromosomes, which will take part in the artificial genetic operations like natural genetics. For this purpose, each gene (component) of a chromosome is generated randomly within the bounds of the corresponding decision variable of the problem. This can be done by any continuous or discrete probability distribution corresponding to continuous or discrete decision variables respectively. In this work, we have used the uniform distribution. In this way, population size numbers of chromosomes V1,V2,y,VN are generated randomly. These chromosomes are considered as initial population in the first round of tournament. In other rounds, the initial population will be the selected population (improved) of the previous round by considering any one option of tournamenting discussed earlier. After getting a population of potential solution of the optimization problem through initialization or from any one generation of GA, we need to find out how good they are. For this purpose, we have to calculate the fitness value of each chromosome. In our work, the objective function due to the chromosome Vi is taken as the fitness value and is denoted by F(Vi). Selection process is an important operator in GA. The basis of this process is the well-known principle of evolution ‘‘Survival of the fittest’’ by Charles Darwin. The primary objective of the process is to emphasize good solutions and eliminate bad solutions from the population for the next generation. In our work, we have used steadystate selection where worst parents in each generation are replaced by an equal number of better offspring, keeping the population size as constant. The objective of crossover is to exchange the information between randomly selected parent chromosomes (individuals) by recombining their features. Actually, it operates on two or more parent chromosomes (individuals) at a time and generates offspring for the next generation by combining both parent chromosomes features. In this operation, expected M (the integer value of pc*N (* denotes the product), pc being the crossover rate) number of chromosomes will take part. This operation is

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performed in each generation of GA and it can be done in different ways. In our study, the well-known whole arithmetical crossover is applied. The different steps of this operation are as follows: Step 1: Find the integer value of pc*N and store it in M. Step 2: Select the chromosomes Vk and Vi randomly from the population for crossover. Step 3: Generate a random real number l in [0,1]. Step 4: Produce two offspring VN+j, VN+j+1 (j ¼ 1,2,y, M1) by V Nþj ¼ lV k þ ð1  lÞV i , V Nþjþ1 ¼ lV i þ ð1  lÞV k , Step 5: Repeat Steps 2–5 for M/2 times. The mutation operation introduces random variations into the population. Generally, it is applied to a single chromosome only and performed with lower probability. In this work, we shall use non-uniform mutation whose action is dependent on the age of population. If the gene (element) Vik of chromosome Vi is selected for this operation and if the domain of Vik is an interval [lk0, lk1] then the reduced value of Vik is given by 1

V 0ik ¼ V ik þ Dðt; lk  V ik Þ

if a random digit is 0

0 lk Þ

if a random digit is 1

¼ V ik  Dðt; V ik 

where k 2 f1; 2; . . . ; ng and D(t, y) returns a value in the range [0, y]. In our method, we have taken

t b Dðt; yÞ ¼ yr 1  , T where r is a random value in [0, 1], T and t represent maximum generation number and the current generation respectively and b is called the non-uniform mutation parameter which is constant. Regarding the termination condition of the algorithm any one condition of the following three is used: (i) the best individual does not improve over specified generations, (ii) the total improvement of the last certain number of best solutions is less than a pre assigned small positive number (iii) the number of generations reaches MGen. In our experiment, we have used the third condition as termination condition of the algorithm. 7. Numerical illustrations In this section, the computational results of our proposed approach TGA on a realistic production-inventory problem have been presented. This approach has been coded in C programming and implemented on a Pentium IV 3.0 GHz with 512 MB RAM PC in LINUX environment. In all the experiments, 40 independent runs

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have been performed for different sets of random numbers. In each experiment of 40 runs, the best value of the profit function has been taken. Here, the GA parameters used are as follows: Population size (N) ¼ 100, maximum generation (MGen) ¼ 70, probability of crossover (pc) ¼ 0.8, probability of mutation (pm) ¼ 0.15 and non-uniform mutation parameter (b) ¼ 4. To illustrate our developed model, we have considered three examples of production-inventory problem with shortage, without shortage and non-deterioration cases. For these examples, the values of the inventory parameters have not been selected from any case study, but these values considered here are feasible. Example-1. (With shortage case) Let Cr ¼ $70, L ¼ $1600, Ch ¼ $5, C0 ¼ $200, a ¼ 250, k ¼ 0.7, a ¼ 0.6, g ¼ 1.5, k1 ¼ 0.02, Cb ¼ $7, Cp ¼ $11, h ¼ 0.08, g ¼ 0.03 and n ¼ 0.1. To test the performance of TGA, we have solved the maximization problem (22) with y(t) ¼ h+gt, hgt and h (constant) for different options of tournamenting. In each case, 40 independent runs are performed for different sets of random numbers and the results with CPU time have been collected. After statistical analysis of these results, the following characteristics are formed out.

 Best objective function value.  Mean, median and standard deviation (S.D.).  Average CPU time. The results are displayed in Table 1. From Table 1, it is seen that the values of S.D. for all options are closed to zero. It means that the results are stable. Again, all the results except CPU time for all options are quite same. However, from the mathematical point of view, option-3 is best in cases of mean or median whereas in case of best objective value option-1 is best. On the other hand, the CPU time for option-3 is larger than others. Again, solving the same problem for option-1 with y(t) ¼ h+gt, hgt and h (constant), the best found values of m, R, t1, t2, t3, T, D(A,p) and the corresponding maximum average profit p for d ¼ 1.5 are obtained. Results are displayed in Table 2. From Table 2, it is observed that average profit for all the options are quite same. However, from the mathematical point of view, average profit of option-1 is best. Example-2. (Without shortage case) Let Cr ¼ $70, L ¼ $1600, Ch ¼ $5, C0 ¼ $200, a ¼ 250, k ¼ 0.7, a ¼ 0.6, g ¼ 1.5, k1 ¼ 0.02, h ¼ 0.08, g ¼ 0.03, n ¼ 0.1. Solving the problem (25) with variable and constant deterioration for option-1, the values of m, A, R, t1, t2, D(A,p) and the corresponding average profit p are obtained. Results are given in Table 3. Example-3. (Non-deterioration case) Let Cr ¼ $70, L ¼ $1600, Ch ¼ $5, C0 ¼ $200, a ¼ 250, k ¼ 0.7, a ¼ 0.6, g ¼ 1.5, k1 ¼ 0.02, h ¼ 0.08, n ¼ 0.1.

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Table 1 Comparison of results for different options of tournamenting and different values of y(t).

y(t)

Option

Mean

Median

S.D.

Best value

Average CPU time

h+gt

1 2 3 4 5

5246.426500538 5246.425704474 5246.426634865 5246.426431176 5246.426378194

5246.426669417 5246.426332408 5246.426670792 5246.426585829 5246.426593578

0.000398663 0.001817168 0.000315044 0.000527782 0.000585340

5246.426952425 5246.426942527 5246.426946835 5246.426902526 5246.426946298

8.93 8.94 11.35 9.01 8.98

hgt

1 2 3 4 5

5311.824831402 5311.823181997 5311.824799169 5311.824714401 5311.824725973

5311.824873709 5311.824416332 5311.824908464 5311.824916045 5311.824863835

0.000265128 0.003874614 0.000342647 0.000574133 0.000422308

5311.825144086 5311.825121831 5311.825129668 5311.825105147 5311.825143905

8.95 9.92 11.33 8.95 8.95

h

1 2 3 4 5

5268.115021174 5268.113864099 5268.115163421 5268.115032120 5268.114799479

5268.115184990 5268.114403015 5268.115242404 5268.115215653 5268.115003979

0.000398680 0.001571557 0.000247251 0.000435330 0.000677124

5268.115386390 5268.115407147 5268.115386144 5268.115368719 5268.115358559

9.06 9.08 11.55 9.13 9.08

Table 2 Results of shortage case for different y(t) for different options of tournamenting.

y(t)

Option

t1

t2

t3

T

M

R

A

D(A,p)

p

h+gt

1 2 3 4 5

0.3396 0.3396 0.3397 0.3396 0.3396

0.6399 0.6399 0.6401 0.6399 0.6400

0.6988 0.6989 0.6989 0.6987 0.6988

0.7582 0.7582 0.7583 0.7580 0.7582

1.4201 1.4201 1.4201 1.4201 1.4201

135.0932 135.0829 135.0788 135.0955 135.0862

8.0991 8.1002 8.1001 8.1020 8.1012

70.7075 70.7052 70.7080 70.7070 70.7083

5246.426952425 5246.426942527 5246.426946835 5246.426902526 5246.426946298

hgt

1 2 3 4 5

0.5181 0.5181 0.5178 0.5180 0.5180

0.9555 0.9554 0.9550 0.9554 0.9553

1.0056 1.0055 1.0052 1.0055 1.0054

1.0592 1.0590 1.0587 1.0590 1.0590

1.4163 1.4163 1.4163 1.4163 1.4163

133.0654 133.0639 133.0641 133.0627 133.0647

8.1575 8.1636 8.1575 8.1602 8.1574

71.2471 71.2519 71.2478 71.2493 71.2475

5311.825144086 5311.825121831 5311.825129668 5311.825105147 5311.825143905

h

1 2 3 4 5

0.3784 0.3783 0.3783 0.3784 0.3784

0.7094 0.7094 0.7093 0.7094 0.7094

0.7654 0.7654 0.7654 0.7655 0.7654

0.8227 0.8227 0.8227 0.8227 0.8227

1.4192 1.4192 1.4191 1.4192 1.4192

134.6075 134.6165 134.6176 134.6094 134.6096

8.1129 8.1239 8.1149 8.1171 8.1169

70.8414 70.8427 70.8475 70.8434 70.8449

5268.115386390 5268.115407147 5268.115386144 5268.115368719 5268.115358559

Table 3 Results of no-shortage case for different y(t).

y(t)

t1

t2

m

R

A

D(A,p)

p

h+gt hgt h

0.3657 0.5860 0.4088

0.6898 1.0698 0.7650

1.4218 1.4154 1.4201

134.8483 131.9832 134.1827

8.0468 8.1542 8.0924

70.4313 71.3441 70.6826

5201.803978540 5289.545360660 5231.164627472

Solving the corresponding problem with variable and constant deterioration for option-1, the values of m, A, R, t1, t2, t3, T, D(A,p) and the corresponding average profit p for different values of d are obtained. Results are given in Table 4.

8. Sensitivity analysis For the numerical Example-1 mentioned earlier, sensitivity analysis is carried out numerically for option-1 to study the effect of discrete changes of backlogging parameter (d) on the best found values of m, A, R, t1, t2,

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119

Table 4 Results of non-deterioration case.

d

t1

t2

t3

T

M

R

A

D(A,p)

p

1.5 2.5 3.5

0.7722 0.7776 0.7803

1.5002 1.5106 1.5158

1.5286 1.5281 1.5285

1.5575 1.5459 1.5414

1.4209 1.4209 1.4210

137.1491 137.1293 137.1223

8.1741 8.1730 8.1753

70.5990 70.5928 70.5882

5502.454560541 5500.618001931 5499.793603583

Table 5 Results of shortage case with different y(t) and d.

y(t)

d

t1

t2

t3

T

m

R

A

D(A,p)

p

h+gt

1.5 2.5 3.5

0.3396 0.3492 0.3536

0.6399 0.6580 0.6664

0.6988 0.6954 0.6938

0.7582 0.7329 0.7211

1.4201 1.4206 1.4208

135.0932 134.9917 134.9551

8.0991 8.0824 8.0724

70.7075 70.6229 70.5776

5246.426952425 5230.525000864 5222.983176745

hgt

1.5 2.5 3.5

0.5181 0.5411 0.5523

0.9555 0.9945 1.0134

1.0056 1.0258 1.0362

1.0592 1.0594 1.0607

1.4163 1.4160 1.4159

133.0654 132.6968 132.5214

8.1575 8.1555 8.1558

71.2471 71.2781 71.2915

5311.825144086 5303.441629145 5299.637352894

h

1.5 2.5 3.5

0.3784 0.3893 0.3944

0.7094 0.7294 0.7387

0.7654 0.7648 0.7646

0.8227 0.8009 0.7909

1.4192 1.4195 1.4197

134.6075 134.4631 134.3852

8.1129 8.1009 8.0954

70.8414 70.7860 70.7610

5268.115386390 5254.771730816 5248.510756353

1.200

0.180

Ch

g C0

Cb

0.600

0.090

0.000 -20

-10

0.000

0

10

20

-20

-10

0

-0.600

-0.090

-1.200

-0.180

Fig. 1. Effect of changes of parameters g and C0 (C0) on profit function for increasing rate of deterioration.

t3, T, and D(A,p) and the corresponding maximum average of p. For this purpose, results are computed performing 40 independent runs for three different values of d and displayed in Table 5. Again the same analysis is performed graphically for option-1 to study the effect of under or over estimation of parameters C0, Ch, Cb, Cp, h, g and d on profit function. This can be done by changing (increasing and decreasing) the parameters from 20% to +20%, taken one parameter at a time and making the remaining parameters at their original values. In each case, the best found values of 40 independent runs with different sets of random numbers are considered. Results are shown graphically in Figs. 1–6. Figs. 1–3 depict the effect of

10

20

Fig. 2. Effect of changes of parameters Ch (Ch), Cb (Cb) on profit function for increasing rate of deterioration.

changes of parameters on the average profit for increasing rate of deterioration whereas Figs. 4–6 indicate the effect of changes of same parameters for decreasing rate of deterioration. From all the figures, it is observed that in both cases of deterioration, average profit is insensitive with the changes of parameters g, Cb, Cp. On the other hand, the optimal profit is less sensitive with respect to the parameters C0, Ch, d and h. 9. Concluding remarks In this paper, we have developed an EPL model for deteriorating items incorporating partial backlogged

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0.540

1.000

h

h Cp Del

Cp 0.360

Del

0.500 0.180

0.000 -20

-10

0.000 0

10

20

-20

-10

0

10

20

-0.180

-0.500 -0.360

-0.540

-1.000

Fig. 3. Effect of changes of parameters h, delta (d), Cp (Cp) on profit function for increasing rate of deterioration.

Fig. 6. Effect of changes of parameters h, delta (d), Cp (Cp) on profit function for decreasing rate of deterioration.

0.800 g C0 0.400

0.000 -20

-10

0

10

20

-0.400

-0.800 Fig. 4. Effect of changes of parameters g and C0 (C0) on profit function for decreasing rate of deterioration.

0.300 Ch Cb

shortages with a rate dependent on the length of the waiting time up to the starting of production. The model is formulated as a constrained optimization problem and is solved by a hybrid algorithm TGA containing the advanced GA and tournamenting. The role of tournamenting is to stabilize the search process from the larger search space and to minimize the CPU time as well as the memory requirements for running the program in a computer. This TGA with eight team tournamenting is equivalent to the advanced GA with eight times the population size of each team of TGA. Again, another feature is that it takes less CPU time than the advanced GA with eight times population size as in each round of tournamenting, the number of teams or the total population size of all teams is half of the preceding round. For future research, one may develop TGA considering round robin league tournamenting or the combination of round robin league and knock-out tournamenting. Again, as the proposed production problem has been formulated in crisp environment, it can be modelled in stochastic, fuzzy, fuzzy–stochastic and interval environments taking demand, deterioration rate and/or inventory parameters to be imprecise.

0.150 Acknowledgements

0.000 -20

-10

0

10

20

-0.150

The authors wish to thank the anonymous referees for their constructive comments and suggestions for the improvement of this paper. The authors also would like to acknowledge the support of Major Research Project provided by the University Grants Commission, India, for conducting this research work. References

-0.300 Fig. 5. Effect of changes of parameters Ch (Ch), Cb (Cb) on profit function for decreasing rate of deterioration.

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