# 5 Scheduling. Operations Planning and Control

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1 5 Scheduling Operations Planning and Control

2 Some Background Machines (resources) are Machines process jobs (molding machine, x ray machine, server in a restaurant, computer ) Machine Environment Single Machine Parallel Machines (identical vs. different) Flow Shops: different machines (e.g. assembly lines) Each job must be processed by each machine exactly once All jobs have the same routing A job cannot begin processing on the second machine until it has completed processing on the first Job Shops Each job may have its own routing Open Shops (e.g. car repair shop) Jobs have no specific routing 2

3 Scheduling Algorithms Instance: particular set of data for the model Exact algorithm: Optimum solution for every instance why Heuristic algorithm: a good solution, we hope, optimal or close to optimal for every instance 3

4 Measures Completion time of job i, Ci Flow time of Job i: Fi=Ci ri, ri as release date Lateness of job i: Li=Ci di, di as due date Tardiness of job i: Ti = max {0;Li}, Li>0 Earliness of job i: Ei=max{0, Li} Number of tardy jobs: Ni 4

5 Some Application Gantt Charts Developed by Henry Gantt (1911) Consider the following four job, three machine jobshop scheduling problem Assume the following sequences: on M on M on M3 5

6 Some Application Gantt Charts (cont.): Example The Gantt Chart oriented towards machines is: Last operation of job 1 is on machine 3 and is completed at time 14. So, completion time, Ci : C 1 = 14 (on machine 3) Also: C 2 = 11 (on machine 3); C 3 = 13 (on machine 1) and C 4 = 10 (on machine 1) 6

7 Some Background Gantt Charts (cont.): Example The makespan is: Since F i = C i r i but r i = 0 in this example for all i, then flowtime and completion time are the same. Total flowtime is: Min 7

8 Some Background Gantt Charts (cont.): Example The lateness and tardiness of a job The total lateness is Number of tardy jobs is (δ i = 1) The total tardiness is The maximum tardiness is 8

9 Common Sequencing Rules FCFS. First Come First Served. Jobs processed in the order they come to the shop SPT. Shortest Processing Time. Jobs with the shortest processing time are scheduled first EDD. Earliest Due Date. Jobs are sequenced according to their due dates CR. Critical Ratio. Schedule the next job with the shortest CR value. Compute the ratio of remaining time until the due date and processing time of the job

10 Common Sequencing Rules A machining center in a job shop for a local fabrication company has 5 unprocessed jobs (1 to 5). Given the processing times and due dates, apply the sequencing rules to determine scheduling options Mean Flow time, Average Tardiness and # tardy jobs Job Number Processing Time Due Date

11 Single Machine Scheduling Introduction Applicability Single Machine Aggregated Machines Bottleneck Machines Methods Simple Methods Target on performance measures Optimization procedures (heuristic, optimal) Sequence dependent setup times Static vs. Dynamic scheduling 11

12 Single Machine Scheduling Minimizing Flow time What if your in process inventory costs dominate? Minimize total flow time tends to minimize total holding costs Example Proposed sequence: Total flowtime (F)=? F = p 1 + (p 1 +p 2 ) + (p 1 +p 2 +p 3 )+...+(p 1 +p p n ) F = n p 1 + (n 1) p p n For this problem F = High value P and delivers asap

13 Single Machine Scheduling Minimizing Flow time Shortest Processing Time (SPT) Sequence of jobs ordered from the smallest to largest processing times Is this optimal? Theorem. SPT sequencing minimizes total flowtime on a single machine with zero release times. Proof. We assume an optimal schedule is not an SPT sequence. S is optimal sequence (assumed) with iand then j S is a schedule where j comes before I The set of jobs B comes before i/j or j/i and A comes after p i > p j TF(S) = TF(B) + (t+p i ) + (t+p i +p j ) + TF(A) TF(S ) = TF(B) + (t+ p j ) + (t+ p j +p i ) + TF(A) TF(S) TF(S )= p i p j > 0» t is the completion of the last job in B, TF(A) and TF(B) are total flowtimes of jobs in A and B 13

14 Single Machine Scheduling Minimizing Flow time Example (cont.) Optimal Schedule is Completion times C 1 = 11, C 2 = 2, C 3 = 7, C 4 = 4 and C 5 =15 Total Flow time = Total Completion Time = 39 Remarks on the SPT Rule Minimizes total time jobs spend in the system (because all release times are 0) Minimizes the mean number of jobs waiting to be processed (mean work in progress) Also minimizes Total Lateness (Li=Ci di). Why? 14 Professor penalty Parkinson Law

15 Single Machine Scheduling Maximal Tardiness and Maximal Lateness Due date oriented measure Earliest due date sequence (EDD) Minimizes the Maximal Tardiness (T max ), T max =max{0;li} Minimizes the Maximal Lateness (L max ), Li=Ci di Example EDD sequence is Tardiness of the jobs is (0, 0, 2, 1, 0) 15 Customer satisfaction Minmax

16 Single Machine Scheduling Number of Tardy Jobs EDD may have several jobs somewhat tardy If the fixed cost component of jobs being tardy dominates we wish to have the most of them on time Moore s Algorithm Step1. Compute the tardiness for each job in the EDD sequence. Set N T =0, and let k be the first position containing a tardy job. If no job is tardy go to step 4. Step 2. Find the job with the largest processing time in positions 1 to k. Step 3. Remove job j* from the sequence, set N T =N T+1, and repeat Step1. Step 4. Place the removed N T jobs in any order at the end of the sequence. This sequence minimizes the number of tardy jobs 16

17 Single Machine Scheduling Number of Tardy Jobs Example EDD sequence Step 1: The tardiness is (0, 0, 2, 1, 0) Job 4 in the third position is the first tardy job; Step 2: The processing times for jobs 5, 3 and 4 are 4, 3, 2, respectively; largest processing time for job 5 Step 3: Remove job 5, go to step 1 Step 1: EDD sequence is ; completion times (3, 5, 7, 11) and tardiness (0, 0, 0, 0) Go to step 4. Step 4: schedule that minimizes the number of tardy jobs is / and has 1 tardy job: Jobs 5 17

18 Single Machine Scheduling Precedence Constraints: Lawler s Algorithm Objective function g i is a non decreasing function of the flow time F i Examples = minimizing maximum lateness ;0 minimizing maximum tardiness 18

19 Single Machine Scheduling Precedence Constraints: Lawler s Algorithm Concept Back scheduling At each step determine the set of jobs V not required to precede any other Among V choose job k that satisfies and corresponds to the processing time of the current sequence Job k is scheduled last Determine again V and is reduced by 19

20 Lawler s Algorithm Example An automotive painting and repair has 6 cars waiting to repairs. Three (1,2,3) from car rental and he agreed to finish these cars based on the due dates. Cars 4,5,6 from a retailer dealer, he agreed that car 4 be completed first (customer is waiting). The processing times and due dates are available for each job. How should be the schedule to minimize the maximum tardiness? Job Pi Due Date

21 Single Machine Scheduling Minimizing Set up Times Sequence dependent set up times The time to change from one product to another may be significant and may depend on the previous part produced p ij = time to process job j if it immediately follows job i Examples: Electronics industry Paint shops Injection molding Minimizes makespan, since it also considers set up Problem is equivalent to the traveling salesman problem (TSP) Plastic Tops 21

22 Single Machine Scheduling Minimizing Set up Times SST = Shortest Set up Time Heuristic A metal products manufacturer has contracted to ship metal braces each day for four customers. Each brace requires a different set up on the rolling mill: *Job C cannot follow job D, because of quality problems SST heuristic:» Step 1 starting arbitrarily by choosing one Job: A» Step 2 B has the smallest set up time following A; A B» Step 3 C has the smallest set up time of all the remaining jobs following B; A B C» Step 4 D is the last remaining job; A B C D A with a makespan of =14 Starting point 22

23 Single Machine Scheduling Minimizing Set up Times A regret based algorithm A regret is a penalty for a decision that was not made Each job must be included once: at least one element from each row Pick the smallest element in each row and their sum is the lower bound on makespan Reduced matrix Row reduction Column reduction Sum of reduced coef = lower bound Find the reduced matrix! Has a 0 in each column and each row 23

24 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) The reduced matrix is 1 2 If job B does not follow (come after) job A, some other job must follow A» C adds no set up time (is already in the lower bound) Some job must precede B: D has 0 set up» Thus, we have 0 regret not to chose B to follow A A B, A?;? B; 24

25 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) The reduced matrix is (cont.) 1 2 Zero cell C D» If C does not precede D then we must select other job to precede D (? D)» A can precede D : A D with a regret of 1 time unit» C? : C A with zero regret time unit» Select the jobs pair based on the highest regret value 25

26 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Find the cycle sequence that minimizes the set up time Data p ij element of the set up time matrix (even if reduced) R ij regret for element ij, where p ij = 0 C max makespan of the partial sequence L iteration n jobs 26

27 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 0: C(max) = 0 and L = 1 Step 1: Reduce the Matrix 3 27

28 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2: Calculate the regret 3 Step 3 Choose the largest regret : 17 Step 4 Assign a job pair: Job 2 immediately follows job 5 (5 2) L = 1+1; We prohibit

29 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) New Matrix Step 1: reduce the matrix C(max) = = 24 29

30 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2 Calculate the regret Step 3 Chose the largest regret: 9 Step 4 Assign a job pair: 3 1 Prohibit 1 3 Step 1 Reduce the matrix: not possible 30

31 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2 Calculate regret Step 3 Choose the largest regret: 3 Step 4 Assign job pair : 1 4; Partial sequence: 5 2, Prohibit 4 1 and 4 3 (to keep from being chosen) Final Matrix Choose 2 3 and 4 5 > sequence The total set up time is 24 31

32 Parallel Machines Introduction A job can be processed in any of the machines The time to process a job is the same on any machine A job consists of a single operation Decision Which machine processes the job? In what order? 32

33 Parallel Machines Introduction List Schedule Sequence of all jobs Assign the next job on the list to the machine with the smallest amount of work assigned Procedure Step 0. Let H i =0, i=1,2,...,m be the assigned workload on machine i, L=([1],[2],...,[n]) the ordered list sequence, C j =0, j=1,2,...,n, and k=1 Step 1. Let j*= L k and H i* =mini =1,m {Hi}; Assign job j* to be processed on machine i*, C j* =H i* +p j*,hi*=h i* +p j* Step 2. Set k=k+1, if k>n, stop. Otherwise go to step 1. 33

34 Parallel Machines Flow Time Consider a facility with 3 identical machines and 15 jobs that need to be done as soon as possible; Processing times(after SPT): 34

35 Parallel Machines Makespan Use the longest processing time list (LPT) Truck Assign the next job on the list to the machine with the least total processing time assigned (heuristic) 35

36 Flow Shops Introduction Jobs processed sequentially on multiple machines All jobs processed in the same order Makespan on a Two Machine Flow Shop Johnson s Algorithm Example Machines Jobs Total Time

37 Flow Shops Makespan on a Two Machine Flow Shop Johnson s Algorithm Example (cont.) Natural schedule for example has a total makespan of 22 h Sequence : makespan of 17» Is this optimal? The makespan must be as large as the sum of the processing times on either machine Makespan must account for unavoidable idle times» For each machine, add the minimum processing time of a job in the other machine» Example: the bound becomes 17, so is optimal 37

38 Flow Shops Makespan on a Two Machine Flow Shop Johnson s Algorithm Step 1» Select the job with the lowest processing time on each machine from the schedulable job list» If the list is empty, the procedure is finished» If the processing time of the job selected is from machine 1, go to step 2, otherwise, go to step 3 Step 2» Schedule the job in the earliest position of the sequence and remove it from the schedulable job list» Return to step 1 Step 3» Schedule the job in the latest position of the sequence and remove it from the schedulable job list» Return to step 1 Johnson s algorithm provides the optimal solution 38

39 Flow Shops Heuristics CDS Heuristic Convert a m machine problem into a two machine problem. How? Procedure Start with: k=1 and l=m; then k=2 and l=m 1; until: k=m 1 and l=2 m 1 schedules are generated Use the best of these m 1 schedules 39

40 Flow Shops Heuristics CDS Heuristic Data Use the CDS to solve the problem First use the Johnson s algorithm for machines 1 and 4 40

41 Flow Shops Heuristics CDS Heuristic Second combine M1 with M2 to pseudomachine 1 and M3 with M4 to pseudomachine 2 Finally combine M1+M2+M3 into pseudomachine 1 and M2+M3+M4 into pseudomachine 2 41

42 Flow Shops Heuristics CDS Heuristic Gantt Chart for CDS schedule 42

43 Job Shops Introduction Different routings for different jobs Difficult to schedule precedence constraints (n!) m possible schedules 43

44 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Job Sets with Machines A and B Machine A: {AB}, {A}, {BA} Machine B: {BA}, {B}, {AB} Why in this order? The order of jobs within the set is to be determined Procedure Machine A: {AB} jobs ordered by Johnson s Algorithm, then {A} in any Shortest Processing Time {BA} jobs in reverse Johnsons order Machine B: {BA} jobs reverse Johnsons order, then {B} in SPT {AB} jobs in ordered by Johnson s Algorithm 44

45 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Example 45

46 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Example 46

47 Stochastic Scheduling: Static Case Single machine case. Suppose that processing times are random variables. If the objective is to minimize average weighted flow time, jobs are sequenced according to expected weighted SPT. That is, if job times are t 1, t 2,..., and the respective weights are u 1, u 2,... then job i precedes job i+1 if E(t i )/u i < E(t i+1 )/u i+1.

48 Stochastic Scheduling: Static Case (continued) Multiple Machines. Requires the assumption that the distribution of job times is exponential, (memoryless property). Assume two parallel machines processing n jobs. Then the optimal sequence is to schedule the jobs according to LEPT (longest expected processing time first). Johnsons algorithm for scheduling n jobs on two machines (flow shop) in the deterministic case has a natural extension to the stochastic case as long as the job times are exponentially distributed.

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