Integrated approach in solving parallel machine scheduling and location ( ScheLoc ) problem

Article history: Received November 4 2015 Received in Revised Format April 6 2016 Accepted April 22 2016 Available online April 24 2016 Scheduling and layout planning are two important areas of operations research, which are used in the areas of production planning, logistics and supply chain management. In many industries locations of machines are not specified, previously, therefore, it is necessary to consider both location and scheduling, simultaneously. This paper presents a mathematical model to consider both scheduling and layout planning for parallel machines in discrete and continuous spaces, concurrently. The preliminary results have indicated that the integrated model is capable of handling problems more efficiently. © 2016 Growing Science Ltd. All rights reserved


Introduction
While in many scheduling problems, the locations of machines are fixed it is possible to show how to consider location and scheduling problems simultaneously.Obviously, this integrated method enhances the modeling power of scheduling for different real-life problems (Heßler & Deghdak, 2015).Hennes and Hamacher (2002) are believed to be the first who introduced the idea of scheduling and layout planning.In their study, schedules and location are examined on a graph consisting of n nodes and m edges where, each node is considered as a storage location for a job.The primary objective was to find the optimal location and schedule of a single machine by minimizing the maximum completion time in two different scenarios.The first scenario considers location of facility could be considered only on edges while in the second scenario, locations can be considered on other parts.Elvikis et al. (2007Elvikis et al. ( , 2009) ) presented some polynomial solution methods for the planar ScheLoc makespan problem, which incorporates an special kind of a scheduling and a rather general, planar location problem, respectively.Kalsch and Drezner (2010) integrated both the location of the machine and the scheduling of the jobs executed by the machine by analyzing two different objectives of the makespan and the total completion time.They considered some important properties of the models and provided a lower bound for the objective functions.The single machine ScheLoc problems with Euclidean, rectilinear and general ℓq norms were solved by the "big triangle small triangle" branch-and-bound technique (Lawler, 1973;Drezner & Suzuki, 2004).According to Scholz (2012a,b), geometric branch-and-bound approaches with mixed continuous and combinatorial variables are more suitable solution methods for ScheLoc problems.Table 1 shows different single machine ScheLoc problems tackled by various authors.

The proposed study
ScheLoc is associated with scheduling and layout planning of a certain jobs on some unique machines with determined processing times.All jobs are available on machines according to the following, Distance between storage and machine The time jobs are available for processing on machine = The time jobs are available at storage+ Speed of transportation facilities The primary objective is to minimize the makespan.The problem can be considered in two forms of discrete and continues.

Discrete ScheLoc
In this case, machines can be located on special places.For instance, as shown in Fig. 1, five alternative locations are considered for two machines and four jobs are stored in inventory with specified processing times.The mathematical problem is stated as follows, As we can observe from the proposed model, the objective function given in Eq. (2) minimizes the makespan.Eq. ( 3) insures that all machines are assigned while Eq.(4) determines that each job has to be executed only on single machine.Eq. ( 5) specifies that job i can only be started when it reaches to machine j.Eq. ( 6) and Eq. ( 7) specify that job i can be started when the process of the previous job was already finished.According to Eq. ( 8), a job can be processed only on a particular machine.Eq. ( 9) computes the makespan and finally Eq. ( 10) demonstrates the type of variables.The problem can be examined under two scenarios.For the first scenario, layout planning is accomplished by minimizing sum of the times for all jobs assigned to machines and then scheduling of jobs on machines are determined.This problem is first formulated as follows, When all yj are determined, the optimal values are denoted as j y and the following problem is solved, constraints (4)(5)(6)(7)(9) ) 14 ( For the second scenario, we first solve the following problem Now, the jobs are sorted according to non-decreasing order of availability times.In case, there are two jobs with the same availability times, the one with less processing time is considered and then the jobs are assigned to machines.

Continous ScheLoc
In this case, machines can be assigned anywhere in job floor as shown in Fig. 2 as follows, The proposed study is formulated as follows, The objective function minimizes the makespan, Eq. ( 21) ensures that each job is processed only on one machine, Eq. ( 22) specifies that job i can only be started when it reaches to machine j.Eq. ( 23) and Eq. ( 24) specify that job i can be started when the process of the previous job was already finished.Eq. ( 25) computes the makespan and finally Eq. ( 26) demonstrates the type of variables.In this study, distances are computed as follows, and Eq. ( 27) is linearalized as follows, Therefore, the following equations are added to the problem statement, The resulted problem formulation is still nonlinear because of nonlinear terms given in Eq. ( 32) and Eq. ( 33).However, since only ki u or ki v appears nonzero in the final solution and ki g or ki h appears in nonzero form in the objective function, we may disregard them in the problem formulation (Lawler, 1973) and the problem is formulated as follows, , z 0,1 To solve the resulted model in continuous form, we consider two scenarios.For the first scenario, the following problem formulation has to be solved.
Using Eqs.(28-35) yields the following problem statement, Therefore we have, ) 46 ( For the second scenario, we first solve the following Here if ik AT receives a value one, it means that job i is assigned to machine k and the assignment of jobs to machines are the same as discrete form.

The results
In this section, we present the results of the implementation of the proposed study using some randomly generated numbers.All problems are coded in GAMS using personal computer with 2.2 GHz core i7 CPU and 6 GB RAM.Table 2 shows the input parameters.

Discrete ScheLoc
In this case, we consider additional input data for the proposed study as given in Table 3 as follows, Table 4 shows the results of the implementation of the proposed study.In addition, Fig. 3 shows the results.In order to have a better understanding on the performance of the proposed method we have generated 10 sample test problems, solved with GAMS software package and in case GAMS software could not reach optimal solution, we have reported the best solution after 1000 seconds shown in (*).Table 5 shows the results of the problem.For all problems relative gaps are computed as follows,    As we can observe from the results of Tables (5-7), the proposed integrated model has relatively performed better than alternative methods.

Continous model
For the case of continuous model, we have solved the problem and Table 8 and Fig. 4 show the results of the proposed study.Again, as we can observe from the results of Table 9, Table 10 and Table 11, the proposed integrated model has been able to solve the randomly generated problems in less amount of time.In most cases, the proposed model has provided better objective function values.In summary, in both cases, there have been some improvement on the performance of the ScheLoc problem using both discrete and continuous spaces.

Conclusion
In this paper, we have presented a mathematical model for ScheLoc problem in discrete and continuous spaces.The proposed study has been formulated under different conditions and the implementations were examined using various randomly generated numbers.The preliminary results have indicated that the proposed study of this paper could provide promising results.As future study, one may consider the problem under uncertain conditions using fuzzy numbers and we leave it as a future study for interested researchers.

Aknowledgement
The authors would like to thank the anonymous referees for constructive comments on earlier version of this paper.

Fig. 3 .
Fig. 3.The locations of the machines and job schedule for discrete case

Fig. 4 .
Fig. 4. The locations of the machines and job schedule for continous case

Table 1
Different studies in machine scheduling and location (ScheLoc) problem The discrete parallel machine ScheLoc problem

Table 2
Input data used for testing different instances

Table 3
Input data for discrete ScheLoc

Table 4
The summary of the results of the optimal solution

Table 5
The results of the optimal solution for 10 sample test problems

Table 6
The results of the optimal solution for 10 sample test problems

Table 7
The results of the optimal solution for 10 sample test problems

Table 8
The summary of the results of the optimal solution Again, we have generated 10 addional test problems and Table9, Table10and Table11present details of our findings.

Table 9
The results of the optimal solution for 10 sample test problems

Table 10
The results of the optimal solution for 10 sample test problems

Table 11
The results of the optimal solution for 10 sample test problems