Operational Research & ManagementOperations Scheduling Economic Lot Scheduling 1.Summary Machine Scheduling 2.ELSP (one item, multiple items) 3.Arbitrary Schedules 4.More general ELSP

Operational Research & ManagementOperations Scheduling Topic 1 Summary Machine Scheduling Problems

Operational Research & ManagementOperations Scheduling3 Machine Scheduling  Single machine  Parallel machine 3-partition partition

Operational Research & ManagementOperations Scheduling4 Why easy?  Show polynomial algorithm gives optimal solution  Often by contradiction – See week 1  By induction – E.g.

Operational Research & ManagementOperations Scheduling5 Why hard?

Operational Research & ManagementOperations Scheduling6 Why hard?

Operational Research & ManagementOperations Scheduling7 Why hard?

Operational Research & ManagementOperations Scheduling8 Why hard?  Some problem are known to be hard problems. – Satisfiability problem – Partition problem – 3-Partition problem – Hamiltonian Circuit problem – Clique problem  Show that one of these problems is a special case of your problem – Often difficult proof

Operational Research & ManagementOperations Scheduling9 Partition problem  Given positive integer a 1, …, a t and b with ½  j a j = b do there exist two disjoint subsets S 1 and S 2 such that  j  S a j = b ?  Iterative solution: – Start with set {a 1 } and calculate value for all subsets – Continue with set {a 1, a 2 } and calculate value for all subsets; Etc. – Last step: is b among the calculated values?  Example: a = (7, 8, 2, 4, 1) and b = 11

Operational Research & ManagementOperations Scheduling10 Some scheduling problems  Knapsack problem Translation: n = t; p j = a j ; w j = a j ; d = b; z = b; Question:exists schedule such that  j w j U j  z ?  Two-machine problem Translation: n = t; p j = a j ; z = b; Question: exists a schedule such that C max  z ?

Operational Research & ManagementOperations Scheduling11 3-Partition problem  Given positive integer a 1, …, a 3t and b with ¼b < a j < ½b and  j a j = t b do there exist pairwise disjoint three element subsets S i such that  j  S a j = b ?

Operational Research & ManagementOperations Scheduling Topic 2a Economic Lot Scheduling Problem: One machine, one item

Operational Research & ManagementOperations Scheduling13 Lot Sizing  Domain: – large number of identical jobs – setup time / cost significant – setup may be sequence dependent  Terminology – jobs = items – sequence of identical jobs = run  Applications – Continuous manufacturing: chemical, paper, pharmaceutical, etc. – Service industry: retail procurement

Operational Research & ManagementOperations Scheduling14 Objective  Minimize total cost – setup cost – inventory holding cost  Trade-off  Cyclic schedules but acyclic sometimes better

Operational Research & ManagementOperations Scheduling15 Scheduling Decisions  Determine the length of runs – gives lot sizes  Determine the order of the runs – sequence to minimize setup cost  Economic Lot Scheduling Problem (ELSP)

Operational Research & ManagementOperations Scheduling16 Overview  One type of item / one machine – without setup time  Several types of items / one machine – rotation schedules – arbitrary schedules – with / without sequence dependent setup times / cost  Generalizations to multiple machines

Operational Research & ManagementOperations Scheduling17 Minimize Cost  Let x denote the cycle time  Demand over a cycle = Dx  Length of production run needed = Dx / Q =  x  Maximum inventory level = (Q - D) Dx / Q = (Q - D)  x = (1 -  ) D x Inventory Time x idle time

Operational Research & ManagementOperations Scheduling18 Optimizing Cost book shorter  Solve  Optimal Cycle Time  Optimal Lot Size

Operational Research & ManagementOperations Scheduling19 With Setup Time  Setup time s  If s  x(1-  ) above optimal  Otherwise cycle lengthis optimal

Operational Research & ManagementOperations Scheduling Topic 2b Economic Lot Scheduling Problem: Lot Sizing with Multiple Items With Rotation Schedule

Operational Research & ManagementOperations Scheduling21 Multiple Items  Now assume n different items  Demand rate for item j is D j  Production rate of item j is Q j  Setup independent of the sequence  Rotation schedule: single run of each item

Operational Research & ManagementOperations Scheduling22 Scheduling Decision  Cycle length determines the run length for each item  Only need to determine the cycle length x  Expression for total cost / time unit

Operational Research & ManagementOperations Scheduling23 Optimal Cycle Length  Average total cost with  Solve as before

Operational Research & ManagementOperations Scheduling24 Example

Operational Research & ManagementOperations Scheduling25 Data of example 7.3.1

Operational Research & ManagementOperations Scheduling26 Solution

Operational Research & ManagementOperations Scheduling27 Solution  The total average cost per time unit is (alternative 1: ) (alternative 2: )  How can we do better than this?

Operational Research & ManagementOperations Scheduling28 With Setup Times  With sequence independent setup costs and no setup times the sequence within each lot does not matter  Only a lot sizing problem  Even with setup times, if they are not job dependent then still only lot sizing

Operational Research & ManagementOperations Scheduling29 Job Independent Setup Times  If sum of setup times < idle time then our optimal cycle length remains optimal  Otherwise we take it as small as possible

Operational Research & ManagementOperations Scheduling30 Job Dependent Setup Times  Now there is a sequencing problem  Objective: minimize sum of setup times  Equivalent to the Traveling Salesman Problem (TSP)

Operational Research & ManagementOperations Scheduling31 Long setup  If sum of setups > idle time, then the optimal schedule has the property: – Each machine is either producing or being setup for production  An extremely difficult problem with arbitrary setup times

Operational Research & ManagementOperations Scheduling Topic 3 Arbitrary Schedules

Operational Research & ManagementOperations Scheduling33 Arbitrary Schedules  Sometimes a rotation schedule does not make sense (remember problem with no setup cost)  For the example, we might want to allow a cycle 1,4,2,4,3,4 if item 4 has no setup cost  No efficient algorithm exists

Operational Research & ManagementOperations Scheduling34 Problem Formulation  Assume sequence-independent setup  Formulate as a nonlinear program

Operational Research & ManagementOperations Scheduling35 Notation  Setup cost and setup times  All possible sequences  Item k produces in l-th position  Setup time s l, run time (production) t l, and idle time u l

Operational Research & ManagementOperations Scheduling36 Inventory Cost  Let x be the cycle time  Let v be the time between production of k  Total inventory cost for k is

Operational Research & ManagementOperations Scheduling37 Mathematical Program Subject to

Operational Research & ManagementOperations Scheduling38 Two Problems  Master problem – finds the best sequence  Subproblem – finds the best production times, idle times, and cycle length  Key idea: think of them separately

Operational Research & ManagementOperations Scheduling39 Subproblem (lot sizing) Subject to But first: determine a sequence

Operational Research & ManagementOperations Scheduling40 Master Problem  Sequencing complicated  Heuristic approach  Frequency Fixing and Sequencing (FFS)  Focus on how often to produce each item – Computing relative frequencies – Adjusting relative frequencies – Sequencing

Operational Research & ManagementOperations Scheduling41 Step 1: Computing Relative Frequencies  Let y k denote the number of times item k is produced in a cycle  We will – simplify the objective function by substituting – drop the second constraint  sequence no longer important

Operational Research & ManagementOperations Scheduling42 Rewriting objective function Assumption: for each item production runs of equal length and evenly spaced

Operational Research & ManagementOperations Scheduling43 Mathematical Program Subject to Remember: for each item production runs of equal length and evenly spaced

Operational Research & ManagementOperations Scheduling44 Solution  Using Lagrange multiplier:  Adjust cycle length for frequencies  Idle times then = 0  No idle times, must satisfy

Operational Research & ManagementOperations Scheduling45 Step 2: Adjusting the Frequencies  Adjust the frequencies such that they are – integer – powers of 2 – cost within 6% of optimal cost – e.g. such that smallest y k =1  New frequencies and run times

Operational Research & ManagementOperations Scheduling46 Step 3: Sequencing  Variation of LPT  Calculate  Consider the problem with machines in parallel and jobs of length  List pairs in decreasing order  Schedule one at a time considering spacing  Only if for all machines assigned processing time < then equal lot sizes possible

Operational Research & ManagementOperations Scheduling Topic 4 Lot Sizing on Multiple Machines

Operational Research & ManagementOperations Scheduling48 Multiple Machines  So far, all models single machine models  Extensions to multiple machines – parallel machines – flow shop – flexible flow shop

Operational Research & ManagementOperations Scheduling49 Parallel Machines  Have m identical machines in parallel  Setup cost only  Item process on only one machine  Assume – rotation schedule – equal cycle for all machines

Operational Research & ManagementOperations Scheduling50 Decision Variables  Same as previous multi-item problem  Addition: assignment of items to machines  Objective: balance the load  Heuristic: LPT with

Operational Research & ManagementOperations Scheduling51 Different Cycle Lengths  Allow different cycle lengths for machines  Intuition: should be able to reduce cost  Objective: assign items to machines to balance the load  Complication: should not assign items that favor short cycle to the same machine as items that favor long cycle

Operational Research & ManagementOperations Scheduling52 Heuristic Balancing  Compute cycle length for each item  Rank in decreasing order  Allocation jobs sequentially to the machines until capacity of each machine is reached  Adjust balance

Operational Research & ManagementOperations Scheduling53 Further Generalizations  Sequence dependent setup  Must consider – preferred cycle time – machine balance – setup times  Unsolved  General schedules  even harder!  Research needed :-)

Operational Research & ManagementOperations Scheduling54 Flow Shop  Machines configured in series  Assume no setup time  Assume production rate of each item is identical for every machine  Can be synchronized  Reduces to single machine problem

Operational Research & ManagementOperations Scheduling55 Variable Production Rates  Production rate for each item not equal for every machine  Difficult problem  Little research  Flexible flow shop: need even more stringent conditions