A hybrid method for solving stochastic job shop scheduling problems

Applied Mathematics and Computation 170 (2005) 185 206 www.elsevier.com/locate/amc A hybrid method for solving stochastic job shop scheduling problems R. Tavakkoli-Moghaddam a, F. Jolai a, F. Vaziri a, P.K. Ahmed b, A. Azaron c, * a Department of Industrial Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran b Business School, University of Wolverhampton, Shropshire, United Kingdom c Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima 739-8527, Japan Abstract This paper presents a nonlinear mathematical programming model for a stochastic job shop scheduling problem. Due to the complexity of the proposed model, traditional algorithms have low capability in producing a feasible solution. Therefore, a hybrid method is proposed to obtain a near-optimal solution within a reasonable amount of time. This method uses a neural network approach to generate initial feasible solutions and then a simulated annealing algorithm to improve the quality and performance of the initial solutions in order to produce the optimal/near-optimal solution. A number of test problems are randomly generated to verify and validate the proposed hybrid method. The computational results obtained by this method are compared with lower bound solutions reported by the Lingo 6 optimization software. The compared results of these two methods show that the proposed hybrid method is more effective when the problem size increases. Ó 2005 Elsevier Inc. All rights reserved. * Corresponding author. E-mail addresses: tavakoli@ut.ac.ir (R. Tavakkoli-Moghaddam), azaron@msl.sys.hiroshima-u. ac.jp (A. Azaron). 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.11.036

186 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Keywords: Stochastic job shop scheduling; Neural networks; Simulated annealing 1. Introduction Scheduling problem is the allocation of resources to perform a set of activities in a period of time [1]. The job shop scheduling is one of the most typical and complicated tasks in scheduling problems. The aim of job shop scheduling problem is to allocate n jobs to m machines in order to optimize a special factor [2]. Traditionally, there are three approaches for solving job shop scheduling problems, namely priority rules, combinational optimization, and constraints analysis [3]. Recently, scheduling systems based on intelligence knowledge have been proposed and presented [4,5]. Job shop is a production system with the capability of producing products with a number of jobs and different operation times for each job. Due to different operations on a product and machine requirements to process each step of production, it is so hard to find an efficient scheduling solution. Traditional approaches often consider small-sized problems with deterministic parameters. The real world of industry is deterministic free and production attributes are stochastic. Stochastic variables and constraints are not available in the application of traditional approaches. In this paper, we consider stochastic parameters and present a flexible algorithm to adjust to the real-world industry situations. Job shop scheduling problem (JSSP) is a class of combinational optimization problems known as NP-Hard one. The above problem is a key issue in production management and combinational optimization. In the last three decades, many researchers have become interested in such problems. Job shop scheduling problem is one of the most significant issues in production planning. Job shop scheduling is significant because it determines process maps and process capabilities for most of industries. For a job shop scheduling problem with n jobs and m machines, there are (n 1 )!(n 2 )!...(n m )! sequences, where n k is the number of operations that must be done by machine k. Of course, not all solutions are feasible. The best sequence has to satisfy the sequence(s) and/or resource(s) constraints and to optimize one or more criteria. However, it is impossible to evaluate all solutions in a reasonable amount of time. Thus, a great number of heuristic algorithms have been developed by researchers to achieve the suitable job sequences in a manufacturing process. These algorithms can be classified into static and dynamic problems based on the time of decision. In static problems, the priority of jobs is identified during the processing. On the other hand, in a dynamic mode the pre-identified job sequence may change from one machine to another based on different situations. Dynamic rules are more applicable than static ones in real-world industry. Suppose that there are m machines and n jobs in a job shop sequencing problem. Each job or part has a special process and the operation sequence

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 187 to complete each job is identified. Each operation in one job has its own processing time. In this problem, the k operation of each job is done on the m machine. This problem may have some constraints at the start and delivery time for each job. The start time constraint exists because the first operation start time cannot be earlier than a certain point of time for reasons such as order placement. The delivery time constraint means incurring a penalty for the delivery earlier or later than a pre-identified time. Earliness or tardiness penalty can be different for various jobs. The objective of solving this problem is to identify the job sequences on machines in order to optimize the performance criteria. One of the following objectives may occur in shop scheduling problems: Maximize the utilization of systems or resources, in other words to minimize the floating time of jobs. Minimize the operational or idle time in each machine, caused by earliness of operation process or tardiness in receiving the jobs by a machine. These two are parallel objectives and in some cases the optimal solution changes according to the objective type. This case is more obvious when there are different penalties for each machine. Another instance arises from a combination of two mentioned objectives, in which it would be necessary to minimize the floating process time of job and idle or operational cost in each machine, in the case of tardiness or earliness of operation processing. The third one is bicriteria scheduling problems having a special penalty based on the time of tardiness or earliness of jobs. In this paper, the third one is considered as the objective of such a scheduling problem. Scheduling problems can be considered and classified according to the main criteria such as requirements, process complexity, scheduling criteria, parameter changes, and scheduling environment [6]. Under the first criterion, namely requirement, there are two states: open and closed production shop based on production requirements. There is no stock in open shop in which production planning is based on the amount of order. In a closed shop, the order is supplied by stock. In this paper, we consider the closed shop. The second criterion, process complexity, is dependent upon the stages of processes and the number of working stations. These can be sub-categorized as follows: (1) one stage, one process (2) one stage, multi-process (3) multistage, production line and (4) multi-stage, job shop production. In the first type, there is one processor and one stage. The second category means one stage and multi-processors that can be carried out in one or several machines. In the third sub-category, each job consists of many operations that need to be processed on one or a number of machines, however all jobs have the same root of operations. In the fourth sub-category, it is possible to allocate a number of machines and route of operations to one job. This situation is used for

188 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 producing different types of products. In this paper, we concentrate on the fourth category, i.e., job shop scheduling with a multi-production stage. The third criterion, scheduling, describes the considered objectives that need to be taken into account in resolving the problem; often these criteria are many, complex and with interaction effects. For instance, some of the scheduling criteria are to decrease the total time of tardiness, to decrease the number of tardy jobs, to increase the ease and utilization of production systems or resources, to decrease the work in process, to balance the usage of resources, and to increase the production rate. In this study, the combination of resource utilization and associated costs is minimized. The fourth criterion, changeable parameters, includes the degree of uncertainty of different parameters in scheduling problems. These parameters include such factors as the characteristics, operation process times, sequencing, precedence constraints, delivery times, start time of jobs. If this uncertainty is not significant to the problem, then the scheduling problem is a deterministic one. In the other case, the scheduling problem is considered as a stochastic one. The last criterion, scheduling environment, can be identified into two categories: static or dynamic. The scheduling problem with the identified number of jobs and a ready time for them is a static problem. On the other hand, a scheduling problem with variable number of jobs and characteristics changing with time is dynamic. In this paper, we consider a static environment for job shop scheduling problems [6]. Therefore, we consider the stochastic and static categories. In the last three decades, a great deal of research works has been conducted on job shop scheduling problems. However, very few have considered stochastic parameters in scheduling problems. Job shop scheduling problems with stochastic process time in normal, exponential, and uniform distributions have been proposed [7]. Ginzburg and Gonik [7] have considered three sets of costs in an allocation problem of n jobs to m machines. These costs are named as penalty cost for each tardy job, delay cost for each unit of time, and storage cost for each unit of early job. The above problem is to identify the earliest start time in order to minimize the average cost of storage and tardiness from the delivery time. Yang and Vang [8] presented a number of heuristic algorithms to solve general job shop scheduling problems. In the neural network model, the weights of bias can be adjusted during the processing time based on sequence and resources constraints. A combination of heuristic algorithms and neural networks improves the quality and performance of feasible solutions generated by neural network. In this study, simulation was carried out for four problems to demonstrate that the proposed neural networks and combinational approaches are highly effective. The main point in this paper is to improve the final solution quality via a combination of the neural network model with heuristic algorithms resulting a suitable sequence.

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 189 In the last few decades, a number of traditional algorithms have been put forward to solve job shop problems. The focus of these algorithms was on JkC max with deterministic constraints of jobs. Stochastic process time was not considered in these problems neither were problems with uncertainty in precedence constraints. A research has been conducted on such type of problems with precedence uncertainty constraints [9]. This research was based on combination of the GERT network and shifting bottleneck algorithm. The second procedure generates an optimal or near-optimal solution based upon priority rules of Giffler Thompson. Finally, the performance of these two heuristic algorithms was investigated. Identification of job delivery times considering tardy costs has been studied [10]. The innovation of this research is the consideration of load work in an identification of internal delivery time to job sequencing in shops and either for identifying the external delivery times requiring probability distribution functions for flow time. Based on the simulation results, they concluded that consideration of workload in identifying the delivery time causes lower costs. Research on scheduling problems focusing on stochastic process time is one of the newest issues that have been interested recently with regard to the need of flexible manufacturing systems. Therefore, in the recent years, stochastic process time problems modeling with various objectives have been presented. The earliest start time as a decision variable and delivery times with a confidence level of different objectives have been investigated. Dynamic programming for one or multi-processors had been proposed with the relation to the less calculation [11]. 2. Stochastic programming problem In this section, a mathematical model of the job shop scheduling problem in stochastic and static environments is presented. In order to solve such a scheduling problem with neural network concepts, mixed and pure integer programming is used for modeling the problem [12 16]. In this paper, we use a pure integer programming model to transmute the sequencing constraints, resource constraints, start processing time, delivery time constraint, processing times without time overlaps and a percent confident interval for process time to integer linear in equivalent. This model has the capability of transmuting the job shop scheduling to a neural network design. 2.1. Definitions and assumptions In this model, assumptions are considered as follows: Each product combination is identified by a special set of parts.

190 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Processing time of all parts on each machine is followed by a special stochastic distribution. Type of products (process combination) is identified in every stage randomly. Tardy cost of each part type is identified. Delivery time of each part type is identified. Operational cost of each machine type is identified. Idle cost of each machine in a unit of time is identified. Number of parts, operations and machines are identified in all stages and are stable during time. Constraints in number of machines must be identified and fixed during time. Each operation can process only one operation in a point of time and each operation in a point of time can be done by just one machine. There is no setup time. There is no delayed ordering. There is no break time. Machine efficiency is 100%. All machines are available at zero in the usage time. The money value is not considered. Flexibility of machines in different processes is not considered. According to above suggestions, parameters and decision variables of the problem are identified as follows: 2.1.1. Symbols M = {1,..., m}: set of machines, where m is the number of machines, P = {1,..., p}: set of parts, where p is the number of parts, j: operation index needed for part p, where j = 1,2,...,O p. 2.1.2. Parameters E tjpm : expected time needed to process operation j of part p on machine m, V tjpm : variance of time needed to process operation j of part p on machine m, 1 if it is available to process operation j of part p on machine m; a jpm : 0 otherwise; D p : delivery time of part p, C m : operational cost of machine m in each time unit, I m : idle cost of machine m in each time unit, O jpm : operation j of part p on machine m.

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 191 2.1.3. Decision variables 1 if operation j of part p is allocated on machine m in sequence s; X jpms : 0 otherwise; Y jpms : start processing time of operation j of part p on machine m in sequence s, t jpma : optimal time requiring for operation j of part p in machine m according to a confidence interval. 2.2. Mathematical model Min Z ¼ XM s:t: : m¼1 þ XM X P m¼1 þ XM X M p¼1 m¼1 m¼1 s¼1 X P X Op p¼1 X S j¼1 X Op j¼1 C m I m Y jmax pms þ X jmax pms t jmax pm D p X S s¼1 X P p¼1 X S 1 X P s¼1 p¼1 X Op j¼1 X Op j¼1 X jpms t jpm!! MAX Y jpmðsþ1þ Y jpms X jpms t jpm ; 0 ð1þ a jpm X jpms ¼ 1 8j; p; ð2þ X jpms 6 1 8m; s; ð3þ Y jpms 6 RX jpms 8j; p; m; s; ð4þ X M X S m¼1 s¼1 X P X Op p¼1 j¼1 X M X S Y jpms þ X jpms t jpm 6 m¼1 s¼1 X P Y jpms þ X jpms t jpm 6 p¼1 X Op j¼1 Y ðjþ1þpms 8j; p; ð5þ ðy jpmðsþ1þ Þ 8m; s; ð6þ E tjpm Z a=2 V tjpm 6 t jpm 6 E tjpm þ Z a=2 V tjpm 8j; p; m; ð7þ X 2½0; 1Š; Y P 0; R 0: Objective function (Eq. (1)) is nonlinear integer equivalence aimed at minimizing the sum of variation of actual processing time and planned process time, operational costs, and idle costs for each machine in a planning horizon. The first statement calculates the sum of actual processing time variations from planned time in time series. This summation is equal to the start processing time of operation j of part p on machine m in sequence s and to optimize time required for processing operation j of part p on machine m with regard to

192 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 percent of a confidence interval if this operation has been allocated to the right machine. The second statement calculates the operational costs of each machine. This cost is equal to the sum of mutilation of required times for all kinds of machines in operational cost of that machine. The third statement of the objective function calculates the idle time of machines. In this case if there is no part to be allocated to a machine and the machine is idle, there will be idle cost for the machine. The first constraint (Eq. (2)) guarantees that each operation for each part must be allocated to just one machine in a sequence. The third equation shows that only one operation can be in each sequence on a machine. The fourth equation guarantees that the start processing time is finite. The fifth equation guarantees the operation sequences for each part. The sixth equation shows that each operationõs processing time does not have any overlap with any other. The seventh equation considers a confidence interval for processing of operations. 3. Proposed algorithm for the model Scheduling job shop production systems is complex in calculation and it is only possible to solve small-scale problems with existing solution algorithms optimally. The proposed model, dealing with machines flexibility in processing different kind of products, is highly difficult and time consuming to solve optimally for large-scale and real-world problems. This problem highly arises from resources constraints (time, memory, computers and so forth). In this section, we propose and present a hybrid algorithm to solve job shop scheduling problems that enable overcoming many of these resource constraints. Thus, in this section we present a heuristic algorithm for the solution of the mathematical model with respect to machines flexibility in processing different kind of products. As mentioned before, there is flexibility for machines to process various parts. We propose a hybrid method, namely neural networks and a heuristic algorithm based on simulated annealing, to solve job shop scheduling problems. In this method, a simulated annealing algorithm can be used individually or accompany with neural network [8]. The neural network model is used for eliminating resource and sequence variations resulting in nonfeasible solutions in a specific problem. The feasible solutions are then optimized through a heuristic algorithm based on simulated annealing. Fig. 1 shows the optimal structure of the hybrid approach by the use of the simulated annealing algorithm. The solution procedure includes three stages as follows: (1) To allocate machines to jobs at random with respect to machine flexibility in processing different products. (2) To generate the initial or feasible solution by the neural network model.

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 193 Initial solutions Neural network Feasible solutions Feasible improved solutions Heuristic algorithm Fig. 1. Combinational approach structure. (3) To improve the quality and performance of the initial solution generated by neural network model and by simulated annealing algorithm. Detail descriptions of different stages of heuristic algorithms are presented below. Step 1: Allocation jobs to machines at random Some special processes can be processed on different machines and the ability of processing various parts makes for the possibility of random scheduling at zero time. The generated solutions caused by random allocation of machines to parts are the input solutions to the neural network model. This solution is used to generate the feasible schedule in the next stage. Step 2: Generating the initial or feasible solution by the neural network model The proposed nonlinear mathematical model is converted to a neural network design to solve the job shop scheduling problem [8]. To formulate the suggested job shop model to a neural network, there is a need for three calculation units (neuron) sets according to the following items: (1) Units related to parts processing time overlap constraint with respect to the presented sequence of that unit (SC units). (2) Units related to machines processing time overlap constraint (RC units). (3) Units related to start processing time for each operation (ST units). The output of these unites will be a feasible solution of the model. The ST unit receives inputs from two related SC and RC units and sends output to both of the units. The ST units have the feedback too. Activation of SC and RC networks sets is implemented by data entry based on Yang and Wang [8]. The procedure to generate the production sequence for various processes is as follows: (1) Generating the initial or feasible situation: LB numbers of unequal parts are allocated to each machine randomly. Then, the related operation in each part is allocated to each machine based on the process requirement. Machines are allocated to each job at random because of machines flexibility in processing the different operations. In this case, there will be an

194 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 initial feasible solution S ikp (0) for each O ikp operation that (i 2 N, k 2 {1,...,n i }) and this solution can be the net input, I STikp, for each ST unit. (2) Each SC unit, SC ikl, of neural SC set activates and calculates the activation A SCikl ðtþ based on related equivalences. If A SCikl ðtþ 6¼ 0, the sequence constraints are not satisfied. In this case, activations will be adjusted by activating related feedbacks, until the constraints are satisfied. (3) Each SC unit, RC qikjl, of neural RC set activates and calculates the activation A RCqikjll ðtþ based on related equivalences. If A RCqikjll ðtþ 6¼ 0, the resources constraints are not satisfied. In this case, activations will be adjusted by activating S ikq (t + 1), S jlq (t + 1) feedbacks, until the constraints are satisfied. (4) Steps 2 and 3 are reiterated until all the units become stable without any change. When sequence and resource constraints are satisfied then the feasible solution is generated. (5) The initial solution is used as an input for the heuristic algorithm. Solution procedure in the heuristic algorithm is developed in the discussion that follows below. At the end of this step of the proposed method, the initial feasible solution will be obtained with regard to the sequence and resource constraints. The next step after defining the initial solution is to extract an optimal or near-optimal solution. This is carried out through the use of the heuristic algorithm, as elaborated in step 3. Step 3: Improving the quality and performance of initial solution generated with neural network model, by simulated annealing (SA) algorithm In this step, we apply SA to improve the initial sequence obtained from the previous step [16]. The improvement procedure is implemented by changing the pair of operations among the machines. In other words, the neighbor solutions are found by changing the parts among machines. At first, one operation of a job is selected randomly, then one operation from another part is selected randomly and these two operations change with each other on the machine. Using SA requires identifying the value of parameters in SA algorithm. The initial temperature considers the maximum difference of cost between neighbor solutions and minimum difference of cost between neighbor solutions. The neighbor solution is obtained from the initial sequence. The cooling rate considered as 0.05. The number of iterations in each temperature differs from another temperature. Its initial value is equal to five and increases with arithmetic gradient of five. The algorithm stops when the final temperature reaches to 0.05. Clearly, the SA program is flexible enough to change all these parameters and has ability to implement sensitivity analysis on each of these parameters.

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 195 The initial symbols and parameters in SA algorithm are as follows: n: number of accepted movements in each temperature, r: number of temperature transactions, T 0 : initial temperature, T f : final temperature, e = AT min : minimum accepted states, which uses for identifying the balance situations. It is a control variable to check whether the system is on balance or not, d: positive small number to check the cooling rate, e i : positive small number to identify balancing of system in a special temperature T r, e r : positive small number to check the freezing point, C i (T r ): cost of the i situation when temperature is T r, C e ðt r Þ: the objectives average of accepted status in start point at T r temperature, C G ðt r Þ: the objectives average of accepted status at T r temperature, CðT r Þ: the objectives average for balancing at T r temperature, V(T r ): the objectives accepted status variances for balancing at T r temperature that is equal to V ðt r Þ¼ 1 n P n i¼1 ðc iðt r Þ CðT r ÞÞ 2. Fig. 2 shows the proposed flowchart. The SA algorithm steps are as follows: 1. Set initial value as r =0,n =0. 2. Set initial temperature value as T r. 3. Use the initial or feasible solution obtained by step 2 of hybrid algorithm. 4. Energy calculation. (C i (T r ): the objective value with the i solution in T r temperature) calculated by (8) statement. 5. Variance pattern feasible space movement. (a) Generate the j feasible solution then calculate the changes of objective function based on DCðT r Þ¼C j ðt r Þ C i ðt r Þ: If DC(T r ) 6 0, then go to step 5(a). (b) Choose a random variable y! u(0, 1). DCðT rþ If y > e T r ; then go to step 5ðaÞ: (c) Accept the new solution because of the objective improvement. Put n ¼ n þ 1 and if n < e then go to step 5ðaÞ:

196 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Start 1. Random allocation of jobs to machines. 2. Generate the initial feasible solution by neural network model. No Is the initial solution feasible? Yes Random allocation. of jobs to machines Initial solution. with neural network 3. Improve the generated sequence No Are all the jobs being scheduled? Yes Finish Fig. 2. Flow chart of the hybrid algorithm. 6. Test for balancing: Put n =0,ifjC e ðt r Þ C Ce ðt Þ r j > 1 or if the system was not static then go to step 5(a). 7. Test for freezing point: (a) Calculate V(T r ), CðT r Þ.Ifr = 0 then go to step 7(a). If V ðt Þ T ðcðt 0 Þ CðT ÞÞ 6 e 2: (b) Stop; because of satisfaction. (c) Update the temperature based on T rþ1 ¼ T r then go to step 7(a). 1þ lnð1þoþt r 3V ðtrþ. Put r = r + 1 and

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 197 4. Computational results To evaluate the proposed algorithm, the computer programming in Visual Basic language was prepared and all the calculations were conducted on a Pentium IV 1800 MHz. For this reason, five sample problems, based on the literature review, are generated. Due to the complexity of job shop scheduling problems requiring a large amount of time to generate the optimal solutions, we consider the lower bound of the solution instead of the optimal solution. In this paper, we use the Lingo 6 software to get the lower bound of the problem. In this case, the lower bound is less than or equal to the optimal solution. 4.1. Manufacturing systems factors To generate a series of selected problems, scheduling factors are chosen among a set of affected variables in sequence. These variables are: (1) Types of parts: the set of parts needs to be produced. Each part has a special operation sequence. (2) Types of machines: the set of machines requires producing types of products. Each machine is able to process more than one operation in different point of times (i.e., machine flexibility). (3) Operation sequence of a part: the order of machines that the part needs to pass on them. The required order and number of operations to produce a part is identified in this sequence. (4) Process time: the required time, which need for processing one part on a specific machine, which is a random variable. (5) Machine capacity: availability time of each machine to process the parts. (6) Operational cost: the machine usage cost to process the parts in a unit of time. (7) Idle cost: the cost of idle machine in a unit of time. (8) Tardy cost: the cost of delays in completing the processing of each operation for each unit of delay time. (9) Delivery time: specific delivery time for each part. In the proposed algorithm, all above characteristics are considered. The problem is investigated under five selected test problems. Table 1 shows the characteristics of selected problems with factors generated at random. 4.2. Value identification of problem parameters The next step after generating the selected problems is to identify the values of required parameters in the problem. The summary of generating value procedures for selected problems is presented in Table 2.

198 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Table 1 Characteristics of the selected problems to evaluate the proposed algorithms Run No. of parts No. of machines No. of operations 1 3 2 3 2 5 3 3 3 6 6 6 4 10 10 5 5 20 5 5 Table 2 Parameter characteristics of the selected problems Scheduling problem factor Amount Description Maximum of flexibility for each operation 2 Range of process time for each operation [1 10] Uniform distribution Range of idle cost for each machine [1 5] Uniform discrete distribution Range of operational cost for each machine [10 15] Uniform discrete distribution Range of delivery time for each part [10 50] Uniform distribution Range of tardy cost for each part [1 10] Uniform discrete distribution 4.3. Lower bound generation We utilize the lower bound solution of the proposed model to compare the computational results obtained by the proposed hybrid method. It is extremely difficult to achieve optimal solution within reasonable computational timeframes due to high complexity of job shop scheduling problems. In this paper, the Lingo 6 software was used to obtain the lower bound solutions. The closeness of the optimal solution generated with hybrid method to lower bound is not an evaluation of the performance of the hybrid method. Therefore, comparing the lower bound with the optimal solution is not verified for making a decision on the quality performance of the hybrid method. In this section, we present a test problem solved with the hybrid method. The Gantt chart and comparison of computational results with the lower bound are presented. There are some selected problems and their solutions obtained by the hybrid method as well as a comparison of results with the solution generated by the Lingo 6 software. 4.4. An example problem (3 2) We consider a job shop scheduling problem with three types of parts, 2 machines, and 3 operations for each job. The completion data of the example is presented in Tables 3 6. The computational results generated by the hybrid method and the lower bounds reported by Lingo 6 software are compared in Table 7.

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 199 Table 3 Input data for operation sequence Operation 1 Operation 2 Operation 3 Part 1 1 2 3 Part 2 0 0 1 Part 3 2 3 1 Table 4 Input data for parts delivery Part 1 Part 2 Part 3 Tardy cost 7 7 5 Delivery time 34 44 20 Table 5 Input data for machines costs Machine 1 Machine 2 Idle cost 5 3 Operational cost 13 11 As shown in Table 7, the gap in the objective value between the two methods is 27.53%. As we considered, the solution obtained by Lingo as a lower bound of our solution, this value shows the suitable performance of the hybrid method in generating the near-optimal solution. One of the most important criteria in generating an optimal or near-optimal solution is the runtime. The hybrid method has 89.5% time saving than Lingo 6. The amount of saving is highly significant, especially in large-scale problems. An important factor in real-world production situations is the makespan of jobs, C max, which is considered in this section with respect to the solutions generated by Lingo 6 and the hybrid method. We present the Gantt chart of makespan in each solution procedure in Figs. 3 and 4. Gant Chart M1 O 23 O 11 O 31 22 62 115 137 M2 O 33 O 21 O13 73 115 130 226 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 Fig. 3. Gantt chart of JSSP 3 2 using the hybrid method.

Table 6 Input data for operation time (E t,v t ) Part 1 Part 2 Part 3 Operation 1 Operation 2 Operation 3 Operation 1 Operation 2 Operation 3 Operation 1 Operation 2 Operation 3 Machine 1 (41.3, 3) (0, 0) (23.7, 7) (0, 0) (0, 0) (0, 0) (0, 0) (24.3, 9) (0, 0) Machine 2 (29.2, 2) (41.7, 7) (98.2, 2) (15.8, 8) (0, 0) (0, 0) (99.4, 4) (54.8, 8) (50.2, 2) Table 7 Comparison of results by the hybrid method and Lingo 6 Solution procedure No. of iterations Objective value Runtime (s) Hybrid algorithm 42 3811 729.63 Lingo 6 24 2988.32 6953 Percent of difference to Lingo 6 27.53% 89.5% 200 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 201 Gant Chart M1 O 31 O 11 O 23 9 60 129 135 M2 O 11 25 O 21 O 31 O 33 52 146 192 283 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 Fig. 4. Gantt chart of JSSP 3 2 using Lingo. As shown in Fig. 3, the makespan is equal to 226 time unit, according to the hybrid method. This value, according to the solution from Lingo 6, is equal to 283 (see Fig. 4). However the objective value by the hybrid method is 27.53% more than the lower bound while decreasing in runtime and makespan. These are the main factors to compare and evaluate the efficiency of the proposed method significantly. 4.5. Calculations of the selected problems This section presents the solutions of selected problems and the compression of the associated solutions with the lower bound as well as the analysis of the related comparison. The aim of these experiments is to identify the performance of the hybrid method in different situations. Observations of solutions and the performance trend of the hybrid method in various problem settings help us to find the best attributes for using this method in production schedules. The solutions of selected problems are shown in Table 8. The solutions show that the hybrid method exhibits good performance in different situations at most of time and the runtime has been decreased significantly in compare with the runtime of Lingo 6. To check problem size effects on the final solution quality, some of the problems were recalculated based on the sequencing of the Lingo software, and it was observed that the solutions are improved. The solution comparisons are presented in Fig. 5. The use of Lingo 6 to generate the lower bound for the objective function is observed. The difference or gap between the objective function value by the hybrid method and Lingo 6 increases as the problem size increases. According to the results, it is clear that the hybrid method generates near-optimal solution in less computational time. It is also clear that the hybrid method decreases the calculations involved in solving the selected problems to generate near-optimal solutions. As shown in Fig. 6, the difference or gap between runtimes and two methods increases as the problem size increases. This fact shows that the runtime is improved by using the hybrid method to generate the near-optimal solution.

202 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Table 8 Computational results of selected problems for the hybrid and the lower bound methods Experiment Types of parts Types of machines Types of operations Solution algorithm No. of iterations Objective values Runtime (s) 1 3 2 3 Hybrid 42 3811 729.63 Lingo 6 24 2988.32 6953 2 5 3 3 Hybrid 35 8780 1266.8 Lingo 6 5277 3740.28 117,874 3 6 6 6 Hybrid 20,040 20,040 4456.99 Lingo 6 6581.96 6581.96 162,179 4 10 10 5 Hybrid 362 29,547 8547.35 Lingo 6 576 10,352.6 190,800 5 20 5 5 Hybrid 473 37,140 13,548.94 Lingo 6 680 18,254.21 219,600 40000 Objective value difference 25000 20000 15000 10000 5000 0 3*2*3 5*3*3 6*6*6 10*5*5 20*5*5 Problem size (a) Difference in objective values * Objective value difference Objective values 35000 30000 25000 20000 15000 10000 5000 0 Hybrid alg. lingo6 3*2*3 5*3*3 6*6*6 10*5*5 20*5*5 Problem size (b) Objective values Fig. 5. Comparison of objective value for the hybrid and the lower bound methods. (a) Difference in objective values; (b) objective values. Runtime difference 250000 200000 150000 100000 50000 0 3*2*3 5*3*3 6*6*6 10*5*5 20*5*5 Prob. size (a) Runtime difference Runtime 240000 220000 200000 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 Runtime1 Runtime2 3*2*3 5*3*3 6*6*6 10*5*5 20*5*5 Prob. size (b) Runtime Fig. 6. Comparison of runtime for the hybrid and the lower bound methods. (a) Runtime difference; (b) runtime.

Objective value Lingo6 20000 19000 18000 17000 16000 15000 14000 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 110000 120000 130000 140000 150000 160000 17000 180000 190000 200000 210000 220000 230000 240000 Time Fig. 7. Increasing trend on time based on the objective values using Lingo 6. R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 203

204 R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 Objective value Hybrid 40000 35000 30000 25000 20000 15000 10000 5000 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 Time Fig. 8. Increasing trend on time based on objective values from the hybrid algorithm. As can be observed from Fig. 5, the difference in objective values based on the hybrid method and Lingo 6 increases as problem size increases, and this is because of various constraints to solve the large parameters problems by the optimization software. According to Fig. 6, the difference of runtimes between two procedures increases exponentially, when the problem size increases. The case for using the hybrid method is obvious in problem likely to consume amount of time in resolution. Fig. 7 shows that the runtime increases exponentially as the objective value increases in the Lingo 6 software. However, in hybrid method the slope of the runtime curve for finding the optimal or near-optimal solution decreases when the objective values increase, as depicted in Fig. 8. 5. Conclusion The main reason for this study was to present a procedure for stochastic job shop scheduling minimizing the difference between the delivery and the completion times of jobs as well as related operational or idle cost of machines. For this case, a mathematical programming model was presented after reviewing the literature survey. To solve the model within reasonable time, a hybrid method consisting of neural network and simulated annealing (SA) was proposed and presented. This method generates the initial feasible solution by neural network and then simulated annealing algorithm improves the performance quality of the initial solution. To evaluate the proposed method, we generated five problems at random and solved them with the proposed method and Lingo 6. Computational results showed that the hybrid method exhibits high performance in generating optimal or near-optimal solutions, especially for large-scale problems. As the problem parameters increase, there is an increase in runtime of Lingo 6 exponentially to generate the lower bound for the given

R. Tavakkoli-Moghaddam et al. / Appl. Math. Comput. 170 (2005) 185 206 205 problem. This time consuming task makes these types of software packages inapplicable for the solution of large-scale problems. The computational results show the suitable performance of the hybrid method in extracting acceptable scheduling solutions within acceptable timeframes. As a result, the proposed method generates acceptable and better solutions than Lingo 6 for all largescale scheduling problems. Thus, the proposed method is recommended for all stochastic scheduling problems with large numbers of machines, parts, and operations. As increasing the problem size increases runtime exponentially, it is obvious that solving large-scale problems is a hugely time consuming task. The proposed method generates the optimal or near-optimal solutions easier and in less time for large-scale problems containing a large number of constraints. Realworld manufacturing situations typically feature lots of constraints and parameters, so the proposed method is highly relevant for many different industries. The summary of results is as follows: (1) There is an exponential increase in runtime of Lingo 6, as the problem size increases. This runtime is almost constant when using the hybrid method in large-scale problems. (2) According to the results, the percentage difference in the objective value is equal to 27.53% in small-sized problems with low number of parts and machines. This criterion increases for larger-sized problems, containing large number of machines and parts. Therefore, it seems that the problem size has a high effect on solution quality. Moreover, the computational time increases exponentially, as the problem size increases. (3) The makespan obtained form the hybrid method (226 time units) is much better than makespan reported by Lingo 6 (283 time units). This fact is a highly significant in order to evaluate the performance of scheduling procedures in real-world production situations. The results show a good performance of the hybrid method in lowering makespan in comparison with Lingo 6. (4) The difference or gap percent increases between these two methods indicating the relatively superior performance of the hybrid procedure, especially in resolving larger complex problems. References [1] K.R. Baker, Introduction to Sequence and Scheduling, John Wiley, NY, 1974. [2] R.W. Conway, W.L. Maxwell, Theory of Scheduling, Addison-Wesley, MA, 1967. [3] D. Dubois, H. Fargier, H. Prade, Fuzzy constraint in job shop scheduling, J. Intell. Manuf. 6 (1995) 215 234. [4] P.V. Hentenryck, Constant Satisfaction and Logic Programming, MIT Press, MA, 1989. [5] M.S. Fox, M. Zweben, Knowledge Based Scheduling, Morgan Kaufman, 1993.

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