2 Useful tools to assist in production scheduling Assignment problem (which job to which work center?)Sequencing rules (which job to process next?)Next Best rule (minimize setup costs)Sequencing through two successive work centers (minimize the completion time for the last job through the process)
3 Assignment problem Objective is to minimize total process time Number of jobs to be loaded = number of work centers or machinesEach machine is assigned one and only one jobLinear programming problemObjective function (maximize or minimize) subject to constraint conditionsManual solution procedureExcel Solver
4 Example of assignment problem The metalworking work center of a job shop has been assigned the task of processing four orders. Four machines in the work center are all capable of processing any of the orders. The foreman must decide which job to allocate to each machine.machine 1machine 2machine 3machine 4job 113 min162114job 2171319job 31220job 411
5 Manual solution procedure to assignment problem (1) Column reduction: Subtract the lowest cost in each column from every cost element in that column. Do this for every column. This new matrix of opportunity costs is now used in the next step.(2) Row reduction: Subtract the lowest cost in each row from every cost element in that row. Do this for every row. This new matrix of opportunity costs is now used in the next step.
6 Manual solution procedure to assignment problem (cont.) (3) Cover the zeroes: Cover all the zero elements in the matrix with horizontal and/or vertical lines. Find the minimum number of lines necessary to cover all zeroes. If the number of lines required is equal to the number of machines available, an optimal solution has been reached. The optimal assignments are found by examining the zero elements in the matrix. If the number of lines is fewer than the number of machines, proceed to step (4).
7 Manual solution procedure to assignment problem (cont.) (4) Create new zeroes: Begin with the matrix from step (3). Find the smallest uncovered cost element (not covered by a line) and subtract it from all uncovered cost elements, including itself; add it to all cost elements at the line intersections. All other cost elements remain unchanged. Now erase all horizontal and vertical lines and return to step (3).
8 Excel solution to assignment problem Load Excel Solver (click on “File” tab and go to “Options,” choose “Add-ins” from the left menu, select “Solver Add-in” and then click the “go” button, check “Solver Add-in” in the checkbox)Solver is now available under the “Data” tabDefine changing cellsCalculate total cost (or profit)Set up resource usageSet up Solver Parameters (Target Cell, Max/Min, Changing Cells, Constraints, and Options)For metalworking center example
9 Another assignment problem example Merchants Bank has four new tellers (A-D) with varying skills who are to be assigned to the main bank or one of the branches (locations 1-4). The criterion for assigning tellers to locations is minimized customer waiting time. An index of customer waiting time is shown in the table below for each of the four locations and varying teller skills. Make teller assignments such that the overall waiting time will be minimized. What is the total waiting time for the optimal assignment?teller Ateller Bteller Cteller Dlocation 140 min605030location 2location 340location 4
10 Other applications of linear programming Transportation problemObjective function subject to resource constraintsReports (Answer, Sensitivity, or Limits) can be generated by clicking on any of the choices in the window
11 Sequencing Which job to process next? Methodology to follow Shortest processing time: select the job having the least processing timeEarliest due date: select the job that is due the soonestFirst-come, first served: select the job that has been waiting the longest for this workstationFirst-in-system, first-served: select the job that has been in the shop the longestSlack per remaining operations: select the job with the smallest ratio of slack to operations remaining to be performed
12 Performance evaluation criteria Choice of sequencing methodology to choose is dependent on the performance evaluation criteria to be appliedTotal job completion timeAvg job completion timeAvg job waiting timeAvg job latenessAvg number of jobs in the systemAvg number of jobs waitingSet-up costsIn-process inventory costs
13 job (in order of arrival) Sequencing exampleFive jobs waiting, one workstation/machinejob (in order of arrival)job time (days)due dateA46B1720C1418D912E11
14 Sequencing example evaluation Using shortest processing timeSequence is A-D-E-C-BConstruct timelineTotal job completion time: 55 daysAvg job completion time: ( )/5 = 26.8 days/jobAvg job waiting time: ( )/5 = 15.8 days/jobAvg job lateness: ( )/5 = 13.6 days/jobAvg number of jobs in the system: (4x5+9x4+11x3+14x2+17x1)/55 = 2.44 jobsAvg number of jobs waiting: (4x4+9x3+11x2+14x1)/55 = 1.44 jobs
15 Sequencing example evaluation (cont.) Using earliest due date: could be same sequence as shortest processing time, A-D-E-C-BUsing first-come, first servedTotal job completion time: 55 daysAvg job completion time: ( )/5 = 31.8 days/jobAvg job waiting time: ( )/5 = 20.8 days/jobAvg job lateness: ( )/5 = 18.6 days/jobAvg number of jobs in the system: (4x5+17x4+14x3+9x2+11x1)/55 = 2.89 jobsAvg number of jobs waiting: (4x4+17x3+14x2+9x1)/55 = 1.89 jobs
16 Comparison of SPT, EDD, and FCFS evaluation criteriaSPTEDDFCFStotal job completion time (days)55same as SPTavg job completion time (days/job)26.831.8avg job waiting time (days/job)15.820.8avg job lateness (days/job)13.618.6avg number of jobs in system (jobs)2.442.89avg number of jobs waiting (jobs)1.441.89“For jobs to be processed in one work center, it (shortest processing time rule) is consistently superior to other rules; it is optimal for minimizing avg. (job) completion time, avg. number of jobs in the system, and avg. job lateness.” (Adam and Ebert, page 493)
17 Setup cost minimization Next Best approach – given that a job (job i) is being processed, select next the unassigned job (job j) for which the setup cost is minimized
18 Next Best rule exampleAssuming A is being processed, the NB approach asks “What next?”Setup costs ($):PredecessorJob iFollower Job jABCDE-29201824141915353726101640
19 The least costly sequence? Two sequences possible using NB approach:A-D-C-E-B ($18+$10+$26+$18=$72)A-D-E-C-B ($18+$10+$16+$35=$79)Enumeration of all 24 possible sequences yields an optimal solution of A-D-E-B-C with a cost of $18+$10+$18+$14=$60NB approach is a heuristic – a rule or a set of rules that generally leads to the optimal solution, but does not guarantee the optimal solutionAssuming the NB approach and say, SPT, sequences do not coincide then one must consider relative importance placed on costs of machine setup (NB) versus the value of gaining overall shop effectiveness and customer satisfaction (SPT)
20 Sequencing through two work centers All jobs have to be processed through two successive work centersJobs must be processed in the same order on both work centersThe following is an optimal procedure to minimize the completion time for the last job through the process
21 Sequencing through two work centers example Processing times (days):JobWork Center 1(sheet metal)Work Center 2(paint)A45B177C1412D92E116
22 Sequencing through two work centers procedure (1) Determine the minimum of all processing times(2) If the minimum is associated with work center 1, place the corresponding job in the earliest available position in the sequence(3) If the minimum is associated with work center 2, place the corresponding job in the latest remaining position in the sequence(4) Cross out the times of the job just assigned across all work centers to omit that job from further consideration(5) Now return to step (1) and repeat the procedure by identifying the minimum of all remaining process times
23 A second example of sequencing through two work centers Data Systems, Inc., processes all incoming jobs through two successive work centers, A and B (each job goes first through A, then through B). Five jobs await processing (see below). Assign a job sequence that minimizes the completion time of the last job processed. What is the total flow time for this sequence?JobSTUVWProcessing time in A (minutes)14322317Processing time in B (minutes)8941712
24 Summary of sequencing methods “… an abundance of sequencing methods is available and they can affect shop performance in different ways. … in choosing among these methods, you should carefully evaluate them in terms of the criteria that are of greatest importance for your organization’s competitive posture.”