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Chapter 15 Short-term Scheduling 1

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Outline ► Introduction ► Loading Jobs: Assignment Method ► Sequencing Jobs ► One-machine sequencing ► Two-machine sequencing

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INTRODUCTION 3

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Introduction Scheduling is the process by which the production or service plan is implemented. Scheduling determines where and when a job is to be processed. A scheduling is a timetable for performing activities, utilizing resources, or allocating facilities. 4

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Introduction 5 Two issues Loading: to which machine a job should be assigned Sequencing: in what order the jobs assigned to a machine should be processed

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Introduction Five printing jobs (1, 2, 3, 4, and 5) are received from local customers and each of them may be done on either of the two high-speed copiers (A and B) in the shop. Scheduling the five jobs calls for a two-step procedure. 1.Determine to which copier each job should be assigned (i.e., loading). 2.Determine the order in which the jobs assigned to each copier should be processed (i.e., sequencing). 1 2 3 4 5 jobcopier A B Loading: assign jobs 1, 2 and 4 to copier A; assign jobs 3 and 5 to copier B Sequencing: 2 then 1 then 4 on A; 5 then 3 on B 6

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Importance of Short-Term Scheduling ▶ Effective and efficient scheduling can be a competitive advantage ▶ Faster movement of goods through a facility means better use of assets and lower costs ▶ Additional capacity resulting from faster throughput improves customer service through faster delivery ▶ Good schedules result in more dependable deliveries

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LOADING JOBS Assignment Method 8

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Loading 9 Assignment method: Quantitative technique for identifying the optimal job-machine pairings Goal is to allocate one and only one job to one and only one machine such that the total cost is as low as possible Arrange information into a square matrix called the assignment matrix

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Loading Procedure 10 1. For each row in the original matrix, subtract the smallest number from each of the numbers 2. For each column in the new matrix, subtract the smallest number from each of the numbers 3. Determine the minimum number of (horizontal or vertical) lines needed to cover all the zeros in the existing matrix. If this is equal to the number of rows (or columns) of the square matrix, go to Step (5); otherwise, go to Step (4) 4. Two sub-steps a) Subtract the smallest uncovered number from each of the uncovered numbers in the matrix b) Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3) 5. Start with any row or column containing only one zero (a tie can be broken arbitrarily), match the "job" and the "machine" associated with the zero and then exclude them from further consideration by drawing a horizontal line and a vertical line. Repeat the same process until the complete assignments have been made.

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Example 1 11 The XYZ Corporation produces four products (A, B, C, and D), which can be manufactured by any of the four machines (1, 2, 3, and 4) in the Production Department. Because of expensive set-up costs, each product is usually produced by one and only one machine. Given the following cost data (in dollars), determine the product-machine assignments such that the total cost is minimized. What is the minimum total cost? 1234 A30504030 B20103020 C304020 D4030 40 Assignment Matrix

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Solution 12 1234 A30504030 B20103020 C304020 D4030 40 Assignment Matrix Step 1: For each row in the original matrix, subtract the smallest number from each of the numbers. 1234 A B C D 020100 02010 2000 1000 Step 2: For each column in the new matrix, subtract the smallest number from each of the numbers 1234 A020100 B 02010 C 2000 D1000 Step 3: Determine the minimum number of (horizontal or vertical) lines needed to cover all the zeros. If this is equal to the number of rows (or columns) of the square matrix, go to Step (5); otherwise, go to Step (4) 1234 A020100 B 02010 C 2000 D1000 4 lines to cover all zeros Matrix includes 4 lines Thus, 4 = 4 Step 5: Starting with any row or column containing only one zero, match the "job" and the "machine" associated with the zero and then exclude them from further consideration. Repeat the same process until the complete assignments have been made 1234 A020100 B 02010 C 2000 D1000 A 1 B 2 D 3 C 4 Total cost = 30 + 10 + 30 + 20 = $90

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13 Example 2 At Valley Hospital in Tyler, TX, nurses beginning a new shift report to a central area to receive their primary patient assignments. Not every nurse is as efficient as another with particular kinds of patients. Given the following patient roster, list of nurses, and estimated task times (in minutes), which nurse should take care of which patient so that the total amount of time needed to complete the routine tasks on this shift will be as small as possible? What is the minimum total time requirement? Nurse 1Nurse 2Nurse 3Nurse 4 Jones70110115130 Hawkins1010513550 Becker30907535 Sweeney55251565 Assignment Matrix

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Solution 14 N1N2N3N4 J70110115130 H1010513550 B30907535 S55251565 Assignment Matrix Step 1 N1N2N3N4 J H B S 0404560 09512540 060455 4010050 Step 2 N1N2N3N4 J0304555 H08512535 B050450 S400045 Step 3 N1N2N3N4 J0304555 H08512535 B050450 S400045 3 lines to cover all zeros Matrix includes 4 lines Thus, 3 ≠ 4 Step 4(a): Subtract the smallest uncovered number from each of the uncovered numbers in the matrix N1N2N3N4 J H B S 01525 55955 0 0 3050450 700045 Step 4(b): Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3). Go back Step 3 N1N2N3N4 J001525 H055955 B3050450 S700045 4 = 4 Step 5 N1N2N3N4 J001525 H055955 B3050450 S700045 B N4 S N3 J N2 H N1 Total time = 35 + 15 + 110 + 10 = 170

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EX 1 in class 15 Coach Terry Young is putting together a relay team for a 400-meter relay. The four swimmers are Gary Hall (GH), Mark Spitz (MS), Jim Montgomery (JM), and Chet Jastremski (CJ) and each of them must swim 100 meters of breaststroke (BR), backstroke (BA), butterfly (BU), or freestyle (FR). Based on the past records, the coach believes that each swimmer will attain the times (in seconds) given in the following table: 1)To minimize the team’s total time for the race so that it has the best chance to win, which swimmier should swim which stroke? What is the minimum total time for the team to complete the relay BRBABUFR GH55 5254 MS53585552 JM51545556 CJ57555654 2)Repeat (1) by assuming that it takes Gary Hall and Mark Spitz, respectively, 52 seconds and 54 seconds, to swim 100 meters of freestyle. By how many seconds does the minimum total swimming time differ from the one in (1) in this case?

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Loading 16 Summary of traditional loading problem The number of jobs is equal to the number of machines Each of the jobs is processed by any of the machines The goal is to minimize the total cost Three special loading problems in reality Case I: unequal numbers of jobs and machines Case II: prohibited job-machine combination Case III: maximization objective

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Loading (special case I) 17 Case I: Unequal numbers of jobs and machines Solution A dummy row (job) or column (machine) of zero costs should be introduced to make the assignment matrix a square one The standard method discussed before should then be applied in the usual manner. Be careful in interpreting the final optimal solution obtained since the dummy job or machine added does not exist.

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Example 3 18 Four professors are responsible for three projects funded by the National Science Foundation. The table presented below shows the time (in weeks) needed for each professor to finish each project. Determine the optimal assignments as well as the minimum total time requirement. 123 Thompson506040 Johnson627836 Nelson577438 Parkinson426645 Assignment Matrix

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Solution 19 1234 T5060400 J6278360 N5774380 P4266450 Assignment Matrix Step 1 1234 T5060400 J6278360 N5774380 P4266450 Step 2 1234 T8040 J201800 N151420 P0690 Step 3 1234 T8040 J201800 N151420 P0690 4 = 4 Step 5 1234 T8040 J201800 N151420 P0690 P 1 T 2 J 3 Minimum total time = 42 + 60 + 36 = 138 weeks Note that Professor Nelson will not be responsible for any project since "4" is a dummy project Dummy machine

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Loading (special case II) 20 Case II: prohibited job-machine combination A job cannot be processed by a machine for some reason Solution Replace the cost corresponding to the prohibited job-machine pair in the matrix with an exceedingly large positive number (e.g., 9,999) The standard method discussed before should then be applied in the usual manner.

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Example 4 21 Three managers (1, 2, and 3) are to be assigned to three fast-food outlets (A, B, and C). The following table indicates the expense (in dollars) to be incurred for each manager to run each outlet. Suppose that Manager 3 is not willing to work at outlet B due to travel distance. What are the optimal personnel assignments? What is the minimum total expense? ABC 14,0003,5003,400 24,8005,0004,700 34,1003,2004,400 Assignment Matrix

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Solution 22 Assignment Matrix ABC 14,0003,5003,400 24,8005,0004,700 34,1003,2004,400 Manager 3 is not willing to work at outlet B due to travel distance, thus, replace 3,200 with 99,999. Revised Assignment Matrix ABC 14,0003,5003,400 24,8005,0004,700 34,10099,9994,400 Step 1 ABC 16001000 2 3000 3095,899300 Step 2 ABC 160000 21002000 3095,799300 Step 3 ABC 160000 21002000 3095,799300 3 = 3 Step 5 ABC 160000 21002000 3095,799300 3 A 1 B 2 C Total cost = 4,100 + 3,500 + 4,700 = $12,300

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Loading (special case III) 23 Case III: maximization objective Solution Replace every number with the difference between the largest number in the entire matrix and the number under consideration. The standard method discussed before should then be applied in the usual manner.

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Example 5 24 The following table shows the expected sales volumes (in units) if four salespersons are assigned to four different marketing territories. Determine the optimal assignment plan and the maximum total sales. 1234 Mary1,5001,4001,600 Jim1,6001,5001,450 Lori1,5501,450 1,350 Tom1,6501,5001,4501,500 Assignment Matrix

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Solution 25 Assignment Matrix 1234 M1,5001,4001,600 J 1,5001,450 L1,5501,450 1,350 T1,6501,5001,4501,500 Revised Assignment Matrix 1234 M15025050 J 150200 L100200 300 T0150200150 Step 1 1234 M10020000 J0100150 L0100 200 T0150200150 Step 2 & 3 1234 M100 00 J00150 L00100200 T050200150 3 ≠ 4 Step 4 & 3 1234 M J L T 50 0100 50 200 00 00 00 050 4 = 4 Step 5 1234 M200 00 J0050 L000100 T05010050 M 4 L 3 J 2 T 1 Total sales = 1,650 + 1,500 + 1,450 + 1,600 = 6,200 units

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EX 2 In Class 26 The following table contains information on the cost to run three jobs on four available machines. Determine an assignment plan that will minimize costs Machine ABCD 112161410 Job298137 31512911

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SEQUENCING JOBS One-machine sequencing Two-machine sequencing 27

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Sequencing 28 To determine the order in which the jobs assigned to a machine are to be processed One-machine sequencing Two-machine sequencing

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One-machine sequencing 29 Each of the jobs needs to go through only one machine in a certain order Applications Books ("jobs") borrowed by students need to be checked out at the circulation desk ("machine") of a university library. Patients ("jobs") are treated by a doctor ("machine") in the local hospital. Passengers ("jobs") check in at an airline's ticket counter ("machine") in the airport. Research manuscripts ("jobs") are typed by a secretary ("machine"). Tasks ("jobs") are performed by a worker ("machine") in a manufacturing plant

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30 Objectives – minimize m: total time needed to process the set of jobs ACT: average completion time (or mean flow time) AJL: average job lateness AWT: average waiting time NLJ: number of late jobs One-machine sequencing

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31 Dispatching (or priority) rules FCFS: First come, first served, i.e., the job that arrives first is processed first EDD: The job with the earliest due date is processed first SPT: The job with the shortest processing time is processed first One-machine sequencing

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32 Example 6 Five machines in a shop have failed and will be repaired by a maintenance mechanic. The estimated repair times (in days) along with the due times (in days) for the machines are given below. Assume that the machines were down in the order shown in the table. Apply each of the FCFS, EDD, and SPT rules to dispatch the jobs and then compute m, ACT, AJL, AWT, and NLJ for each of the resulting sequences MachineABCDE Repair time47263 Due date15168239

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33 MachineABCDE Repair time47263 Due date15168239 Solution (1) FCFS: First come, first served Job (i) Processing time for job i (P i ) Waiting time of job i (W i ) Completion time of job i (C i ) Due time for job i (d i ) Lateness of job i (L i ) total A B C D E 4 7 2 6 3 15 16 8 23 9 Sum of the processing time of preceding jobs w i = p 1 + p 2 +... + p i-1 0 4 4 + 7 =11 4 + 7 + 2 =13 4 + 7 + 2 + 6 =19 waiting time plus its processing time c i = w i + p i 4 7 + 4 = 11 2 + 11 = 13 6 + 13 = 19 3 + 19 = 22 L i = max {c i - d i, 0} 0 0 13 – 8 = 5 0 22 – 9 = 13 22 476918 m = 22 days ACT = 69/5 = 13.8 days AJL = 18/5 = 3.6 days AWT = 47/5 = 9.4 days NLJ = 2 machines

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34 MachineABCDE Repair time47263 Due date15168239 Solution (2) EDD: The job with the earliest due date is processed first Job (i) Processing time for job i (P i ) Waiting time of job i (W i ) Completion time of job i (C i ) Due time for job i (d i ) Lateness of job i (L i ) total C E A B D Order the due date from the smallest to the largest: 8 (C), 9 (E), 15 (A), 16 (B) and 23 (D) 8 9 15 16 23 2 3 4 7 6 0 2 2 + 3 = 5 2 + 3 + 4 = 9 2 + 3 + 4 + 7 = 16 2 + 0 = 2 3 + 2 = 5 4 + 5 = 9 7 + 9 = 16 6 + 16 = 22 0 0 0 0 0 223254 m = 22 days ACT = 54/5 = 10.8 days AJL = 0/5 = 0.0 days AWT = 32/5 = 6.4 days NLJ = 0 machines

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35 MachineABCDE Repair time47263 Due date15168239 Solution (3) SPT: The job with the shortest processing time is processed first Job (i) Processing time for job i (P i ) Waiting time of job i (W i ) Completion time of job i (C i ) Due time for job i (d i ) Lateness of job i (L i ) total C E A D B Order the repair time from the smallest to the largest: 2 (C), 3 (E), 4 (A), 6 (D) and 7 (B) 8 9 15 23 16 2 3 4 6 7 0 2 2 + 3 = 5 2 + 3 + 4 = 9 2 + 3 + 4 + 6 = 15 2 + 0 = 2 3 + 2 = 5 4 + 5 = 9 6 + 9 = 15 7 + 15 = 22 0 0 0 0 6 223153 m = 22 days ACT = 53/5 = 10.6 days AJL = 6/5 = 1.2 days AWT = 31/5 = 6.2 days NLJ = 1 machine 6

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Solution 36 Applying the SPT rule leads to the sequence with the minimum ACT and AWT Applying the EDD rule leads to the sequence with the minimum AJL and NLJ The FCFS rule should be used if fairness is the major concern mACTAJLAWTNLJ FCFS2213.83.69.42 EDD2210.806.40 SPT2210.61.26.21 Summary

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EX 3 in Class 37 Processing time (including setup times) and due date for six jobs waiting to be processed at a work center are given in the following table. Determine the sequence of jobs, the average flow time (ACT), average job lateness (AJL), average waiting time (AWT), and the number of late jobs (NLJ) at the work center, for each of these rules: FCFS, SPT and EDD. (assume jobs arrived in the order shown) JobABCDEF Processing time (day)28410512 Due date7164171518

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Two-Machine Sequencing 38 Each of the jobs to be processed has to go through two machines in the same order Applications Athletes ("jobs") competing in the Olympic Games must be subject to a series of two drug tests ("two machines") after each event Cars ("jobs") in a garage must be sanded ("first machine") and then painted ("second machine")

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39 Objectives Minimize the total time needed to process the set of jobs Johnson's rule Step 1: Select the job that has the shortest processing time among all processing times on both machines. If the time is associated with the first machine, then the job should be processed first. If the time is associated with the second machine, however, then the job should be processed last. Step 2: Schedule the job identified in Step (1) and then eliminate it along with its processing times on both machines. If all of the jobs have been scheduled, stop; otherwise, go to Step (1) and work toward the center of the sequence Two-Machine Sequencing

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40 Example 7 Hirsch Products, Inc., produces certain computer connectors, which first require a shearing operation and then need a punch press operation. The company currently has orders for five jobs with estimated processing times (in days) given below: JobPQRST Shearing341272 Punch press516104 Questions 1.Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. 2.Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations 3.What is the total idle time on both operations?

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41 Solution JobPQRST Shearing341272 Punch press516104 Sequencing 1.Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. QTPRS 2.Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations TPSRQShearing Punch press TPSR Q 25122428 2 61112 22 243031 3.What is the total idle time on both operations? There is no idle time on "Shearing" Idle time on "Punch Press" = (2 - 0) + (12 - 11) + (24 - 22) = 5 days Total idle time = 0 + 5 = 5 days.

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EX 4 In Class 42 A group of six jobs is to be processed through a two-machine flow shop. The first operation involves cleaning and the second involves painting. Determine a sequence that will minimize the total completion time for this group of jobs. Processing times are as follows: JobABCDEF Work Center 16482612 Work Center 25397815 Questions 1.Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. 2.Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations 3.What is the total idle time on both operations?

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