Solving stochastic job shop scheduling problems by a hybrid method

Transcription

1 Solving stochastic job shop scheduling problems by a hybrid method By Pervaiz Ahmed, Reza Tavakkoli- Moghaddam, Fariborz Jolai & Farzaneh Vaziri Working Paper Series 004 Number WP006/04 ISSN Number Professor Pervaiz Ahmed Professor of Management University of Wolverhampton, UK Tel: +44 (0) P.Ahmed@wlv.ac.uk

2 Copyright University of Wolverhampton 004 All rights reserved. No part of this work may be reproduced, photocopied, recorded, stored in a retrieval system or transmitted, in any form or by any means, without the prior permission of the copyright holder. The Management Research Centre is the co-ordinating centre for research activity within the University of Wolverhampton Business School. This working paper series provides a forum for dissemination and discussion of research in progress within the School. For further information contact: Management Research Centre Wolverhampton University Business School Telford, Shropshire TF 9NT Fax All Working Papers are published on the University of Wolverhampton Business School web site and can be accessed at choosing Internal Publications from the Home page.

3 Abstract This paper presents a non-linear mathematical programming model for a stochastic job shopscheduling problem. Due to the complexity of the proposed model, traditional algorithms have low capability in producing a feasible solution. In order to deal with this in this paper a hybrid method is proposed that enables definition of a 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. We assume that machine flexibility in processing operations to decrease the complexity of the proposed model. 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 optimisation software. The compared results of these two methods show that the proposed hybrid method is more effective when the problem size increases.

4 The authors Professor Pervaiz Ahmed Pervaiz Ahmed is Professor of Management at the University of Wolverhampton Business School. His key research interests include processes for strategy implementation, innovation, knowledge and learning, and social responsibility and business ethics. Reza Tavakkoli-Moghaddam Reza Tavakkoli-Moghaddam is an Associate Professor of Industrial Engineering at the University of Teran. His current research interests are involved in scheduling, facilities layout and location, vehicle routing, and metaheristic methods such as genetic algorithms, simulated annealing, tabu search, fuzzy set theory as well as neural networks. Fariborz Jolai Fariborz Jolai is an Assistant Professor of Industrial Engineering at the University of Teran. His current research is the field of scheduling, inventory control, decision theory, and project control. Farzaneh Vaziri Farzaneh Vaziri received an MSc in Industrial Engineering at the University of Teran. Her current areas of research interest are scheduling, six sigma, quality management systems, and risk management. 4

5 Solving stochastic job scheduling problems by a hybrid method Introduction The problem of scheduling is one that involves allocation of resources to perform sets of activities in a given period of time (Baker, 974). Job shop scheduling is one of the most typical and complicated tasks in scheduling problems. The aim of job shop scheduling problems is to allocate m machine(s) to n job(s0 in order to optimize a special factor (Conway & Maxwell, 967). There are three traditional approaches for solving job shop-scheduling problems, namely priority rules, combinational optimization, and constraints analysis (Dubois et al., 99). Recently, scheduling systems based on intelligence knowledge have been proposed and presented (Hentenryck, 989; Fox & Zweben, 99). Job shop is a production system with the capability of producing products with various numbers of jobs and different operation time for each job. Due to different operations on a product and machine requirements to process each step of production, it is often very difficult 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 situations encountered in the real world of industry. Job shop scheduling problem (JJSP) is a class of combinational optimization problems known as NP- Hard one. This is a central 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 industries. In a job shop scheduling problem with n jobs and m machines, there is (n )!(n )!...(n m )! sequences. n k is the number of operations that must be executed by machine k. Of course, there is probability that all the solutions are not feasible. The best sequence is to satisfy the sequence(s) and/or resource(s) constraints and to optimize one or more criteria. However, it is impossible to evaluate all the optimal solutions within a reasonable amount of time. To overcome this, a number of heuristic algorithms have been developed by researchers to define 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 pre-processing. On the other hand, in a dynamic model the pre-identified job sequence may change from one machine to another based on different situations. Dynamic rules are more applicable than static ones to real world realworld situations in industry. Suppose that there is m machines and n jobs in a job shop-sequencing problem. Each job or part has a special process and the operation sequence 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 machine m. This problem may have some constraints for 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. An 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 exist in job shop scheduling problems: to maximize the utilization of systems or resources, in the other words to minimize the floating time of jobs. to minimize the operational or idle time in each machine, caused by earliness of operation process or tardiness in receiving the jobs by a machine.

6 These two are parallel objectives and in some cases the optimal solution changes depending on the objective type. This issue is obvious when there are different penalties for each machine. Another instance of this arises from a combination of the two mentioned objectives, in which it is necessary to minimize the floating process time of job and idle or operational cost in each machine, for the case of tardiness or earliness of operation processing. The third instance involves bi-criteria scheduling problems having a special penalty based on the time period of tardiness or earliness of jobs. In this paper, the third case is adopted as the objective of the scheduling problem. Literature review Scheduling problems can be considered and classified according to the main criteria such as requirements, process complexity, scheduling criteria, parameter changes, and scheduling environment (Johns & Rabelo, 998). Under the first criterion, namely requirement, there are two states: open and closed production shop based on production requirements. There is no stock in an open shop in which production planning is based on the amount of order. In a closed shop, the order is supplied from stock. In this paper, we consider the closed shop. The second criterion, process complexity, is dependent upon the stages of the process and the number of working stations. These can be sub-categorized as follows: ) one stage, one process ) one stage, multi- process ) multi-stage, line production and 4) multi-stage, job shop production. In the first type, there is one processor and one stage. The second category involves one stage and multiprocessors 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 the jobs have the same route 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 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 progress; 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 time, 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 alternative 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 that change with time is dynamic. In this paper, we consider a static environment for job shop scheduling problems (Johns & Rabelo, 998). Next, we consider the stochastic and static categories. In the last three decades, a great deal of research work has been conducted on job shop scheduling problems. However, very few studies have considered stochastic parameters in scheduling problems. Job shop scheduling problem with stochastic process time in normal, exponential, and uniform distributions has been proposed (Golenko-Ginzburg & Gonik, 00). Golenko-Ginzburg and Gonik (00) considered three sets of costs in an allocation problem of n jobs to m machines. These costs 6

7 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. CSANN (Yang & Wang, 000) presents a number of heuristic algorithms to solve general job shop scheduling problems. In their neural network model, the weights of bias can be adjusted during processing time based on sequence and resource constraints. A combination of heuristic algorithms and neural networks improve the quality and performance of feasible solutions generated by neural network. In their study, simulation was carried out for four problems to demonstrate that the proposed neural networks and combinational approaches are highly effective. The main point of this paper was to improve the final solution quality via a combination of the neural network model with heuristic algorithms resulting in a suitable sequence (optimal or approximate solutions). 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 J C with deterministic constraints of jobs. max Stochastic process time was not considered in these problems, and neither were problems with uncertainty in precedence constraints. Research has been conducted on such types of problems with precedence uncertainty constraints (UBKA)[9]. This research was based on combination of the GERT network and shifting bottleneck algorithm. The second procedure to generate an optimal or approximate solution was 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 (Van Ooijen & Bertrand, 00). The innovation of this research was consideration of load work in an identification of internal delivery time to job sequencing in shops and for identifying the external delivery times requiring probability distribution functions for flow time. Based on simulation results, Van Ooijen and Bertrand (00) conclude that consideration of workload in identifying the delivery time leads to lowering of costs. Research on scheduling problems focusing on stochastic process time is one of the issues that has received interest with regard to the need of flexible manufacturing systems. Thus 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 has been proposed in terms of reduced requirement for calculation (Elmaghraby, 00). Stochastic mathematical model 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 (Thomalla, 00; Veral, 00; Jain & Meran, 998; Smith, 999; Steninhofel et al., 999). In this paper, we use a pure integer mathematical model to transmute the sequencing constraints, resource constraints, start processing time, delivery time constraint, processing times without time overlaps and α percent confidence 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. Definitions and assumptions In this model, assumptions are considered as follows: There are suitable numbers of product combinations that can be happen. Each combination of product is identified by a special set of parts. 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. 7

8 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. Machines efficiency is 00%. All the 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 the above, parameters and decision variables of the problem are identified as described below. Symbols M={,..m} : Machines set, where m is the number of machines. P={,..p} : Parts set, where p is the number of parts. j : Operation index needed for part p, where j=,,., O p Parameters Et jpm : Mean of time needed to process operation j of part p on machine m. Vt jpm : Variance of time needed to process operation j of part p on machine m. If it s 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. Decision variables 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 α confidence interval. 8

9 Mathematical model Min Z = M P Op ( Yj pms + X ) max jmax pms t jmax pm D p m= p= j= + + M m= C m m= s= p= j= M I m S P S P Op Op s= p= j= X jpms MAX t jpm ( Y Y X t, 0) jpm( s+ ) jpms jpms jpm )( s.t.: M m= s= P p= j= Y M S M S ( Yjpms + X jpmst jpm ) ( Y(j + )pms ) m= s= P Op P Op ( Yjpms + X jpmst jpm ) ( Yjpm(s+ ) ) p= j= Et S Op jpms jpm a X jpm jpms RX Z X α jpms Vt jpms jpm X [0,], Y 0 = t, R jpm j, p m,s j, p, m,s Et >> 0 m= s= p= j= jpm + Z α Vt jpm j, p m, s j, p, m Objective function (eq. ) 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 the 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 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 mutation 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. ) 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 operation s processing time does not have any overlap with another. The seventh equation considers a confidence interval for processing of operations. 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 largely arises from resources )( )( )4( )( )6( )7( 9

10 constraints (time, memory, computers and so forth). In this section of paper we propose and present a hybrid algorithm to solve job shop scheduling problems that enables overcoming some 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, utilizing a neural network 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 as an accompaniment to the neural network [8]. The neural network model is used for eliminating resource and sequence variations resulting in non-feasible solutions in a specific problem. The feasible solutions are then optimized through a heuristic algorithm, in this case based on simulated annealing. Figure () shows the optimal structure of the hybrid approach by the use of the simulated annealing algorithm. Initial solutions Feasible solutions Feasible improved l ti Neural network Heuristic algorithm Figure. Combinational approach structure The solution procedure contains three stages as follows: ) Allocate machines to jobs at random with respect to machine flexibility in processing different products. ) Generate the initial or feasible solution by the neural network model. ) Improve the quality and performance of the initial solution generated by neural network model and by simulated annealing algorithm. Detailed discussion of different stages of the heuristic algorithms is presented below. Step : Allocation of machines to jobs 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 : Generating the initial or feasible solution by the neural network model The proposed nonlinear mathematical model converts to a neural network design to solve the job shopscheduling problem [8]. To formulate the suggested job shop model to a neural network there is a need for three calculation unit (neuron) sets according to the following: ) Units related to parts processing time overlap constraint with respect to the presented sequence of that unit (SC units). ) Units related to machines processing time overlap constraint (RC units). ) Units related to start processing time for each operation (ST units). The output of these units 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 feedback. Activation of SC and RC network 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: ) 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 initial feasible 0

11 solution S ikp (0) for each O ikp operation that ( i N, k {,..., ni}) net input, I ST ikp, for each ST unit. and this solution can be the ) Each SC unit, SC ikl, of neural SC set activates and calculates the activation ASC ikl (t) based on related equivalences. If ASC ikl ( t) 0, the sequence constraints are not satisfied. In this case, activations will be adjusted by activating related feedbacks, until the constraints are satisfied. ) Each SC unit, RC qikjl, of neural RC set activates and calculates the activation ARC qikjll (t) based on related equivalences. If ARC qikjll ( t) 0, the resources constraints are not satisfied. In this case, activations will be adjusted by activating Sikq ( t + ), S jlq ( t + ) feedbacks, until the constraints are satisfied. 4) Steps and re-iterated until all the units become stable without any change. When sequence and resource constraints are satisfied then the feasible solution is generated. ) The initial solution is used as an input for the heuristic algorithm. The solution procedure in the heuristic algorithm is developed in the discussion that follows below. At the end of the fourth step of the proposed method, the initial feasible solution(s) will have been obtained with regard to the sequence and resource constraints. The next step after defining the initial solution is to extract an optimal or approximate solution. This is carried out through the use of the heuristic algorithm, as elaborated in step below. Step : To improve the quality and performance of initial solution generated with neural network model, by simulated annealing (SA) algorithm In this step, we apply simulated annealing (SA) to improve the initial sequence obtained from the previous step [6]. The improvement procedure is implemented by changing the pair of operations among the machines. In the other words, the neighbor solutions are found by changing the parts among machines. t 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. To use SA, requires identifying the value of parameters in the 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 is 0.0. The number of iterations in each temperature can differ at each temperature. Its initial value is equal to five and increases with an arithmetic gradient of five. The algorithm stops when the final temperature reaches to a value of 0.0. The SA program is flexible enough to change all these parameters and also has the ability to conduct a sensitivity analysis on each of these parameters. The initial symbols and parameters in SA algorithm are as follows: n: The number of accepted movements in each temperature. r : The number of temperature transactions. T : The initial temperature. 0 T : The final temperature. f e = AT min : Start, minimum accepted states that uses for identifying the balance situations. It is a control variable to check whether the system is on balance or not. δ: Positive small number to check the cooling rate. ε : Positive small number to identify balancing of system in a special temperature T r i ε r : Positive small number to check the freezing point. C i ( T r ) : The cost of i situation when temperature is T r.

12 C e ( T r ) : The objectives average of accepted status in start point at T r temperature. C T ) : The objectives average of accepted status at T r temperature. G ( r C ( T r ) : 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 n V ( Tr ) = ( Ci ( Tr ) C ( Tr )). n i= Figure () depicts the proposed flowchart. The SA algorithm steps are as follows:. Initial value as r=0,n=0.. Initial temperature value as T r.. To use the initial or feasible solution obtained by step from the hybrid algorithm. 4. Energy calculation. ( C i ( T r ) : the objective value with the i solution in T r temperature) calculated by (8) statement.. Variance pattern- feasible space movement. a) Generate the j feasible solution then calculate the changes of objective function based on Δ C( Tr ) = C j ( Tr ) Ci ( Tr ). If ΔC ( Tr ) 0, then go to (a) step. b) Choose a random variable y u(0,). ΔC( T ) r Tr If y > e, then go to step (a). c) Accept the new solution because of objective improvement. Put n=n+ and if n < e then go to (a). 6.Test fo balancing: Put n=0, if C e ( Tr ) CCe ( T ) r > or if the system was not static then go to (a). 7. Test for freezing point: a) Calculate V ( T r ), C ( T r ). If r=0 then go to 7(a). V ( T ) If ε T ( C ( T0 ) C ( T )) b) Stop because of satisfaction. Tr c) Update the temperature based ontr + = ln( + ) Tr + V ( Tr ) Put r=r+ and then go to 7(a). Analysis of computational results To evaluate the proposed algorithm, a computer program in Visual Basic language was prepared and all the calculations were run on a Pentium 800 MHz. For this purpose, five sample problems based on the literature review were generated. Due to the fact that complex job shop scheduling problems requires a large amount of time to generate optimal solutions, we first consider the lower bound of the solution instead of the optimal solution. In this paper, we used Lingo 6 software to obtain the lower bound of the problem. Once the scheduling problem is assumed solved, on satisfaction of constraints, the sum of costs in different sequences allows the definition of the lower bound of the problem. In this case the lower bound is less than or equal to the optimal solution. Start. Random allocation of

13 Figure. Hybrid algorithm proposed flow chart Manufacturing systems factors To generate a series of selected problems, scheduling factors are chosen among the set of affected variables in sequencing. These variables are: ) Parts type: the set of parts needs to be produced. Each part has a special operation sequence. ) Machines types: the set of machines required to produce the required types of products. Each machine is able to process more than one operation at different points of times (i.e. machine flexibility). ) Operation sequence of a part: the order of machines that the part needs to passed. The required order and number of operations to produce a part is identified in this sequence. 4) Process time: the required time, needed for processing one part on a specific machine, is a random variable. ) Machine capacity: availability time of each machine to process parts. 6) Operational costs: the machine usage cost to process parts in a unit of time.

14 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 the above characteristics are considered. The J m /recrc/l problem is investigated under five selected test problems. Table () shows the characteristics of the selected problems with factors generated at random. Table. Characteristics of selected problems to evaluate the proposed algorithms Run No. of parts No. of machines No. of operations Value identification of problem parameters The next step after generating the selected problems is to identify the values of parameters required in the problem. The summary of generating value procedures for the selected problems is presented in Table (). Table. Parameter characteristics or selected problems Scheduling problem factor Maximum flexibility for each operation Range of process time for each operation Range of idle cost for each machine Range of operational cost for each machine Range of delivery time for each part Range of tardy cost for each part Amount [-0] [-] [0-] [0-0] [-0] Description Uniform distribution Uniform discrete distribution Uniform discrete distribution Uniform distribution Uniform discrete distribution 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 an optimal solution within a reasonable computational timeframes due to the high complexity of job shop scheduling problems. In this paper, Lingo 6 software was used to obtain the lower bound solution. However, the closeness of the optimal solution generated with hybrid method to the lower bound is not an evaluation of the performance of the hybrid method. So, comparing the lower bound with the optimal solution is not verified for making a decision on the quality of performance of the hybrid method. In the next section, we present a test problem solved with the hybrid method. GANTT charts and comparison of computational results with the lower bound are presented. We note 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

15 An example problem (x) We consider a job shop-scheduling problem with three types of parts, machines, and operations for each job. The complete data of the example is presented in Tables () to (6). The computational results generated by the hybrid method and the lower bounds reported by Lingo 6 software are illustrated in Table (7). Table. Input data for operation sequence. Operation Operation Operation Part Part 0 0 Part Table 4. Input data for parts delivery Part Part Part Tardy cost 7 7 Delivery time Table. Input data for machines costs Idle cost Operational cost Machine Machine Part Table 6. Input data for operation time (E t, V t ) Part Part Operation Operation Operation Operation Operation Operation Operation Operation Operation Machine (4.,) (0,0) (.7,7) (0,0) (0,0) (0,0) (0,0) (4.,9) (0,0) Machine (9.,) (4.7,7) (98.,) (.8,8) (0,0) (0,0) (99.4,4) (4.8,8) (0.,) Table 7. Comparison of results by the hybrid method and Lingo 6 software Solution Procedure No. of iterations Objective value Run time (Second) Hybrid algorithm Lingo Percent of difference to Lingo 6 7.% 89.% As shown in Table (7), the gap in the objective value between the two methods is 7.%. As we noted earlier, the solution obtained by Lingo is a lower bound of our solution. Table 7 values show an acceptable performance of the hybrid method in generating an approximate solution. One of the most important criteria in generating an optimal or near-optimal solution is run time. The hybrid method has 89.% time saving over Lingo 6. The amount of saving is highly significant, especially if the problem involves large-scale problems. The makespan of jobs, C max, is always an important factor in determining applicability to real-world production situations. Thus it is necessary to examine this with respect to the solutions generated by

16 Lingo 6 and the hybrid method. We present the Gantt chart of makespan for each solution procedure in Figures () and (4). Gant Chart M O O O 6 7 M O O O Figure. Gannt chart of JSSP x from the hybrid method Gant Chart M O O O M O O O O Figure 4. Gannt chart of JSSP x from Lingo 6 As shown in Figure (), the makespan is 6 times unit based for the hybrid method. The value based on the solution from Lingo 6 is equal to 8 (see Figure 4). However, the objective value by the hybrid method is 7.% more than the lower bound while decreasing in run time and makespan. These are the main factors to significantly compare and evaluate the efficiency of the proposed method. The calculation solution of 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 compression. 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 the selected problems are shown at Table (8). The solutions show that the hybrid method exhibits good performance in different situations most of the time and the run time is decreased significantly in comparison to the run time 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 improved. The solution comparisons are presented in Figure (). 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 this hybrid method reduces the calculations involved in solving the selected problems to generate near-optimal solutions. As shown in Figure (6), the difference or gap between run times for the two 6

17 methods increases as the problem size increases. This fact shows that the run time is improved by using the hybrid method to generate a near-optimal solution. Table 8. Computational results of selected problems for the hybrid and lower bound methods No. of experiments No. Types of parts No. Types of machines No. Types of operations Solution algorithm No. of iterations Objective values Run time (second) Hybrid Lingo Hybrid Lingo Hybrid Lingo Hybrid Lingo Hybrid Lingo Objective value difference Value objective Objective values ** ** 6*6*6 0** 0** ** ** 6*6*6 0** 0** Objective value difference Problem size ت Hybrid alg. lingo6 Problem size a) Difference in objective values b) Objective values Figure. Comparison of objective value for the hybrid and lower bound methods Runtime Runtime ** ** 6*6*6 0** 0** Prob. size ** ** 6*6*6 0** 0** Runtime Runtime Prob. size a) Run time difference b) Run time Figure 6. Comparison of run time for the hybrid and lower bound methods 7

18 Lingo6 Objective Time Figure 7. Increasing trend on time based on objective values from Lingo 6 SA Objective Time Figure 8. Increasing trend on time based on objective values from the hybrid algorithm As can be observed from Figure (), the difference in objective values based on the hybrid method and Lingo 6 increases as problem size increases; and this increase in objective values is because of the various constraints to solve a large number parameter problem by optimization software. According to Figure (6), the difference of runtimes between two procedures increases exponentially when the problem size increases. The case for using the hybrid method seems obvious for problems likely to consume a large amount of time in computational resolution. Figure (7) shows that the run time increases exponentially as the objective value increases in the Lingo 6 software. However, in SA the slope of the run time curve for finding the optimal or approximate solution decreases when the objective values increase, as depicted in Figure (8). Discussion and conclusion The computational results show that the hybrid method exhibits high performance in generating optimal or near-optimal solutions, especially for large-scale problems. As the number of problem parameters increase there is an increase in the run time of Lingo 6 exponentially to generate the lower bound for the given problem. This time consuming task makes these types of software packages inapplicable for the identification of a solution for large-scale problems. The computational results show a suitable performance of the hybrid method in extracting acceptable scheduling solutions within an acceptable timeframe. As a result the proposed method generates acceptable and better solutions than the software package for all large-scale scheduling problems. Thus, from the evidence 8

19 from this paper the proposed method is recommended for all stochastic scheduling problems with large numbers of machines, parts, and operations. Since increasing the problem size increases run time exponentially it is obvious that solving largescale problems is a hugely time consuming task. The proposed method generates the optimal or nearoptimal solutions easier and in less time for large-scale problems containing a large number of constraints. Real-world manufacturing situations typically features a lot of constraints and parameters, so the proposed method is highly relevant for many different industries. The main reason for this study was to present a procedure for stochastic job shop scheduling minimizing the difference between delivery time and completion time 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. To solve the model within reasonable time, a hybrid method consisting of neural network and simulated annealing (SA) models was proposed and presented. This method generates the initial feasible solution by a neural network and then a 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 software. The results are evaluated and analyzed. The analysis of results indicates good performance of the hybrid procedure under different situations. The summary of results is as follows: ) There is an exponential increase in run time of Lingo 6, as the problem size increases. This run time is almost constant in the use of the hybrid method for large-scale problems. ) According to the results, the difference percent is 7.% in small-sized problems with low number of parts and machines. This figure increases for larger-sized problems, containing large numbers of machines and parts. So, it seems that the problem size has a high effect on solution quality and also the computational time increases exponentially as the problem size increases. ) The proposed method exhibits acceptable levels of performance in different situations and there is high decrease in comparative time to generate optimal or near-optimal solutions. 4) Makespan obtained form the hybrid method (6 time units) is much better than makespan reported by Lingo 6 (8 time units). This fact is highly significant in evaluating the performance of scheduling procedures in real-world production situations. The results show a positive performance of the hybrid method in lowering makespan in comparison to Lingo 6. ) 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 Baker, K. R. (974) Introduction to sequence and scheduling (New York: John Wiley). Conway, R. W. & Maxwell, W. L. (967) Theory of scheduling (Reading MA: Addison-Wesley). Dubois, D., Fargier, H. & Prade, H. (99) Fuzzy constraint in job shop scheduling Journal of Intelligent Manufacturing 6 pp.-4. Elmaghraby S. E. (00) On the optimal release time of jobs with random processing times, with extensions to other criteria International Journal of Production Economics 74() pp.0-. Fox, M. S. & Zweben, M. (99) Knowledge based scheduling (San Francisco, CA: Morgan Kaufman Publishing). Golenko-Ginzburg, D. & Gonik, A. (00) Optimal job-shop scheduling with random operations and cost objectives International Journal of Production Economics 76() pp Hentenryck, P. V. (989) Constant satisfaction and logic programming (Cambridge, MA: MIT Press). Jain, S. & Meeran, S. (998) Job-shop scheduling using neural networks International Journal of Production Research 6() pp Johns, A. & Rabelo, L. (998) Survey of job shop scheduling techniques NISTIR, National Institute of Standards and Technology Gaithersburg, MD ( Smith, K. A. (999) Neural network for combinatorial optimization: a review of more than a decade of research Informs Journal on Computing () pp.-4. Steinhofel, K., Albrecht, A. & Wong, C. K. (999) Two simulated annealing-based heuristics for the job shops scheduling problem European Journal of Operational Research 8() pp Thomalla, C. S. (00) Job-shop scheduling with alternative process plans International Journal of Production Economics 74() pp.-4. 9

20 Van Ooijen H. P. G. & Bertrand J. W. M. (00) Economic due-date setting in job-shops based on routing and workload dependent flow time distribution functions International Journal of Production Economics 74() pp Veral E. (00) Computer simulation of due-date setting in multi-machine job shops Computer and Industrial Engineering 4() pp Yang, S. & Wang, D. (000) Constraint satisfaction adaptive neural network and heuristics combined approaches for generalized job-shop scheduling IEEE Transactions on Neural Networks () pp