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**Chapter 15 Short-term Scheduling**

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**Outline Introduction Loading Jobs: Assignment Method Sequencing Jobs**

One-machine sequencing Two-machine sequencing

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INTRODUCTION

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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.

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**Introduction 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. Determine to which copier each job should be assigned (i.e., loading). Determine the order in which the jobs assigned to each copier should be processed (i.e., sequencing). job copier Loading: assign jobs 1, 2 and 4 to copier A; assign jobs 3 and 5 to copier B 1 A 2 Sequencing: 2 then 1 then 4 on A; 5 then 3 on B 3 4 B 5

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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

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**Loading 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 For each row in the original matrix, subtract the smallest number from each of the numbers For each column in the new matrix, subtract the smallest number from each of the numbers 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) Two sub-steps Subtract the smallest uncovered number from each of the uncovered numbers in the matrix Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3) 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 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? Assignment Matrix 1 2 3 4 A 30 50 40 B 20 10 C D

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**Step 1: For each row in the original matrix, subtract the smallest number from each of the numbers.**

Step 2: For each column in the new matrix, subtract the smallest number from each of the numbers Solution Assignment Matrix 1 2 3 4 A 30 50 40 B 20 10 C D 1 2 3 4 A B C D 1 2 3 4 A 20 10 B C D 20 10 10 20 10 10 20 10 10 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) 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 1 2 3 4 A 20 10 B C D 1 2 3 4 A 20 10 B C D 4 lines to cover all zeros A 1 Matrix includes 4 lines B 2 Thus, 4 = 4 D 3 C 4 Total cost = = $90

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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? Assignment Matrix Nurse 1 Nurse 2 Nurse 3 Nurse 4 Jones 70 110 115 130 Hawkins 10 105 135 50 Becker 30 90 75 35 Sweeney 55 25 15 65

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**Solution Step 1 Step 2 Assignment Matrix 40 45 60 95 125 40 60 45 5 40**

J 70 110 115 130 H 10 105 135 50 B 30 90 75 35 S 55 25 15 65 N1 N2 N3 N4 J H B S N1 N2 N3 N4 J 30 45 55 H 85 125 35 B 50 S 40 40 45 60 95 125 40 60 45 5 40 10 50 Go back Step 3 Step 4(a): Subtract the smallest uncovered number from each of the uncovered numbers in the matrix N1 N2 N3 N4 J 15 25 H 55 95 5 B 30 50 45 S 70 Step 3 Step 4(b): Add the same smallest uncovered number to every number where two covering lines intersect. Go to Step (3). N1 N2 N3 N4 J 30 45 55 H 85 125 35 B 50 S 40 4 = 4 N1 N2 N3 N4 J H B S Step 5 15 25 B N4 N1 N2 N3 N4 J 15 25 H 55 95 5 B 30 50 45 S 70 3 lines to cover all zeros S N3 55 95 5 Matrix includes 4 lines J N2 30 50 45 H N1 Thus, 3 ≠ 4 70 45 Total time = = 170

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EX 1 in class 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: BR BA BU FR GH 55 52 54 MS 53 58 JM 51 56 CJ 57 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 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 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)**

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 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. Assignment Matrix 1 2 3 Thompson 50 60 40 Johnson 62 78 36 Nelson 57 74 38 Parkinson 42 66 45

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**Solution Dummy machine Assignment Matrix Step 1 Step 2 Step 3 Step 5**

4 T 50 60 40 J 62 78 36 N 57 74 38 P 42 66 45 1 2 3 4 T 50 60 40 J 62 78 36 N 57 74 38 P 42 66 45 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9 Step 3 Step 5 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9 1 2 3 4 T 8 J 20 18 N 15 14 P 6 9 P 1 T 2 J 3 4 = 4 Minimum total time = = 138 weeks Note that Professor Nelson will not be responsible for any project since "4" is a dummy project

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**Loading (special case II)**

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 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? Assignment Matrix A B C 1 4,000 3,500 3,400 2 4,800 5,000 4,700 3 4,100 3,200 4,400

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Solution Manager 3 is not willing to work at outlet B due to travel distance, thus, replace 3,200 with 99,999. Assignment Matrix Revised Assignment Matrix Step 1 A B C 1 4,000 3,500 3,400 2 4,800 5,000 4,700 3 4,100 3,200 4,400 A B C 1 4,000 3,500 3,400 2 4,800 5,000 4,700 3 4,100 99,999 4,400 A B C 1 600 100 2 300 3 95,899 Step 2 Step 3 Step 5 A B C 1 600 2 100 200 3 95,799 300 A B C 1 600 2 100 200 3 95,799 300 A B C 1 600 2 100 200 3 95,799 300 3 = 3 3 A 1 B Total cost = 4, , ,700 = $12,300 2 C

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**Loading (special case III)**

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 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. Assignment Matrix 1 2 3 4 Mary 1,500 1,400 1,600 Jim 1,450 Lori 1,550 1,350 Tom 1,650

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**Solution Assignment Matrix Revised Assignment Matrix Step 1 Step 2 & 3**

4 M 1,500 1,400 1,600 J 1,450 L 1,550 1,350 T 1,650 1 2 3 4 M 150 250 50 J 200 L 100 300 T 1 2 3 4 M 100 200 J 150 L T Step 2 & 3 Step 4 & 3 Step 5 1 2 3 4 M 100 J 150 L 200 T 50 1 2 3 4 M J L T 1 2 3 4 M 200 J 50 L 100 T 200 200 50 50 100 50 100 50 3 ≠ 4 4 = 4 M 4 L 3 Total sales = 1, , , ,600 = 6,200 units J 2 T 1

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EX 2 In Class 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 A B C D 1 12 16 14 10 Job 2 9 8 13 7 3 15 11

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SEQUENCING JOBS One-machine sequencing Two-machine sequencing

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Sequencing 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**

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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**One-machine sequencing**

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

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**One-machine sequencing**

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

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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 Machine A B C D E Repair time 4 7 2 6 3 Due date 15 16 8 23 9

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**Solution Machine A B C D E Repair time 4 7 2 6 3 Due date 15 16 8 23 9**

(1) FCFS: First come, first served waiting time plus its processing time ci = wi + pi Sum of the processing time of preceding jobs wi = p1 + p pi-1 Li = max {ci - di, 0} Job (i) Processing time for job i (Pi) Waiting time of job i (Wi) Completion time of job i (Ci) Due time for job i (di) Lateness of job i (Li) total A B C D E 4 7 2 6 3 4 15 16 8 23 9 4 7 + 4 = 11 4 + 7 =11 = 13 13 – 8 = 5 =13 = 19 =19 = 22 22 – 9 = 13 22 47 69 18 m = 22 days AWT = 47/5 = 9.4 days ACT = 69/5 = 13.8 days NLJ = 2 machines AJL = 18/5 = 3.6 days

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**Processing time for job i (Pi) Completion time of job i (Ci)**

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

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**Processing time for job i (Pi) Completion time of job i (Ci)**

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

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Solution Summary m ACT AJL AWT NLJ FCFS 22 13.8 3.6 9.4 2 EDD 10.8 6.4 SPT 10.6 1.2 6.2 1 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

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EX 3 in Class 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) Job A B C D E F Processing time (day) 2 8 4 10 5 12 Due date 7 16 17 15 18

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**Two-Machine Sequencing**

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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**Two-Machine Sequencing**

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

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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: Job P Q R S T Shearing 3 4 12 7 2 Punch press 5 1 6 10 Questions Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations What is the total idle time on both operations?

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**Solution Job P Q R S T Shearing 3 4 12 7 2 Punch press 5 1 6 10**

Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. Sequencing T P S R Q Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations 2 5 12 24 28 Shearing T P S R Q Punch press T P S R Q 2 6 11 12 22 24 30 31 What is the total idle time on both operations? There is no idle time on "Shearing" Idle time on "Punch Press" = (2 - 0) + ( ) + ( ) = 5 days Total idle time = = 5 days.

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EX 4 In Class 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: Job A B C D E F Work Center 1 6 4 8 2 12 Work Center 2 5 3 9 7 15 Questions Determine the order in which the jobs should be processed to minimize the total time needed to process the set of jobs. Based on the sequence obtained in Part 1 above, construct a time-phased bar chart (or Gantt chart) for both operations What is the total idle time on both operations?

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