SIMULATION OF JOB SHOP USING ARENA
Mini Project Report
Submitted in partial fulfillment of the requirements for the award of the degree of Master of Technology in Industrial Engineering and Management
by KAILAS SREE CHANDRAN
(Roll No.:M100447ME)
Department of Mechanical Engineering NATIONAL INSTITUTE OF TECHNOLOGY CALICUT April 2011
CERTIFICATE This is to certify that the report entitled “SIMULATION OF JOB SHOP USING ARENA” is a bonafide record of the Mini Project done by KAILAS SREE CHANDRAN (Roll No.: M100447ME), in partial fulfillment of the requirements for the award of the degree of Master of Technology in Industrial Engineering and Technology from National Institute of Technology Calicut.
Dr. R. Sridharan Faculty-in-Charge (ME6194 – Mini Project) Dept. of Mechanical Engineering
Place : NIT Calicut Date : 02/05/2011
ACKNOWLEDGEMENT I am deeply indebted to my guide Dr. R. Sridharan, Professor, Department of Mechanical Engineering, for his invaluable guidance, consistent encouragement and suggestions throughout the course of the work. I wish to express my sincere thanks to Dr. S. Jayaraj, Professor and Head, Department of Mechanical Engineering, for providing the necessary facilities to carry out this work. Last but not the least, I extend hearty thanks to all our teachers and classmates whose constant support and encouragement helped me to complete this mini-project on time.
KAILAS SREE CHANDRAN
ABSTRACT A Job Shop includes jobs which have different sequence of operations. Here a Gear manufacturing Job Shop which produces three types of gears was modeled in ARENA Simulation software to study the Gear flow times (by type), Gear delays at operations locations, Machine utilizations etc. While modeling the system, the sequence of operations for different types of gears, plant layout, distance between different facilities, processing time of each operation, transportation of jobs through the factory, speed of transporter etc. were considered. The model is run for 1 year with three 8 hour shifts per day. The results were analyzed and a suggestion was also made to improve the system.
CONTENTS List of Tables 1
2
3.
i
Introduction
01
1.1
Job Shop Production
01
1.2
Characteristics of Job Shop Production
01
1.3
Importance of Job Shop Production
02
Problem Environment
04
2.1
Problem Statement
04
2.2
Assumptions
05
2.3
Objectives of the Project
06
ARENA Model
07
3.1
Modeling the System
07
3.1.1
Gear Job Arrivals
08
3.1.2
Gear Transportation
09
3.1.3
Gear Processing
12
3.1.4
Gear Departure
15
3.2
Simulating the Model
18
4.
Results
20
5.
Improving the System
23
5.1
Suggestion for Improvement
24
5.2
Comparison of Results
24
6.
Conclusion
Reference
25 26
LIST OF TABLES 2.1
Distances among Job Shop locations
04
2.2
Operation Plan for gears by type
05
5.1
Comparing Gear Delays at Operation Locations
24
5.2
Comparing Resource Utilization
24
5.3
Comparing Gear Flow Time
24
CHAPTER 1 INTRODUCTION 1.1 JOB SHOP PRODUCTION Job Shop production is characterized by the manufacture of one or few numbers of a single product designed and manufactured strictly to customer‟s specifications, within, the given period and wit/tin the price fixed prior to tile contract. Some typical examples of industries engaged in Job Shop production are: general repair shops; special purpose machine tool manufacturers; workshops to manufacture jigs and fixtures for other units; building contractors; tailoring shops manufacturing made-to-measure suite of clothes; manufacturers of ships, cranes, furnaces, turbo-generators, pressure vessels; and others manufacturing articles made to customers orders. 1.2 CHARACTERISTICS OF JOB SHOP PRODUCTION 1.2.1 Disproportionate Manufacturing Cycle Time A considerable amount of pre-planning and organization is necessary in such a venture. Relatively long delays occur at the assembly as well as at the materials processing stages due to lack of materials or components, imbalanced work flow, design changes, design errors detected during manufacture, inaccurate work measurements, etc. which tend to lengthen the manufacturing cycle time. At times, tile time needed to design the product exceeds its manufacturing time. 1.2.2 Large Work-In-Progress The work-in-progress inventory in a Job Shop production is generally very large as detailed scheduling and progress control in this type of production is economically infeasible. For various reasons, jobs get delayed causing temporary work shortages. To overcome work shortages and keep men and machines busy, more work is released to the shop which in turn increases work-in-progress. 1.2.3 Limited Functions Of Production Planning And Control The success of Job Shop production mainly depends on the ability of the engineer in-charge of the contract.
1
1.3 IMPORTANCE OF JOB PRODUCTION 1.3.1 Small Production Runs Job Shop production is characterized by the manufacture of one or few pieces of a product at a time under a separate contract; the production is made strictly to customers‟ specifications. 1.3.2 Discontinuous Flow of Materials The flow of materials and components between different stages of manufacture is highly discontinuous due to imbalance in operation wise work content. 1.3.3 General Purpose Machines and Process Layout Plant and equipment is designed or procured and arranged to obtain maximum flexibility. General purpose machines and handling equipments capable of performing variety of operations with minimum set-tip times are installed in lie of variety in products. Tile machines are arranged to give process layout - layout by function. Similar machines, capable of doing similar type of operations, are grouped together. Presses, for example are kept at one place; milling machines are placed at another place; drilling machines are kept at third place; and so on and so forth. Each group of machines is usually designated as a work centre or a section or a shop. The grouping of machines gives a lot of flexibility in loading and scheduling. Temporary machine breakdowns and operator‟s absenteeism can be taken care of by shifting jobs to another machine or shifting operators from less important jobs to important jobs. 1.3.4 Highly Skilled Labor The labor force is usually highly skilled-highly qualified trade apprentices who are expected to work from minimum instructions. Instructions regarding “what to make” are issued in the form of specifications while instructions as to “how to manufacture” are usually oral. The workmen being highly skilled are expected to work independently and display a great deal of initiative and judgment. They are required to set up their own machines and prepare their OW special tools or production aids in order to further the manufacture of a part or a assembly. 1.3.5 Highly Competent Knowledgeable Supervision Highly competent general engineers are engaged as foreman in the base workshop and a group of site engineers, practical men, with thorough training, capable of taking independent charge of each contract are employed to work at site. Therefore, these engineers (supervisors)
2
in a Job Shop production are the reservoir of job knowledge. The supervisor besides being able administrator is expected to improvise and determine best work methods, determine tool requirements, select the best process and provide management with reliable estimates of labor and materials for specific orders. The span of control - the number of workmen to be supervised by a supervisor - is kept low because of technical nature of the job. 1.3.6 Simple Mechanism Tools control function is simple. Standard tools are stocked while special tools are either made on the shop floor by the operators or purchased on request from supervisor. 1.3.7 Decentralized Process The scheduling activity is more or less decentralized. A schedule is prepared to show the start and completion date of each major component of the product. Job tickets giving completion date of each component are raised and given to the shop. The activity of day to day scheduling is left to the individual shop supervisor.
3
CHAPTER 2 PROBLEM ENVIRONMENT 2.1 PROBLEM STATEMENT Consider a Job Shop producing three types of gears, G1, G2, and G3, for a ship. The job shop is spread out geographically on the factory floor and its layout consists of the following locations:
An arrival dock
A milling workstations with four milling machines
A drilling workstations with three drilling machines
A paint shop with two spray booths
A polishing area with a single worker
A shop exit
The distances among locations are given in Table 2.1 Table 2.1 Distances among Job Shop locations Arrival Dock
Milling Station
Drilling Station
Paint Shop
Polishing Area
Shop Exit
100
100
250
250
550
300
400
150
300
150
400
500
300
400
Arrival Dock Milling Station Drilling Station
100 100
300
Paint Shop
250
400
150
Polishing Area
250
150
400
300
Shop Exit
550
300
500
400
200 200
Gear jobs arrive in batches of 10 units and their inter-arrival times are uniformly distributed between 400 and 600 minutes. Of arriving batches, 50% are of type G1, 30% are of type G2, and 20% are of type G3. A gear job arrives at the arrival dock and from there is dispatched to its specific (type-dependent) sequence of manufacturing operations. A sequence consists of a subset of milling, drilling, painting, and polishing operations. Table 2.2 displays the operations plan showing the sequence of operations and the associated processing times for
4
each gear type. The layout of the job shop and operation sequences of gear types are depicted in Figure 2.1. Table 2.2 Operation Plan for gears by type GEAR TYPE
OPERATION SEQUENCES
PROCESSING TIME(Minutes)
Milling
35
Drilling
20
Painting
55
Polishing
15
Milling
25
Painting
35
Polishing
15
Drilling
18
Painting
35
Polishing
15
G1
G2
G3
Gears are transported among locations by two trucks running at a constant speed of 100 feet/minute. Each truck can carry only one gear at a time. When a job is complete at a location, the gear is placed into an output buffer, a transport request is made for a truck, and the gear waits for the truck to arrive. Once a gear is transported to the next location, it is placed in a FIFO input buffer. Finally, when the polishing operation is completed, the finished gear departs from the job shop via the shop exit. To analyze the performance of the job shop, plan to run a simulation over 1 year of operation. 2.2 ASSUMPTIONS These assumptions were made during modeling the system. a. Transporter (Truck) speed is same for both loaded and empty. b. The freed transporter stays at the destination station until requested by another station. c. The Job Shop works for 24 hours a day in 3 shifts at 8 hours each.
5
Milling Workstation
Painting Shop
Arrival Dock
Shop Exit
G1 Drilling Workstation
G2
Polishing Area
G3 Figure 2.1 Layout of the Job Shop and Operation Sequences by Gear type. 2.3 OBJECTIVE OF THE PROJECT The following statistics are of interest:
Gear flow times (by type)
Gear delays at operations locations
Machine utilizations
The objective of the project is to model the Gear Manufacturing Job Shop in ARENA Simulation Software and find; i. Gear Flow Time (by Type) ii. Gear Delays at operation locations iii. Utilization of Resources iv. Suggest an improvement
6
CHAPTER 3 ARENA MODEL 3.1 MODELING THE SYSTEM The given system is modeled in ARENA Simulation software. Figure 3.1 depicts the Arena model for the gear shop, consisting of three main segments: 1. Gear Job Arrivals 2. Gear Transportation 3. Gear Processing 4. Gear Departure A segment-by-segment walkthrough of the model follows next.
Figure 3.1 Arena Model for the Gear Manufacturing Job Shop
7
3.1.1 Gear Job Arrivals This part includes the arrival section of Gear Jobs. The segment is shown in Figure 3.2.
Figure 3.2 Gear Job Arrivals Segment Gear entities are created in the Create module, called Create Jobs, whose dialog box is displayed in Figure 3.3. The Entities per Arrival field indicates that gear jobs arrive in batches of 10, and the Time Between Arrivals section specifies batch inter-arrival times to be uniformly distributed between 400 and 600 minutes. Following arrival, each incoming gear entity proceeds as a separate entity.
Figure 3.3 Dialog box of the Create module Create Jobs An arriving gear entity next enters the Assign module, called Assign Job Type and Sequence, whose dialog box is displayed in Figure 3.4. Here, a gear entity is assigned a type by sampling it from a discrete distribution, and saving the type code (1, 2, or 3) in its Type attribute. In addition, the ArrTime attribute is assigned the value of the simulation clock, Tnow, for later use in computing the gear entity‟s flow time. Finally, the Arena attribute
8
Entity.Sequence is assigned the value of the Type attribute. This attribute acts as an index that associates a gear type with the corresponding operations sequence.
Figure 3.4 Dialog box of the Assign module Assign Job Type and Sequence. The operations sequences for gear types are specified in the Sequence module from the Advanced Transfer template panel, whose dialog spreadsheet is displayed at the bottom of Figure 3.5. Three sequences (row entries) are defined here, one for each gear type. Each sequence consists of a sequence name (Name column) and a series of steps (Steps column), listed in the order of processing. To specify steps, the modeler clicks the button under the Steps column and pops up the Steps dialog spreadsheet. The five steps of type G1 gears processing are displayed in the middle spreadsheet of Figure 3.5. Each step is a row entry specifying the location name and associated values (under the Assignments column). Clicking the corresponding button pops up the associated Assignments dialog spreadsheet. The assignment of time for the milling time operation is exemplified in the top spreadsheet of Figure 3.5. 3.1.2 Gear Transportation Gear Transportation segment includes jobs arriving to Arrive Dock, Requesting the truck and transporting the jobs to job shop. The segment is shown in Figure 3.6.
9
Figure 3.5 Dialog spreadsheet of the Sequence module (bottom), the Steps dialog spreadsheet for specifying operations steps of type G1 gears (middle), and the Steps Assignment spreadsheet for specifying milling time assignment of type G1 gears (top). Job shop locations are modeled as Station modules. Accordingly, every gear entity proceeds to the Station module, called Arrive_Dock, to model its physical arrival at the job shop‟s arrival dock. From here, gear entities will be transported to the job shop floor to start the first step in their operations sequence.
Figure 3.6 Gear Transportation Segment To this end, a gear entity enters the Request module (from the Advanced Transfer template panel), called Request a Truck, whose dialog box is displayed in Figure 3.7. The Transporter Name field indicates a request for a Fork Truck transporter. If multiple transporters are available, the modeler can specify how to select one in the Selection Rule field. Such selections may be cyclical, random, preferred order (as listed in the Transporter module), smallest distance, largest distance, or a specific transporter. Here the selection rule requests the transporter nearest to the arrival dock. Furthermore, the Save Attribute field specifies that the ID of the selected transporter be saved in the Truck_ID attribute of the requesting gear
10
entity. The saved ID will be used in due time to free that particular truck. Since requesting transporters from multiple locations is a form of contention for resources, the Priority field allows the modeler to assign a priority to requests issued at multiple Request modules (here a high priority is assigned in order to clear the arrival dock as soon as possible). The Entity Location field indicates the location of the requesting entity, and the Velocity field specifies the transporter‟s velocity, which is 100 feet/minute in our case. Finally, gear entities requesting transportation at the same Request module are instructed in the Queue Name field to wait in the queue, called a Truck.Queue, until a transporter becomes available.
Figure 3.7 Dialog box of the Request module Request a Truck As soon as a gear entity grabs a truck, it proceeds to the Transport module, called Transport to Shop Floor, whose dialog box is displayed in Figure 3.8. The Transporter Name and Unit Number fields specify the type and ID of the selected transporter, which here is the truck whose ID is kept in the Truck_ID attribute of the requesting gear entity. The transporter/gear destination is specified in the Entity Destination Type field as the By Sequence option, indicating that the destination is determined by the gear entity‟s sequence number. This field may also specify a Station module name, using the Station option. It can also specify an
11
attribute or expression. The gear entity and the transporter move as a grouped entity at a velocity of 100 feet/minute as specified in the Velocity field. Note that the velocity may depend on trip type, so that an empty truck and a loaded one can be made to move at different velocities.
Figure 3.8 Dialog box of the Transport module Transport to Shop Floor Next, the distances between different facilities are specified. Figure 3.9 displays the dialog spreadsheet of the Distance module (left), as well as a corresponding Stations dialog spreadsheet (right), which pops up on clicking a button under the former‟s Stations column. 3.1.3 Gear Processing This segment includes the actual processing of gears. The gear processing segment encompasses sets of Station modules, each modeling an operation in the sequence, from milling to polishing. Since all sets have the same structure (except for names), only the milling operation logic is explained here. The segment is shown in Figure 3.10. When a gear entity is transported to the milling operation, it enters the Station module, called Milling Station. It then proceeds to the Free module, called Free Truck at Mill, whose dialog box is displayed in Figure 3.11. Here, the Transporter Name and Unit Number fields specify the truck to be freed for use by other gear entities, using the Truck_ID attribute of the freeing gear entity.
12
Figure 3.9 Dialog spreadsheet of the Distance module (left) and the Stations dialog spreadsheet (right).
Figure 3.10 Gear Processing Segment
13
Figure 3.11 Dialog box of the Free module Free Truck at Mill In this case, it enters the Process module, called Milling, whose dialog box is displayed in Figure 3.12. The Seize Delay Release option in the Action field is used to model gear delays at this process. The resource seized is Milling Machine and the processing time is kept in the Milling Time attribute specified in the Sequence module of Figure 3.5. Furthermore, to model four milling machines at the milling workstation, resource Milling Machine has to be declared as having a capacity of four in the spreadsheet view of the Resource module. The capacity of other machine groups is similarly declared. Figure 3.13 shows it.
Figure 3.13 Dialog spreadsheet of Resource module On completing the milling operation, the gear entity proceeds to the Request module, called Request Truck at Milling, whose dialog box is displayed in Figure 3.14. In this module, the gear entity requests transportation to the next operation, similarly to the first request from the arrival dock to the job shop floor (Figure 3.7). Here, it may have to wait in the queue, called Request Truck at Milling.Queue, which serves as the output buffer for the milling process. The transport operation takes place when a truck arrives and both gear and transporter enter the Transport module called Transport From Milling.
14
Figure 3.12 Dialog box of the Process module Milling Gear entities move from one operation to another according to their specified sequences. It should be pointed out that Arena handles all sequencing details at runtime. The internal Arena attribute IS keeps track of each gear entity‟s step number in its sequence. Whenever a sequential transport is requested, Arena increments the IS attribute and indexes into the appropriate Steps module spreadsheet to determine the destination location and travel time. The IS attribute may also be modified by the modeler. 3.1.4 Gear Departure This segment includes the transport of finished gears from job shop to outside. The segment is shown in Figure 3.15. Eventually the gear entity arrives at the Station module, called Shop Exit, which is always the last location in each operations sequence.
15
Figure 3.14 Dialog box of the Request module Request Truck at Milling Next, the transporting truck is freed in the Free module, called Free Truck at Exit, and the finished gear entity is ready to record some statistics and then depart from the model at a Dispose module.
Figure 3.15 Gear Departure Segment Figure 3.16 displays the dialog box of the Record module, called Tally Flow Time. Here, flow times are tallied with the aid of the ArrTime attribute of each finished gear entity. Note that these flow times are tallied by gear type, using the tally set mechanism. The Tally Set Name field indicates that tallies are to be entered in the Flow Times set. Each gear entity indexes into this set using its Type attribute, specified in the Set Index field. The Flow Times set is
16
specified in the Set module spreadsheet from the Basic Process template panel. Figure 3.17 displays the Set spreadsheet and the members of the Flow Times set.
Figure 3.16 Dialog box of the Record module Tally Flow Time
Figure 3.17 Dialog spreadsheet of the Set module (bottom) and the Members dialog spreadsheet of the Flow Times set (top) Arena computes travel times of transporters among Station modules based on their distances and transporter speeds. Figure 3.18 introduces Fork Truck transporters into the model and specifies their parameters in the Transporter module spreadsheet. These include columns for a Name field to specify the transporter set, a capacity field (Number of Units), a Distance Set field for specifying the name of a Distance module allowing the user to specify distances between pairs of Station modules, a Velocity and Units fields that specify the transporter speed (in our case, in feet per minute), and an Initial Position Status column of buttons, which pop up the Initial Position Status dialog spreadsheet. The latter is used to specify the location at which a transporter resides initially (at simulation time 0). Note that the transporter speed is the default speed. Arena allows the modeler to override this value and to
17
further distinguish between the speed of an empty transporter (specified in a Request module) and a loaded transporter (specified in a Transport module).
Figure 3.18 Dialog spreadsheet of Transporter module Finally, the collection of fork truck utilization (a Time-Persistent statistic) and flow-time statistics (Tally statistics) is specified in the Statistic spreadsheet module, as shown in Figure 3.19. Observe that the Arena variable nt(transporter_name) is used to collect transporter utilization, in our case, nt(Fork Truck).
Figure 3.19 Dialog spreadsheet of the Statistic module for collecting fork truck utilization and flow-time tallies. 3.2 SIMULATING THE MODEL The ARENA job shop model was simulated for 1 year. Parameters like Replication Length and Hours Per Day etc. are given in Run Setup. The Figure 3.20 shows it. While simulation, we can see the movement of entities (gears) through different facilities, waiting for processing, transfer between machines etc. We are assuming that the plant works for 24 hours. i.e. Three shifts with eight hours each. After completing the simulation, report will be generated automatically. From the generated report, we can find Gear flow times (by type), Gear delays at operations locations, Machine utilizations etc.
18
Figure 3.20 Dialog box of Run Setup
Figure 3.21 Screen Shot of ARENA Model
19
CHAPTER 4 RESULTS The resulting output report is displayed in Figure 4.1. The Time per Entity section lists the statistics of gear waiting times for each operation. As expected, the average waiting time at the paint shop is very large as compared to the other operation locations, since spray times are quite long. The results are displayed in graphical forms in Figure 4.2, 4.3 and 4.4.
Gear Delays At Operation Locations 50 44.8111
45 Waiting Time(Minutes)
40 35 30 25 20
14.6109
15 10 5
3.0743
0.6656
0 Drilling
Milling
Painting
Polishing
Figure 4.2 Gear Delays at Operation Locations
Resource Utilization 50 44.8
45
37.63
40 Utilization, %
35 29.88
30 25 20 15 10
12.48 8.98
5 0 Drilling M/C
Milling M/C
Paint Booth
Polishing Worker
Figure 4.3 Resource Utilization
20
Fork Truck
Figure 4.1 Simulation results for the gear manufacturing job shop model
The Usage section displays resource utilizations at individual operation locations. For instance, the utilization of a drilling machine is about 0.0898. The User Specified section displays fork truck utilization statistics and flow-time statistics by gear type. The fact that the average flow times are much larger than the total processing time bears witness to excessive delays in resource queues.
21
Gear Flow Time (by Type) 300
Flow Time, Minutes
250
240.46
200 153.2 150 105.16 100 50 0 G1
G2
Figure 4.3 Gear Flow Time
22
G3
CHAPTER 5 IMPROVING THE SYSTEM 5.1 SUGGESTION FOR IMPROVEMENT In Figure 4.2, its visible that Paint shop is having more waiting time compared to other stations. To reduce the wait at the paint booth, we modify the job shop model by increasing the number of paint booth from two to three. The impact of this modification on gear delays and flow times is indicated in the simulation results of Figure 5.1. Clearly, the addition of a paint booth has significantly reduced the delay at the paint shop but slightly increased the delay at the polishing area, because speeding up an operation increases congestion downstream. The overall effect on gear flow times, however, is a slight reduction.
Figure 5.1 Simulation results for the Modified Gear Manufacturing Job Shop Model 5.2 COMPARISON OF RESULTS In the Table 5.1, its visible that Painting operation after modification has a delay of just 16 minutes compared to 44 minutes initially. This improvement is not visible in Resource 23
Utilization, see Table 5.2. Initially Resource Utilization of Paint Booth was 44% but after improvement, it came down to 29% because the number of Paint Booth has increased by one. But the improvement caused to reduce to Gear Flow time of all types of Gears, Table 5.3. GEAR DELAYS AT OPERATION LOCATIONS Initial
Suggestion
Drilling
0.6656
0.6556
Milling
14.6109
14.619
Painting
44.8111
16.082
Polishing
3.0743
8.945
Table 5.1 Comparing Gear Delays at Operation Locations RESOURCE UTILIZATION Initial
Suggestion
Drilling M/C
8.98
8.98
Milling M/C
12.48
12.48
Paint Booth
44.8
29.86
Polishing Worker
29.88
29.86
Fork Truck
37.63
37.42
Table 5.2 Comparing Resource Utilization GEAR FLOW TIME Initial
Suggestion
G1
240.46
203.02
G2
153.2
141.93
G3
105.16
103.93
Table 5.3 Comparing Gear Flow Time
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CHAPTER 6 CONCLUSION The given Job Shop System was modeled in Arena Simulation software and the results were generated. After analyzing the results, it was noticed that the waiting time in paint shop was more compared to other stations. So a suggestion was made to increase the number of paint booth by one. The results before and after improvement were compared. And it was found that the waiting time in paint shop is increasing but the utilization of that station is decreasing. The acceptance or rejection of this suggestion depends on the organization policies. If they are ok with sacrificing the utilization of a machine compared to waiting time, they can go ahead with this suggestion. They have to consider the cost of machine operation and loss due to huge waiting time in a station. The trade-off lies there. Again lot of suggestions can be made in this system, by removing some machines in a station which has less utilization so that we can increase the utilization and we can save money in operating the machine. We can also consider changes in the assumptions „a‟ and „b‟.
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REFERENCE [1]. David Kelton W., Randall P. Sadowski, Deborah A. Sadowski. 2001. Simulation with Arena (2nd ed.), McGraw Hill. [2]. R. Tavakkoli, M., Daneshmand Mehr. 2005. A Computer Simulation Model for Job Shop Scheduling Problems Minimizing Makespan, Computers & Industrial Engineering, 48, 811–823. [3]. Rahime Sancar E. & Arslan O. 2009. Simulation Analysis of Lot Streaming in Job Shops with Transportation Queue Disciplines, Simulation Modelling Practice and Theory, 17,442–453. [4]. Tayfur Altiok & Benjamin Melamed. 2007. Simulation Modeling and Analysis with Arena (2nd ed.), Elsevier Inc.
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