PRODUCTION SCHEDULING WITH SEQUENCE DEPENDENT SETUPS AND JOB RELEASE TIMES

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1 PRODUCTION SCHEDULING WITH SEQUENCE DEPENDENT SETUPS AND JOB RELEASE TIMES PROGRAMACIÓN DE LA PRODUCCIÓN CON TIEMPOS DE PREPARACIÓN DEPENDIENTES DE LA SECUENCIA Y FECHAS DE LLEGADA DE TRABAJOS JAIRO R. MONTOYA TORRES Universidad de La Sabana, Cía, Colombia, airo.montoya@unisabana.edu.co MILTON SOTO FERRA Universidad del Norte, Barranquilla, Colombia FERNANDO GONZÁLEZ SOLANO Universidad del Norte, Barranquilla, Colombia Received for review July 3 t, 2009, accepted Marc 4 t, 2010, final version April, 4 t, 2010 ABSTRACT: Tis paper studies a sort term production sceduling problem inspired from real life manufacturing systems consisting on te sceduling a set of obs (production orders) on bot a single macine and identical parallel macines wit te obective of minimizing te makespan or maximum completion time of all obs. Jobs are subect to release dates and tere are sequence dependent macine setup times. Since tis problem is known to be strongly NP ard even for te single macine case, tis paper proposes a euristic algoritm to solve it. Te algoritm uses a strategy of random generation of various execution sequences, and ten selects te best of suc scedules. Experiments are performed using random generated data and sow tat te euristic performs very well compared against te optimal solution and lower bounds, and requiring sort computational time. KEYWORDS: Sceduling, sequence dependent setup times, release dates, randomness, euristic. RESUMEN: Este artículo estudia un problema de programación de la producción en el corto plazo inspirado de sistemas de fabricación reales en los cuales se tiene un conunto de tareas (órdenes de producción) tanto en una configuración de una máquina como en máquinas paralelas idénticas con el obetivo de minimizar el lapso de fabricación o tiempo máximo de terminación de todos los trabaos. Las tareas están suetas a fecas de disponibilidad diferentes y existen tiempos de preparación de las máquinas dependientes de la secuencia de procesamiento. Puesto que este problema es conocido como fuertemente NP completo, incluso para el caso de una máquina simple, este artículo propone un algoritmo eurístico para resolverlo. El algoritmo emplea una estrategia de generación aleatoria de varias secuencias de procesamiento de los trabaos y luego selecciona el meor de estos programas. Se desarrollaron experimentos computacionales empleando datos generados aleatoriamente. Los resultados muestran que el procedimiento propuesto se desempeña muy bien comparado con la solución óptima o con cotas inferiores, requiriendo un menor tiempo de cálculo. PALABRAS CLAVE: Programación de la producción, tiempos de preparación dependientes de la secuencia, fecas de disponibilidad, aleatoriedad, eurística. Dyna, year 77, Nro. 163, pp Medellin, September, ISSN

2 261 Dyna 163, INTRODUCTION Sceduling is a decision making process tat is used on a regular basis in many manufacturing and services industries. It deals wit te allocation of resources (often simply called macines) to tasks (obs) over given time periods and its goal is to optimize one or more obectives [1]. Efficient production scedules can result in substantial improvements in productivity and cost reductions. Generating a feasible scedule tat best meets management s obectives is a difficult task tat manufacturing firms face every day [2]. In many industries, te decision to manufacture multiple products on common resources results in te need for cangeover and setup activities, representing costly disruptions to production processes. Terefore, setup reduction is an important feature of te continuous improvement program of any manufacturing, and even service, organization. It is even more critical if an organization expects to respond to canges like sortened lead times, smaller lot sizes, and iger quality standards. Every sceduler sould understand te principles of setup reduction and be able to recognize te potential improvements. Setup time, in general, can be defined as te time required to prepare te necessary resource (e.g., macines, people) to perform a task (e.g., ob, operation). Setup times can be of two types: sequence independent and sequence dependent. If setup time depends solely on te task to be processed, regardless of its preceding task, it is called sequence independent. On te oter and, in te sequence dependent type, setup time depends on bot te task and its preceding task [3]. Sceduling problems wit sequence dependent setup times can be found in various production, service, and information processing environments [3]. For example, in a computer system application, a ob requires a setup time to load a different compiler if te current compiler is not suitable. In a printing industry, a setup time is required to prepare te macine (e.g., cleaning) wic depends on te color of te current and immediately following obs. In a textile industry, setup time for weaving and dying operations depends on te obs sequence. In a container/bottle industry, setup time relies on te sizes and sapes of te container/bottle, wile in a plastic industry different types and colors of products require setup times. Similar situations arise in cemical, parmaceutical, food processing, metal processing, paper industries, and many oter industries/areas. As stated by Allaverdi and Sorous [3], in today s manufacturing sceduling problems it is of significance to efficiently utilize various resources. Treating setup times separately from processing times allows operations to be performed simultaneously and ence improves resource utilization. Tis is particularly important in modern production management systems suc as Just in Time (JIT), Optimized Production Tecnology (OPT), Group Tecnology (GT), cellular manufacturing, and time based competition. Te benefits of reducing setup times include [3]: reduced expenses, increased production speed, increased output, reduced lead times, faster cangeovers, increased competitiveness, increased profitability and satisfaction, enabling lean manufacturing, smooter flows, broader range of lot sizes, lower total cost curve, fewer stock outs, lower inventory, lower minimum order sizes, iger margins on orders above minimum, faster deliveries, and increased customer satisfaction. Te importance and benefits of incorporating setup times in sceduling researc as been investigated by many researcers (see for instance [4,5,6,7,8]. Tis paper studies te problem of ob sceduling on bot a single macine and identical parallel macines wit sequence dependent setup times and release dates. Te single macine environment does represent a building block for more complex configurations. Many researcers ave dealt wit ob sceduling problems on a single macine under different constraints. A fundamental issue is te inerent difficulty of one macine sceduling problems tat involve sequence dependent setup times. Pinedo [1] sowed tat te makespan minimization on a single macine wit sequence dependent setup

3 Montoya et al 262 times is strongly NP ard, wic means tat is not possible to find optimal solutions in reasonable computational time for large sized instances. For te parallel macine case, computational results are not encouraging. In tis paper, we are interested in studying suc sceduling problems adding te constraint of obs aving unequal release times. Tis paper is organized as follows. Section 2 is devoted to analyze te problem under a single resource (single macine) environment. Section 3 extends te study for te multiple parallel macines context. Related relevant literature and computational experiments are presented respectively witin bot sections. Te paper ends in section 4 by presenting te conclusions. 2. ANALYSIS OF THE SINGLE MACHINE CASE Formally, we first consider te problem of sceduling a set of n obs on one macine. Job, wit =1,...,n, is caracterized by its integer processing time p and an non negative integer release date r. Sequence dependent macine setup times are also considered. Tat is, if ob k is executed on te macine immediately after ob, a setup time s k is needed during wic te macine cannot process any ob. We consider te obective of minimizing te makespan of te scedule or total completion time of all obs. Using te classical notation in Sceduling Teory, tis problem is noted as 1 r, s C k max. Wen all obs ave equal release dates ( r = 0, ), te one macine sceduling problem, noted as 1 C, is equivalent to te Traveling s k max Salesman Problem (TSP), wic is known to be NP ard [1]. Tis means tat no efficient (polynomial time) algoritm can be found to solve large sized instances. Hence, te problem considered in tis paper is at least tat difficult. Te problem of ob sceduling wit sequencedependent macine setup times ave been largely studied in te literature. State of te art surveys are presented in [9,10,11]. For te one macine case, even toug complexity analysis is not encouraging, researcers ave developed exact approaces based on branc and bound, dynamic programming or integer linear programming. Te obective function under study in tis paper is te makespan and can be expressed as: C max = n = 1 p + s k k Wen setup times are dependent on te sequence, minimizing makespan becomes equivalent to minimizing te total setup time. Tat is because te sum of processing times remains a constant troug te wole sceduling wen all information about obs is deterministic and known at te initial time of sceduling. Tis problem corresponds to wat is usually called te Traveling Salesman Problem (TSP). In a TSP, eac city corresponds to a ob and te distance between cities corresponds to te time required to cange from one ob to anoter. If te setup times for all pairs of obs are indifferent to teir sequencing order wen sceduled consecutively, te sceduling problem is equivalent to a symmetrical TSP, oterwise, it is equivalent to an asymmetrical TSP [9]. One of te pioneering works on te sequencedependent setup time problem was presented by Gilmore and Gomory [12] wo modeled and solved te problem as a TSP. Presby and Wolfson [13] provided an optimization algoritm tat is suitable only for small problems. Bianco et al. [14] formulated te problem 1 r, s C as a mixed integer linear k max program and developed a euristic algoritm using lower bounds and dominance criteria. For te problem 1 prec, s k, He and Kusiak [15] proposed a simpler mixed integer formulation and a fast euristic algoritm of low computational time complexity. Ozgur and Brown [2] developed a two stage traveling salesman euristic procedure for te problem were similar products produced on te macine can be partitioned into families. Tere are several works presented in te literature tat consider oter obective functions. Barnes and Vanston [16] combined branc and bound wit dynamic programming to solve te

4 263 Dyna 163, 2010 problem noted as s k w C 1. For te + s k case of precedence constraints wit a special structure (cains), Uzsoy et al. [8] developed branc and bound algoritm for 1 prec, s k L max and Uzsoy et al. [17] developed dynamic programming algoritms for 1 prec, s k L max and 1 prec, s k U, were te obective function corresponds to te minimization of te number of tardy obs. Tan and Narasiman [18] proposed a simulated annealing algoritm to minimize total tardiness ( 1 s T ). Tan et al. k [19] later compared te performance of branc and bound, genetic searc, simulated annealing and random start pairwise intercange euristics for te same problem. Different versions of genetic algoritms ave also been proposed (e.g. [19,20]. França et al. [21] proposed a memetic algoritm wile Gagne et al. [22] proposed an Ant Colony Optimization (ACO) algoritm for te same problem. Cang et al. [23] proposed a matematical programming model wit logical constraints for te problem 1 r, s k w T. Tey also proposed euristics and conducted computational experiments wic revealed tat te euristics can efficiently solve te problem. Wang [24] studied te single macine sceduling problem wit time dependent learning effect and considerations of setup times wit various obective functions based on completion times of obs. 2.1 Te proposed algoritm Tis paper first analyzes te problem on a single macine noted as 1 r, s C. Te proposed k max randomized euristic algoritm is presented in tis section. Te basics of te procedure are presented next. To scedule a set of n on a single macine, we can observe tat, from a total of n positions in te scedule, we ave to select one position for eac ob. Te euristic proposed ere is based on a random insertion strategy, in wic random numbers are generated from an equilikely distribution between 1 and n, in order to define te position of a ob in te scedule. A certain number of iterations are required so as to improve te initial solution (scedule). Te algoritm is described in detail in figure 1. Algoritm Random Insertion One macine Initialization 1. Enter te number n of obs. 2. For eac ob, enter its processing time p and its release date r. Order obs in a list by increasing order of teir release times. Break ties by increasing order of processing times. 3. Read setup times s k for eac pair of obs and k, wit k. 4. Define te number of iterations (niter). Algoritm 5. Set = 1, te first iteration. Set = Generate an integer random number R from an equilikely distribution between 1 and n. 7. Scedule ob on position defined by R. If tis position is already assigned, go to step Do = + 1 and repeat from step 6 wile n (tat is, until all obs are sceduled). 9. Ensuring tat release dates are respected, compute, te makespan for te scedule of iteration. 10. Do = + 1 and repeat from step 6 wile niter (tat is, until te number of iterations is reaced). 11. Select te scedule wit min (tat is, select te scedule wit minimum makespan over all te iterations). Figure 1. Random insertion algoritm for te singlemacine problem 2.2 Experiments In order to analyze computational performance of proposed algoritm, experimental studies were conducted on a PC Pentium bi processor Dual Core 1.73 GHz. Exact solution metods were programmed using X press IVE wile te proposed euristic was programmed using Visual Basic for Applications (VBA) in MS Excel spreadseets. Data was generated using a similar structure as proposed by Cu [25] and later extended by Nessa et al. [26] to consider setup times.

5 Montoya et al 264 Integer processing times were generated from a uniform distribution [1, 100]. Integer release dates were generated using a uniform distribution [ 0, α n ], were n is te number of obs to be sceduled and α is a real wit values 0.6, 1.5 and 3.0. Integer setup times were generated from a uniform distribution 0, min p ]. Five instances for eac of value of [ α were generated. Problems wit 10, 20, 50 or 100 obs were considered. Experiments were run wit equal and unequal release dates. A full factorial experimental design gave a total of 120 testing scenarios. Because of te random beavior of te proposed algoritm, 10 replications for eac instance scenario were run and te best sequence (i.e., te sequence wit minimum value of te makespan) was registered and compared against te optimum makespan. Te first sets of experiments were performed assuming tat all obs are released to te macine at te same time. Tat is, we are supposing tat r = 0 for all obs. Te second set of experiments, te same values of bot processing and setup times were taken but in addition considering unequal integer nonnegative release dates for obs, tat is wit r 0. OPT For te performance analysis, let and be respectively te makespan obtained using te proposed Random Insertion euristic and te optimum makespan. Te performance of proposed euristic was computed using te deviation from te optimal solution as: C % dev = OPT max OPT 100 % Tables 1 and 2 summarize te results obtained from te experiments for te single macine environment wen all obs ave equal release dates (i.e. respectively wen r = 0, ) and OPT r 0,. In bot tables, represents te average values of te optimal makespan and represents te average value of te makespan applying te proposed euristic. Te last column of bot tables corresponds to te average value of te deviation from te optimal solution for eac set of obs. Table 1. Average makespan for experiments wit r = 0 and m = 1 Average values # obs OPT % % % % Average 3.4% % dev Table 2. Average makespan for experiments wit r 0 and m = 1 # obs OPT Average values % dev % % % % Average 4.4% For te case of equal release dates, our algoritm te average deviation from te optimal solution is 3.4%. Wen unequal release date are present ( r 0 ), te average deviation is 4.4% of te optimal solution. Analyzing te individual instances, in 4% of te cases te euristic obtained te optimal makespan, wile in 29% of te cases te value of te makespan was witin a 2% of te optimal value. Finally, it is important to note tat te running time of te algoritm for small instances (10 ob and 20 ob instances) was less tan 3 seconds, wile te time required to run te experiments for large instances was between 20 and 30 seconds for 50 ob instances and about 55 seconds for instances wit 100 obs. In comparison wit te optimal solution approac, te matematical model required about 30 minutes and 1 our to solve small and large instances, respectively. 3. ANALYSIS OF THE CASE OF M IDENTICAL MACHINES IN PARALLEL In real life, usually discrete manufacturing processes ave several m macines in parallel. In tis section we extend te algoritm previously proposed to solve te problem. Using te classical notation in Sceduling Teory, te problem under study in noted as Pm r, s C k max.

6 265 Dyna 163, 2010 In te literature te problem under study as been very little studied in te literature. Some related works are cited next. Guinet [27] proposed a matematical formulation to minimize te makespan and te total completion time of obs wit identical release times (i.e., r = 0 ) for all obs. Heuristics and metaeuristics procedures ave been proposed for several obective functions, suc as due date related obectives (e.g. [28,29,30] or flowtime related obectives (e.g. [31,32,33]. Guinet [27] also suggested tat makespan minimization problem wen all obs ave equal release dates ( r = 0, ), te problem is equivalent to te Veicle Routing Problem (VRP) wit service time requirements. Te problem wit obs arriving at different release dates as been very little studied in te literature, to te best of our knowledge. Nessa et al. [26] considered te obective of minimizing total completion time of obs (problem Pm r, s C. k Te problem under study in tis paper, Pm r, s C, as only been studied by Kurz and k max Askin [34] wo proposed several euristics algoritms, including multiple insertion and a genetic algoritm. Tese autors also derived a datadependent lower bound for te makespan criterion. Teir compared teir euristics between tem, but tey neiter computed te optimal makespan nor compare te performance of teir euristics against te optimum nor te lower bound Proposed algoritm modified Our algoritm described in section 2 for te single macine case can be easily modified for application in te parallel macine environment. Te first modification consists on computing te number of obs tat can sceduled on eac macine. Ten, obs are randomly selected and assigned to macines respecting te workload balance defined. Te modified algoritm is described in detail in figure Experiments and results A computational study was also performed using te same random generated data as described in section 2.2. We considered ere configurations wit m = 3 and m = 5 identical macines in parallel. As for te single macine case, because of te random beavior of te proposed algoritm, 10 replications for eac instance scenario were run and te best sequence (i.e., te sequence wit minimum value of te makespan) was registered and compared against te optimum makespan. As explained previously, te NP completeness of tis problem unable us to obtain optimal solutions witout excessive computational costs even for small instances [34]. A lower bound on te makespan can be found by looking at te minimum preemptive scedule makespan [35]. Tis lower bound, owever, can be very poor, especially in cases wit a ig range of processing times [34]. Algoritm Random Insertion Parallel Macines Initialization 1. Enter te number n of obs. 2. Enter te number m of identical parallel macines. 3. For eac ob, enter its processing time p and its release date r. Order obs in a list by increasing order of teir release times. Break ties by increasing order of processing times. 4. Read setup times s k for eac pair of obs and k, wit k. 5. Define te number of iterations (niter). Algoritm 6. Compute te number of obs to be sceduled on te macines. For te first m 1 macines. Tis bound is computed as n/ m. Te m t macine as assigned te oter obs. 7. Set = 1, te first iteration. 8. Generate an integer random number R from an equilikely distribution between 1 and n. 9. Scedule ob R on te first macine wit available positions. If tis ob as already been assigned, repeat from step Repeat from step 8 until all obs ave been sceduled. 11. Ensuring tat release dates are respected, compute, te makespan for te scedule of iteration. 12. Do = + 1 and repeat from step 8 wile niter (tat is, until te number of iterations is reaced). 13. Select te scedule wit min (tat is, select te scedule wit minimum makespan over all te iterations). Figure 2. Random insertion algoritm for te identical parallel macines problem

7 Montoya et al 266 As explained previously, Kurz and Askin [34] derived a preemptive type lower bound on te makespan for eac of te individual data sets using te actual data. Tese autors sowed tat teir lower bound performs well. Tis lower bound is tus computed as: LB ( Pm r, s ) max{ LB 1, LB 2 } were LB 1 and LB 2 k = are, respectively: n 1 LB 1 = p + min s m = { 1,..., n } k 1 LB 2 = max r + p + min k { 1,..., n } k s k Results of our experimental study are ence compared against tis lower bound. Let be te makespan obtained using te proposed LB euristic and let be te value of te lower bound. Hence, te performance of te proposed euristic was computed as te deviation from suc lower bound as: C % dev = LB max LB 100 % Tables 3 and 4 present te summary of results, respectively, wit equal and unequal obs release dates. From tese results, we can observe tat te algoritm performs well, wit reference to te percentage deviation from te lower bound of te makespan: te average deviation, regardless of te number of obs, is 9.9% wit equal release dates and 12.2% for te case wit unequal release dates. Tese results are te first results in literature tat sow te performance of a euristic in comparison against a known lower bound. In terms of te computational costs, te iger te number of obs, te iger te time to find a solution. For te large instance in our tests (100 obs), te computational time was never iger tan 12 seconds. For 10 obs and 20 obs instances, te CPU time was less tan 1 second. Table 3. Results for parallel macines experiments wit r = 0 # macines m = 3 m = 5 # obs LB Avg Avg Avg. % dev 7.7% 7.2% 4.9% 2.9% 16.7% 27.2% 7.7% 5.0% Avg. 5.7% 14.1% Table 4. Results for parallel macine experiments wit r 0 # macines m = 3 m = 5 # obs LB Avg Avg Avg. % dev 0.7% 18.3% 5.7% 3.2% 7.3% 21.6% 9.5% 6.7% Avg. 27.9% 11.3% 4. CONCLUSIONS Tis paper considered te problem of sceduling obs on bot a single macine and identical parallel macines environments subect to release dates and setup times. Setup times, de fined in general as te time required to prepare te necessary resource to perform a ob, add complexity for te analysis of sceduling problems. Since te problem is NP ard, a euristic algoritm was proposed.

8 267 Dyna 163, 2010 Te strategy for sceduling is based on a random insertion of obs in te scedule. Computational experiments were performed using random generated data following similar procedures as in literature. Two main cases were considered. Te first test was carried out wit obs aving equal release dates. Te second set of tests considered non negative release dates ( r 0 ). Compared against te optimal solution, te proposed euristic performed very well giving scedules wit a makespan value no greater tan te 10% of te optimum. In average, te proposed procedure was between about 2% and 6% of te optimal solution. Te computational time was less tan 2 seconds for small instances, and never iger tan 1 minute for large instances (100 obs). An extension to te identical parallel macine environment was also considered. Results of te computational experiments sowed tat our algoritm performs well in comparison wit te lower bound of te makespan value. Te average deviation from tis bound was 9.9% wit r = 0 and 12.2% for te case wit r 0, regardless of te number of obs. ACKNOWLEDGEMENTS Tis work was performed under researc proect CEA supported by Researc Funds from Universidad de La Sabana, Cía, Colombia. Autors wis to acknowledge te anonymous reviewers for teir comments tat allow improving te presentation of te paper. REFERENCES [1] PINEDO, M Sceduling: Teory, Algoritms, and Systems. Springer. [2] OZGUR, C.O., BROWN, J.R A two stage traveling salesman procedure for te single macine sequence dependent sceduling problem. Omega, 23, [3] ALLAHVERDI, A., SOROUSH, H.M Te significance of reducing setup times/setup costs. European Journal of Operational Researc, 187, [4] FLYNN, B.B Te effects of setup time on output capacity in cellular manufacturing. International Journal of Production Researc, 25, [5] KOGAN, K., LEVNER, E A polynomial algoritm for sceduling small scale manufacturing cells served by multiple robots. Computers & Operations Researc, 25, [6] KRAJEWSKI, L.J., KING, B.E., TZMAN, L.P., WONG, D.S Kanban, MRP and saping te manufacturing environment. Management Science, 33, [7] LIU, C.Y., CHANG, S.C Sceduling flexible flow sops wit sequencedependent setup effects. IEEE Transactions on Robotics and Automation, 16, [8] TROVINGER, S.C., BOHN, R.E Setup time reduction for electronics assembly: Combining simple (SMED) and IT based metods. Production and Operations Management, 14, [9] ALLAHVERDI, A., GUPTA, J.N.D., ALDOWAISAN, T A review of sceduling researc involving setup considerations. Omega, 27, [10] ALLAHVERDI, A., NG, C.T., CHENG, T.C.E., KOVALYOV, M.Y A survey of sceduling problems wit setup times or costs. European Journal of Operational Researc, 187, [11] ZHU, X., WILHELM, W.E Sceduling and lot sizing wit sequencedependent setup: A literature review. IIE Transactions, 38, [12] GILMORE, P.C., GOMORY, R.E Sequencing a one state variable macine: a solvable case of te traveling salesman problem. Operations Researc, 12,

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