Genetic algorithms for solving a class of constrained nonlinear integer programs

Transcription

1 Intl. Trans. in Op. Res. 8 (2001) 61±74 Genetic algorithms for solving a class of constrained nonlinear integer programs Ruhul Sarker, Thomas Runarsson and Charles Newton School of Computer Science, University of New South Wales, ADFA, Northcott Drive, Canberra, ACT 2600, Australia ruhul@cs.adfa.edu.au Received 7 July 1999; received in revised form 6 April 2000; accepted 4 May 2000 Abstract We consider a class of constrained nonlinear integer programs, which arise in manufacturing batch-sizing problems with multiple raw materials. In this paper, we investigate the use of genetic algorithms (GAs) for solving these models. Both binary and real coded genetic algorithms with six different penalty functions are developed. The real coded genetic algorithm works well for all six penalty functions compared to binary coding. A new method to calculate the penalty coef cient is also discussed. Numerical examples are provided and computational experiences are discussed. Keywords: Nonlinear, integer, penalty function, genetic algorithms, batch sizing 1. Introduction Consider a constrained nonlinear programming problem. Problem P1: Minimize f (x) Subject to g(x) < 0 h(x) ˆ 0 x 2 X where g is a vector function with components g 1,..., g m and h is a vector function with components h 1,..., h k. Here f, g 1,..., g m, h 1,..., h k are functions on E n and X is a nonempty set in E n. The set X represents simple constraints that could be easily handled explicitly, such as lower and upper bounds on the variables. There are several existing conventional methods to solve nonlinear programming problems such as penalty function, barrier function, reduced gradient, and other methods. In the penalty and barrier # 2001 International Federation of Operational Research Societies. Published by Blackwell Publishers Ltd

2 62 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 function methods, the unconstrained subproblem becomes extremely ill-conditioned for extreme values of the penalty/barrier parameters. Reduced gradient methods have dif culties following the boundary of high nonlinear constraints. These methods rely on local gradient information; the optimum is the best in the neighborhood of the current point. To enhance the reliability and robustness of the search technique in constrained optimization, a GA-based search method incorporating several penalty functions is tested in this paper. We consider a manufacturing batch-sizing problem with multiple raw materials. A single raw material unconstrained manufacturing batch-sizing model was developed by Sarker and Khan (1997; 1999). Later, this model was modi ed by Sarker and Newton (2000) to incorporate the transportation module for nished products delivery. In this paper, we reformulated the problem of Sarker and Khan (1997; 1999) to accommodate multiple raw materials and impose constraints to make the situation more realistic. The formulation of the new problem comes from a class of constrained nonlinear integer programming problems. These are known as hard problems in the operational research literature and there is no simple and easy way to implement solution techniques for these problems. Although there are several heuristics and line-search based methods for solving unconstrained joint batch-sizing problems, no specialized algorithm has appeared for constrained problems. However, the constrained problems can be solved using the theory of nonlinear programs, though these are not easy to implement. Our rst attempt was to solve a simple representative problem using a commercial optimization package LINDO/LINGO. The package provides suboptimal solutions for some known problems (Sarker and Newton, 1999). This encouraged us to explore the use of other methods, for example evolutionary algorithms (EAs). In this research, our purpose is to explore the use of GAs for solving this class of constrained nonlinear integer programming problems. We choose a GA as a heuristic because it is well known for its success and ease of implementation. We also use evolution strategies (ES) for the same problem but that has not been reported in this paper because of its poor performance (Runarsson and Sarker, 1999). Earlier we have used simulated annealing (SA) for unconstrained problems (Sarker and Yao, forthcoming). Like the conventional optimization approach, the GAs used the penalty-function method to convert the constrained problems into unconstrained problems. There are a number of methods available in the literature which have not been investigated to nd the appropriate penalty function and the relevant parameters for a class of models considered in this paper. Six penalty-function based GAs are investigated to nd their suitability for solving the problems. We also discuss a new method, which is currently under investigation, to calculate an appropriate penalty coef cient. We use both binary coding and real coded genetic algorithms with different crossovers and mutations. Usually the binary coding is recognized as the most suitable encoding for any problem solution because it maximizes the number of schemata being searched implicitly (Holland, 1975; Goldberg, 1989), but there have been many examples in the evolutionary computation literature where alternative representations have resulted in algorithms with greater ef ciency and optimization effectiveness when applied to identical problems (see for example articles by Back and Schwefel (1993) and Fogel and Stayton (1994)). Davis (1991) and Michalewicz (1996) comment that for many applications real values or other representations may be chosen to advantage over binary coding. There does not appear to be any general bene t in maximizing implicit parallelism in evolutionary algorithms, and, therefore, forcing problems to t into a binary representation may not be recom-

3 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 63 mended. In our case, the real coded genetic algorithm works better than binary coding. The detailed computational experiences are presented. The organization of the paper is as follows: following this introduction, this paper presents a brief introduction on penalty function methods in Section 2, and then on genetic algorithms, in Section 3. In Section 4, the batch-sizing problem is described, and the mathematical formulation of the problem is presented in Section 5. In Section 6, the different transformation methods are provided. The modelsolving approach using GAs is implemented in Section 7. The computational experiences are presented in Section 8. Finally, conclusions are provided in Section Penalty function method The penalty function method converts the problem into an equivalent unconstrained problem and then solves it using a suitable search algorithm. Two basic types of penalty functions exist: exterior penalty functions, which penalize infeasible solutions, and interior penalty functions, which penalize feasible solutions. We will discuss only the exterior penalty functions in this paper as the implementation of interior penalty functions is considerably more complex for multiple constraint cases (Smith and Coit, 1997). Three degrees of exterior penalty functions exist: (i) barrier methods in which no infeasible solution is considered; (ii) partial penalty functions in which a penalty is applied near the feasibility boundary; (iii) global penalty functions that are applied throughout the infeasible region (Schwefel, 1995). In general, a penalty function approach places the constraints into the objective function via a penalty parameter in such a way that it penalizes any violation of the constraints. A general form of the unconstrained problem is as follows: Problem P2: Minimize f p (x) ˆ f (x) ìá(x) Subject to x 2 X where ì. 0 is a large number, and á(x) is the penalty function. We de ne the problem P2 as a penalty problem and the objective function of P2 as the penalized objective function ( f p (x)) which is the simple sum of the unpenalized objective function and a penalty (for a minimization problem). A suitable penalty function incurs a positive penalty for infeasible points and no penalty for feasible points. The solution to the penalty problem can be made arbitrarily close to the optimal solution of the original problem by choosing ì suf ciently large. However, if we choose a very large ì and attempt to solve the penalty problem, we may get into some computational dif culties of ill-conditioning (Bazaraa and Shetty, 1979). With a large ì, more emphasis is placed on feasibility, and most procedures for unconstrained optimization will move quickly toward a feasible point. Even though this point may be far from optimal, premature termination could occur. As a result of the above dif culties associated with large penalty parameters, most algorithms use penalty functions that employ a sequence of increasing penalty parameters. With each new value of the penalty parameter, an optimization

4 64 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 technique is employed, starting with the optimal solution corresponding to the previously chosen parameter value. It is very dif cult to choose an appropriate penalty coef cient (Michalewicz, 1995). A lower value of penalty coef cient means a higher number of iterations is required to solve the penalty problem. Several methods for selecting a penalty coef cient are discussed later. 3. Introduction to GA During the last two decades there has been a growing interest in algorithms which are based on a principle of evolutionðsurvival of the ttest. A common term, accepted recently, refers to such techniques as evolutionary computation (EC) methods. The best known algorithms in this class include genetic algorithms, evolutionary programming, evolution strategies, and genetic programming. There are also many hybrid systems which incorporate various features of the above paradigms, and consequently are hard to classify; they are generally referred to as EC methods (Khouja et al., 1998). The methods of EC are stochastic algorithms whose search methods model some natural phenomena: genetic inheritance and Darwinian strife for survival. GAs follow a step-by-step procedure that mimics the process of natural evolution, following the principles of natural selection and `survival of the ttest'. In these algorithms a population of individuals (potential solutions) undergoes a sequence of unary (mutation-type) and higher order (crossover-type) transformations. These individuals strive for survival. A selection scheme, biased towards tter individuals, selects the next generation. This new generation contains a higher proportion of the characteristics possessed by the `good' members of the previous generation, and in this way good characteristics are spread over the population and mixed with other good characteristics. After a number of generations, the program converges and the best individuals represent a near-optimum solution. The GAs procedure is shown below. begin t 0 initialize Population (t) evaluate Population (t) while (not terminate-condition) do begin t t 1 select Population (t) from Population (t 1) variate Population (t) evaluate Population (t) end end 4. Batch-sizing problem (BSP) Consider a batch manufacturing environment that processes raw materials procured from outside suppliers to convert them into nished products for retailers. The manufacturing batch size is

5 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 65 dependent on the retailer's sales volume (/market demand), unit product cost, set-up cost, and inventory holding cost. The raw-material purchasing lot size is dependent upon the raw-material requirement in the manufacturing system, unit raw-material cost, ordering cost and inventory holding cost. Therefore, the optimal raw-material purchasing quantity may not be equal to the raw material requirement for an optimal manufacturing batch size. To operate the manufacturing system optimally, it is necessary to optimize the activities of both raw-material purchasing and production batch sizing simultaneously, taking all operating parameters into consideration. A larger manufacturing batch size reduces the set-up cost component to the overall unit product cost. The products produced in one batch (/one manufacturing cycle) are delivered to the retailer in m small lots (that is, m retailer cycles per one manufacturing cycle) at xed time intervals. This delivery system is common in a batch manufacturing environment (Sarker and Parija, 1994; 1996). In the pharmaceuticals and chemicals industries, the critical products are not delivered to the retailers until the whole lot is nished and quality certi cation is ready (Sarker and Khan, 1999; Sarker and Newton, 2000). The inventory level of such products increases linearly at the production rate during the production up-time, and it forms a staircase pattern during production down-time in each inventory cycle. The manufacturer procures raw materials from outside suppliers in a lot of xed quantity at an equal interval of time. In this case, the manufacturer has a number of options on how much to receive in each lot as compared to the manufacturing batch size (see Sarker and Khan, 1999). In this paper, we consider that a lot of raw material will be consumed in n manufacturing cycles. This is a typical case of having a higher ordering cost relative to an inventory holding cost. 5. Mathematical formulation The unconstrained models under various situations are presented in Sarker and Khan (1997; 1999) and Sarker and Newton (2000), and the single raw material constrained model has recently been developed by Sarker and Newton (1999). In this paper, we extend the single raw material model into a multiple raw materials constrained model. The notations used in developing the model are as follows: D p ˆ demand rate of a product p, units per year P p ˆ production rate, units per year (here, P p. D p ) Q p ˆ production lot size H p ˆ annual inventory holding cost, $=unit=year A p ˆ set-up cost for a product p ($=setup) r i ˆ amount/quantity of raw material i required in producing one unit of a product D i ˆ demand of raw material i for the product p in a year, D i ˆ r i D p Q i ˆ ordering quantity of raw material i A i ˆ ordering cost of a raw material i H i ˆ annual inventory holding cost for raw material i PR i ˆ price of raw material i

6 66 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 Q i ˆ optimum ordering quantity of raw material i x p ˆ shipment quantity to customer at a regular interval (units/shipment) L ˆ time between successive shipments ˆ x p =D p T ˆ cycle time measured in years ˆ Q p =D p m p ˆ number of full shipments during the cycle time ˆ T=L n i ˆ number of production cycles which consume one lot of raw material i ˆ Q i =Q p s i ˆ space required by one unit of raw material i S p ˆ space required by one unit of nished product TC ˆ total cost of the system The mathematical model is presented below: Minimize TC ˆ Dp A p X! A i m px p m p x p n i i 2 D p 1 H p X P p i r i H i! D p n i 1 P p x p 2 H p Subject to: X n i r i m p x p s i < Raw_s_Cap i X n i r i m p x p s i > Min_Truck_Load i m p x p s p < Finish_S_Cap m p and n i are integers and greater than zero. This is clearly a nonlinear-integer program. The rst constraint indicates that the storage space required by the raw materials must be less than the space speci ed for raw materials. The second constraint represents the lower limit truckload for transportation and the third constraint is for the nished products storage capacity. 6. Transformation methods The transformation method, from a constrained to an unconstrained problem, in evolutionary algorithms is almost the same as the classical approach but with a slight difference. The value of ì k is changed as a function of generation k, but not equal to ì k 1. Several selected transformation techniques used in the evolutionary computation literature for solving constrained optimization problems are presented below. Most, if not all, are of the exterior kind which will allow the initial population of solutions to be partially or completely infeasible. In some of these techniques, the controlling parameters are updated in a predetermined manner. The controlling (or penalty) parameters ì k may be updated each generation or every n generations Static penalties The method of static penalties (Homaifar et al., 1994) assumes that for every constraint we establish a family of intervals that determine the appropriate penalty coef cient. It is clear that the results are parameter dependent. A limited set of experiments reported by Michalewicz (1995) indicate that the

7 method can provide good results if violation levels and penalty coef cients are tuned to the problem. We use three levels of xed penalty coef cients for each constraint: 0.5, 0: and 0: Dynamic penalties Joines and Houck (1994) proposed dynamic penalties. The authors assumed that ì k ˆ (Ck) á, where C and á are constants. A reasonable choice for these parameters is C ˆ 0:5, and á ˆ 2. This method requires a much smaller number (independent of the number of constraints) of parameters than the rst method. Also, instead of de ning several levels of violation, the pressure on infeasible solutions is increased due to the (Ck) á component of the penalty term: towards the end of the process (for high values of the generation number k) this component assumes large values Annealing penalties The method of annealing penalties, called Genocop II (for Genetic Algorithms for Numerical Optimization of Constrained Problems) is also based on dynamic penalties and was described by Michalewicz and Attia (1994) and Michalewicz (1996). In this system, a xed ì k is used for all constraints of a given generation k, where ì k ˆ 1=2ô. An initial temperature ô is set and is used to evaluate the individuals. After a certain number of generations, the temperature ô is decreased and the best solution found so far serves as a starting point for the next iteration. This process continues until the temperature reaches freezing point. Genocop II proved its usefulness for linearly constrained optimization problems; it gave a surprisingly good performance for many functions (Michalewicz et al., 1994; Michalewicz, 1996). Michalewicz and Attia (1994) proposed the control parameters to be ô k 1 ˆ 10:ô k, where ô 0 ˆ 1 and ô k remains constant once it reaches Adaptive penalties Adaptive transformation attempts to use the information from the search to adjust the control parameters. This is usually done by examining the tness of feasible and infeasible members in the current population. Bean and Hadi-Alouane proposed an adaptive penalty method in 1992 which uses the feedback from the search process (see Michalewicz and Schoenauer, 1996). This method allows either an increase or a decrease of the imposed penalty during evolution as shown below. This involves the selection of two constants, â 1 and â 2 (â 1. â 2. 1), to adaptively update the penalty function multiplier, and the evaluation of the feasibility of the best solution over successive intervals of N f generations. As the search progresses, the penalty function multiplier is updated every N f generations based on whether or not the best solution was feasible during that interval. The parameters were chosen to be similar to the above methods with N f ˆ 3, â 1 ˆ 5, â 2 ˆ 10 and ì 0 ˆ 1. In our experiments, we always increased the ì values by Death penalty R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 67 The death-penalty method just rejects infeasible individuals. It can be interpreted as a truncation selection based on á(x) and then followed by a selection procedure based on f (x). In this method, the

8 68 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 initial population must be feasible. In the experiments reported by Michalewicz (1995), it was shown that the method generally gives a poor performance. Also, the method is not as stable as others; the standard deviation of solutions returned is relatively high Superiority of feasible points The method of superiority of feasible points was developed by Powell and Skolnick (1993) and is based on a classical penalty approach, with one notable exception. Each individual is evaluated by the formula: f p (x) ˆ f (x) ì[á(x) è(t, x)], where ì is a constant; however, the component è(t, x) isan additional iteration-dependent function that in uences the evaluation of infeasible solutions. The point is that the method distinguishes between feasible and infeasible individuals by adopting an additional heuristic rule: For any feasible individual x and any infeasible individual y, f p (x), f p (y) (i.e., any feasible solution is better than any infeasible one). The penalties are increased for infeasible individuals. The method performs reasonably well. However, for some case problems, the method may have some dif culty in locating a feasible solution (Michalewicz, 1995). 7. Solving BSP using genetic algorithms Two types of genetic algorithms were tested, for each of the methods discussed in the previous section: a simple genetic algorithm (SGA) and a real coded genetic algorithm (RGA). The basic components for these two algorithms are discussed brie y below. The rst hurdle to overcome in using GAs is problem representation. The representation often relies on binary coding. GAs work with a population of competing strings, each of which represents a potential solution for the problem under investigation. The individual strings within the population are gradually transformed using biological-based operations. In accordance with the law of the survival of the ttest, the best-performing individual in the population will eventually dominate the population. We also use real coding where the variable values are generated as real numbers. The next step is to initialize the population. There are no strict rules for determining the population size. Larger populations ensure greater diversity but require more computing resources. We use a population size of 50, as commonly used by many researchers. Once the population size is chosen, then the initial population must be randomly generated. For binary coding, the random number generator generates random bits (0 or 1). Each individual in the population is a string of n bits. For real coding, the actual numbers are generated randomly. Now that we have a population of potential problem solutions, we need to see how good they are. Therefore, we calculate a tness, or performance, for each string. The tness function is the penalty objective ( f p ) function in our case. For binary coding, each string is decoded into its decimal equivalent. This gives us a candidate value for the solution. This candidate value is used to calculate the tness value. For real coding, we use the variable values directly. Genetic algorithms work with a population of competing problem solutions that gradually evolve over successive generations. We use a binary tournament selection, where two individuals are selected at random from the population pool and the better one is allowed to survive to the next generation. The chromosomes which survive the selection step undergo genetic operations, crossover and mutation. Crossover is the step that really powers the GA. It allows the search to fan out in diverse directions

9 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 69 looking for attractive solutions and permits two strings to `mate'. This may result in offspring that are ` tter' than their parents. We use two-point crossover as it is the general choice in many binary coding genetic algorithms because it minimizes disruptive effects (for more on disruptive effects see Booker (1997)). We use a heuristic crossover for real coding GAs as suggested by Fogel (1997) and Michalewicz (1996). In this crossover, the operator generates a single offspring x 3 from two parents x 1 and x 2 according to the following rule: x 3 ˆ rounded[r:(x 2 x 1 ) x 2 ], where r is a random number between 0 and 1, and the parent x 2 is no worse than x 1. Actually, when the generated value x 3 is out of bounds (upper and lower bounds), it is set equal to the corresponding violated bound. Also, note that x 1 and x 2 are `parent variables' not parents, and x 3 is rounded after crossing to integer. This is not required for binary coding. Rounding is also performed after the mutation operator. Mutation introduces random deviations into the population. For binary coding, mutation changes a 0 into a 1 and vice versa. Mutation is usually performed with low probability, otherwise it would defeat the order building being generated through selection and crossover. Mutation attempts to bump the population gently into a slightly better course. We used Nonuniform mutation for real coding GA. In this mutation, we set the system parameter b (ˆ 6r 1) that determines the degree of non-uniformity (for more see, Michalewicz et al., 1994). All GA runs have the following standard characteristics: Probability of crossover: 1.0 Probability of mutation: 1/(string length of the chromosome) Population size: 50 Number of generations in each run: 200 Number of independent runs: 300 for SGA: binary coding, two point crossover, and bit-wise mutation for RGA: heuristic crossover and nonuniform mutations. 8. Computational results The mathematical model presented in an earlier section is solved for a test problem of a single product with three raw materials using six different penalty functions based on binary and real coded GAs described earlier. The data used for the model are as follows: Sample Data: D p ˆ ; P p ˆ ; A p ˆ $50:00; H p ˆ $1:20, x P ˆ 1,000 units, S p ˆ 1:00: Raw material 1 Raw material 2 Raw material 3 A i 3,000 2,500 4,600 H i s i r i With the RHS values: Raw_S_Cap ˆ 400,000, Truck_Min_load ˆ 200,000 and Fin_S_Cap

10 70 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 Table 1 Results with real coded GAs Method Minimum Mean Std. Dev. Median Maximum Optimal (%) Death penalty Superiority of feasible points Dynamic penalty Annealing penalty Static penalty Adaptive penalty Table 2 Results with binary coded GAs Method Minimum Mean Std. Dev. Median Maximum Optimal (%) Death penalty Superiority of feasible points Dynamic penalty Annealing penalty Static penalty Adaptive penalty ˆ 20,000, the best solution found was: m ˆ 15, n 1 ˆ 15, n 2 ˆ 19, and n 3 ˆ 19 with TC ˆ 3:3471E5. The comparison for the six different penalty functions is shown in Table 1 for real coded GAs and in Table 2 for binary coded GAs. The minimum, presented in column 2 of both tables, means the best objective function value obtained in any of 300 runs and maximum, presented in column 6, indicates the worst value. The statistics (like mean, standard deviation and median) of the objective function values based on 300 runs are also produced to sense the variability of the target value. The far righthand columns represent the percentage of times we hit the optimal solution out of 300 runs. With tighter RHS values: Raw_S_Cap ˆ 130,000, Truck_Min_load ˆ 250,000 and Fin_S_Cap ˆ 13,000, the best solution found was: m ˆ 12, n 1 ˆ 8, n 2 ˆ 11, and n 3 ˆ 12 with TC ˆ 4:2837E5. The comparison for the six different penalty functions is shown in Table 3 for real coded GAs and in Table 4 for binary coded GAs. The results, in terms of percentage of times the optimal solution was obtained, are always better with real coded GAs as compared to binary coded GAs for all six penalty functions considered in this paper. The results are worse with tighter constrained problems. All the penalty functions, except superiority of feasible points, worked extremely well with real coded GAs for the rst test problem. However, the death-penalty method along with superiority of feasible points performed badly for tighter constrained problems, as shown in Table 3. However, all six penalty functions, with real coding GAs, provide unique optimal solutions for both test problems. These solutions are acceptable irrespective of how many times the optimal solutions are generated in a single run. The binary coded GAs failed to produce any optimal solution for the tighter test problem.

11 Table 3 Results with real coded GAs R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 71 Method Minimum Mean Std. Dev. Median Maximum Optimal (%) Death penalty Superiority of feasible points Dynamic penalty Annealing penalty Static penalty Adaptive penalty Table 4 Results with binary coded GAs Method Minimum Mean Std. Dev. Median Maximum Optimal (%) Death penalty Superiority of feasible points Dynamic penalty Annealing penalty Static penalty Adaptive penalty The results of ve test problems are plotted in Figure 1. Problem 1 is a problem with loose constraints, and problem 5 is the problem with tightest constraints. Problems 2 to 4 are in between problems 1 and 5. The performance of superiority of feasible points and death penalty methods are consistently poor compared to other methods. Although the performances of all methods decrease with tighter % Optimum solution obtained P1 P2 P3 P4 P5 Problem number Death penalty Sup./feasible points Dynamic penalty Annealing penalty Static penalty Adaptive penalty Fig. 1. Comparison of performances for six different methods.

12 72 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 constraints, the performances of four methods (static, dynamic, annealing, and adaptive methods) are very close for all cases Calculating penalty coef cient Consider the commonly-used binary tournament selection. In binary tournament selection two individuals are chosen randomly from the population and the better of them is allowed to survive to the next generation. Consider this comparison between two individuals 1 and 2 transformed to take the general form from problem P2: f 1 (x) ìá 1 (x) > f 2 (x) ìá 2 (x), for a given penalty parameter ì. 0. From this equation, we can write ì < [ f 1 (x) f 2 (x))=(á 2 (x) á 1 (x)], or ì c ˆ [ f 1 (x) f 2 (x))=(á 2 (x) á 1 (x)]. Evolutionary algorithms are a population-based search method. Population information can be used to estimate the most appropriate value for ì. This value would be the one that attempts to balance the number of competitions decided by the objective and penalty function. This value may be approximated by simulating the binary tournament selection and computing for each comparison the critical penalty coef cient ì c. Then for the actual selection the ì used will be the average of these values. The results of the two test problems using the new way of calculating penalty coef cient are presented in Table 5. As we can see, the proposed method produced optimal solutions in 100% of cases for test problem 1 and 68% of cases for a tighter problem like test problem 2. Although this approach produces better solutions, in terms of the percentage of optimum solutions obtained, than any existing penalty functions based GA, it requires extensive experimentation to claim its better performance. 9. Conclusions We considered a single product±multiple raw materials joint batch-sizing problem. The mathematical programming model of this problem forms a constrained nonlinear integer program. It is dif cult to solve such a complex model using the existing algorithms. In this paper, we investigate the use of genetic algorithms (GAs) for solving such a model with one product and three raw materials. We used both binary and real coded genetic algorithms with six different penalty functions reported in the literature. We also reported a new algorithm which we developed recently. Two test problems were solved to demonstrate the use of GAs in solving BSP. The results were compared and analysed for all seven penalty functions with binary and real coded GAs. The real coded genetic algorithm works well Table 5 Performance of the proposed method Problem and Code Minimum Mean Std. Dev. Median Maximum Opt. soln. in % times P1 Real P1 Binary P2 Real P2 Binary

13 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 73 compared to binary coding. The new method works very well compared to the existing methods. Among the penalty functions, the death-penalty method and superiority of feasible points were found to be the worst. More experimentation is required to ensure the performance of the new method of calculating penalty coef cient. It also needs to test other types of constrained optimization problems such as models with continuous variables and mixed integer models. Some work should be performed to accelerate the convergence of the algorithm. Optimal parameter selection in GA for a class of problems would be very interesting work. Developing a set of benchmark problems considering all extreme cases to judge the performance of penalty functions based GAs would provide a challenge for the future. Acknowledgement This work is supported by UC special research grant, ADFA, University of New South Wales, Australia, awarded to Dr Ruhul Sarker. We thank Professor Xin Yao for many useful comments during the conduct of this research. References Back, T., Schwefel, H-P An Overview of Evolutionary Algorithms for Parameter Optimization, Evolutionary Computation, 1, 1±24. Bazaraa, M.S., Shetty, C.M Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, Chapter 9. Bean, J.C., Hadj-Alouane, A.B A Dual Genetic Algorithm for Bounded Integer Programs. Technical Report TR92-53, Ann Arbor, MI: University of Michigan, Dept. of I&OE. Booker, L Binary Strings. In Back, T., Fogel, D. and Michalewicz, Z. (Eds.), Handbook of Evolutionary Computation, Oxford University Press, Oxford, UK, C3.3:1±10. Davis, L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York. Fogel, D Real-valued Vectors. In Back, T., Fogel, D. and Michalewicz, Z. (Eds.), Handbook of Evolutionary Computation, Oxford University Press, Oxford, UK, C3.3:11±20. Fogel, D., Stayton, L On the Effectiveness of Crossover in Simulated Evolutionary Optimization. BioSystems, 32, 171± 182. Goldberg, D Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA, USA. Holland, J Adaptation in Natural and Arti cial Systems. University of Michigan Press, Ann Arbor, MI, USA. Homaifar, A., Qi, C.X., Lai, S.H Constrained Optimization Vis Genetic Algorithms. Simulation, April, 242±253. Joines, J.A., Houck, C.R On the Use of Non-Stationary Penalty Functions to Solve Nonlinear Constrained Optimization Problems With GAs. Proceedings of the IEEE ICEC 1994, pp. 579±584. Khouja, M., Michalewicz, Z., Wilmot, M The Use of Genetic Algorithms to Solve the Economic Lot Size Scheduling Problem. European Journal of Operational Research, 110, 509±524. Michalewicz, Z Genetic Algorithms Numerical Optimization and Constraints, Proc.6 th Int. Conf. on Genetic Algorithms (Pittsburgh, PA, July 1995) (ed.) L.J. Eshelman (San Mateo, CA: Morgan Kaufmann), 151±158. Algorithms (Pittsburgh, PA, July 1995). Morgan Kaufmann, San Mateo, CA, 151±158. Michalewicz, Z Genetic Algorithms Data Structures ˆ Evolution Programs, 3rd edn. Springer-Verlag, New York. Michalewicz, Z., Attia, N Evolutionary Optimization of Constrained Problems, In Sebald, A.V. and Fogel, L.J. (Eds.), Proc. of the 3rd Annual Conference on Evolutionary Programming, River Edge, NJ. World Scienti c, 98±108. Michalewicz, Z., Logan, T., Swaminathan, S Evolutionary Operators for Continuous Convex Parameter Spaces, In

14 74 R. Sarker, T. Runarsson, C. Newton / Intl. Trans. in Op. Res. 8 (2001) 61±74 Sebald, A.V. and Fogel, L.J. (Eds.), Proceedings of the 3rd Annual Conference on Evolutionary Programming, River Edge, NJ. World Scienti c, 84±97. Michalewicz, Z., Schoenauer, M Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation 4(1), 1±32. Powell, D., Skolnick, M.M Using Genetic Algorithms in Engineering Design Optimization with Nonlinear Constraints. In Forrest, S. (Ed.), Proceedings of the 5th International Conference on Genetic Algorithms. Morgan Kaufmann, San Mateo, CA, 424±430. Runarsson, T., Sarker, R Constrained Nonlinear Integer Programming and Evolution Strategies. Proceedings of the Third Australia±Japan Joint Workshop on Intelligent and Evolutionary Systems. Canberra, November, 193±200. Sarker, B.R., Parija, G.R Optimal batch size and raw material ordering policy for a production system with a xedinterval, lumpy demand delivery system. European Journal of Operational Research, 89, 593±608. Sarker, B.R., Parija, G.R An Optimal Batch Size for a Production System Operating Under a Fixed-Quantity, Periodic Delivery Policy. Journal Operational Research Society, 45(8), 891±900. Sarker, R.A., Khan, L.R An Optimal Batch Size for a Manufacturing System Operating Under a Periodic Delivery Policy. Conference of the Asia-Paci c Operational Research Societies (APORS), Melbourne, Australia. November± December Sarker, R., Khan, L An Optimal Batch Size for a Production System Operating under Periodic Deliver Policy. Computers & Industrial Engineering, 37(4), 711±730. Sarker, R.A., Newton, C Determination of Optimal Batch Size for a Manufacturing System. In Yang, X., Mees, A., Fisher, M., and Jennings, L. (Eds.), Progress in Optimization. Kluwer Academic Publishers The Netherlands, pp. 315± 327. Sarker, R.A., Newton, C Genetic Algorithms for Solving Economic Lot Size Problems. Proceedings of the 26th International Conference on Computers and Industrial Engineering, December 15±17, Melbourne, Australia. 789±793. Sarker, R., Yao, X. (forthcoming). Simulated Annealing for Solving a Manufacturing Batch-Sizing Problem. Schwefel, H.-P Evolution and Optimum Seeking, Wiley, New York, p.16. Smith, A.E., Coit, D.W Constraint-Handling Techniques: Penalty Functions, Handbook of Evolutionary Computation, Release 97/1, IOP Publishing Ltd and Oxford University Press, C5.2:1±2:6.