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1. Optimal allocation and processing time decisions onnon-identical parallel CNC machines: e-constraint approach Seçil Sözüer

2. Agenda 1-) Introduction 2-) Problem Defn. 3-) Single machine subproblem (Pm) 4-) Cost lower bounds for a partial schedule(for B&B and BS) 5-) Initial solution (IS) (a heuristic for finding IS for B&B) 6-) B&B algorithm (exact algorithm) 7-) Beam search algorithm (BS) (if B&B: not computationally efficient) 8-) Improvement search heuristic (ISH) (improves any given feasible schedule) 9-) Recovering beam search (RBS) 10-) Computational results

3. 1. Introduction • Turning (metal cutting) operation on non-identical parallel CNC machines • Controllable processing times in practice (attaining small proc. time by cutting speed or feed rate ) • Decide on • Processing times of the jobs • Machine / Job Allocation • Bicriteria Problem (Two objectives): COST and TIME Total Manuf. Cost: & Makespan • Converting Bicriteria Problem to Single Criterion Problem: e-constraint Approach: min. s.t. Makespan ≤ K (Upper Limit) • Decision Maker: Interactively specify and modify K and analyze the influence of the changes on solutions

4. 1. Introduction • Controllable process. times: Pioneer: Vickson(1980) Problem: Total Process. Times & Total Weighted Compl. Time on a single machine • Trick(1994): (Linear Process. Cost function) Nonidentical Parallel Mach. – NP-hard problem Problem: min. Total Process. Cost s.t. Makespan ≤ K (This paper considers nonlinear convex manuf. Cost Function) • Kayan and Aktürk(2005): Determining upper and lower bounds for process. Times and manuf. Cost function

5. 2. Problem Definiton Each job isdifferent in terms of its length and diameter, depthof cut, maximum allowable surfaceroughness, andits cutting tool type.Each job must be performed on a single machine without preemption Parameters is a convex fnc and minimized at Each machine is differentin terms of its maximum applicable cutting powerHm, and its unit operating cost, Cm($/minute). Operating Cost + Tooling Cost N: number of jobs, card( J )= N M: number of machines m=1..M

6. 2. Problem Definiton Decision Var. Model Non-convex Mixed Integer Nonlinear Prog. (MINLP) Non-convex Feasible Region Non-convexities: major difficulty for finding global opt. soln EXPLOITING THE STRUCTURE Non-convex obj. Func.

7. 3. Single Machine Subproblem (Pm) • With given Xjm (mach/job allocation), we can find optimal pjm by solving Pm for each machine m: Non-linear convex obj. func. Jm: set of jobs assignedtomachine m. Convex Feasible Region is the Lagr. Dual var. for the makespan constraint (5). ≤ 0 since fjm is non-increasing on the interval (6) For the Lemma 1 proof: replace (6) and use KKT - CS Conditions Pm is convex. Hence local opt = global opt. Lemma 1 is sufficient for opt.

8. 3. Single Machine Subproblem (Pm) Immediate extension of Lemma 1 to the non-ident. Parallel machine

9. 4. Cost lower bounds for partial schedules • Sp: partial schedule • Jp: subset of jobs assigned to machines • Jmp : subset of jobs assigned to machine m • optimal pjmdecisions were made by solving the Pm • We assume that when we add an unscheduled jobto Sp, processing times of previously scheduled jobsmay change, but the machine/job assignments in Spstay the same. • Addingan unscheduled job j tomachine m does not violate the makespanconstraint(1): hence • The processing time decisionsfor the jobs in Jpwere made previously by solvingthePmfor each machine m, we haveat hand the optimal dual price for each for each m.

10. 4. Cost lower bounds for partial schedules lbjm : Cost change–increase- lower bound for adding job j tomach. m Compressing jobs to mach. M. Marg. Cost of decreasing makespan: - Pjmub satisfying Lemma 1. and Pjm* ≤Pjmub Additionalcost of adding job j to m ≥fjm (Pjmub ) IP: By using lbjm , Cost increase lower bound for adding all unschedled jobs (Forming a complete sch. by starting with Sp) sum of lbjmfor thepossibleassignments of unscheduledjobs to the machines. (7): makespan constr. (8): assign unsch. Jobs to machines

11. 4. Cost lower bounds for partial schedules LBIP : lower bound found by solving theIP If an IP for Spturns out to beinfeasible, this means no complete schedule can be achieved from Sp LBLP : Relaxing (9) LBR: Relaxing (7):makespan const.

12. 5. Initial Soln (IS) • IS: will be ub soln. For B&B Scheduling min-cost job first ! Each iteration: Adds a new job to schedule by choosing the best machine (that will give min. cost increase) Performance: N x M : iteration

13. 6. B&B Algorithm • The major difficulty in a B&B algorithm for a non-convex MINLP problem :computinga lbat a node of a B&B tree • Each node: partial schedule (level 0: all jobs: unscheduled) (level k: jobs (j1,...jk) : scheduled) (level N: all jobs: scheduled) • Reducing the size of tree: Pruned by Infeasiblity? • If feasible, solve Pm. Find a lb LBP ; LBIP or LBLP or LBR :Prune by inf? LBC = LBP + FP • If LBC ≥ UBC, Then Prune by Optimality For eah node, we find the opt. Cost and opt pjm by Pm Cost of the partial sch. LB for complete sch. achievable from node Cost incr. lb of the node Cost upperbound: initally found by IS, (then updated if we find a feas. complete sched)

14. 6. B&B Algorithm Modified depth first search: Selects the one with minimum lb as new parent node

15. 7. Beam Search (BS) -near opt.soln- • NP-hard problem and the size of the search tree for the B&Balgorithm increases exponentially as N and Mincrease. • For higher levels of N, M and K, we use BS algorithm. • BS: polynomial time algorithm. Complexity O(m.n.b) • BS: Fast B&B method. Keeps the best b nodes at a level of the tree. Eliminates the rest. Choosing the nodes to be saved: LBLP

16. 8. Improvement Search Heur. (ISH) • Starts with an initial schedule which satisfies the optimality conditionin Corollary 1 so that we assume the Pm is solved for each machine m. (We have at hand the optimal dual price for each m.) • Definetwo moves todescribe the neighborhood of a solution. 1-move :move a job j from its currentmachine m1to another machine m2 2-swap :exchange job j1 onmachine m1 with another job j2 on machine m2. Cost change lb by expanding proc. Times of the remaining jobs Cost of job j in m1 Adding job j to m2 Removing job j from m1

17. 8. Improvement Search Heur. (ISH) If < 0, then it is a promosing move! (obj. Value may not improve)

18. 9. Recovering BS (RBS) • RBS: BS + local search techniques (2-swap moves) • IDEA in recovering step :prevent the elimination of promising nodes (nodesthat could lead to optimal or near optimal solutions)due to errors in the node evaluation step ofbeam search algorithms by applying local searchtechniques to achieve better partial solutions at eachlevel of the beam search algorithm.RBS Complexity: O(m.n2.b) • In Step 3 of the BS, we generatechild nodes for a given level of search tree. K child nodes generated at level l

19. 10. Computational Results • Selecting K: makespan limit (In order to see the effect of K, solving each replication of the problem for 5 different levels of K.) • To find proper K: • First, solving the makespanminimization problem for each replicationfor fixed processing times case where for each j and m.: NP-HARD • Polyn.-Time Algorithm: finding a feasible makespan level K: KDJ • K = k x KDJ k = 0.6, 0.8, 1, 1.2, 1.4 • Solving each B&B : with LBIP , LBLP, LBR • If B&B finds opt. soln: Solving the problem with BS and RBS (LBLP) • kn

20. 10. Computational Results • Critical Step in B&B: Deciding on (j1,... jk) • Reduce Tree Size of B&B by lower bounds Schedule Higher Proc. Time Job Earlier: Catch Infeasibility of schedules earlier Schedule Lower Cost Jobs Earlier: Schedule Higher Cost Jobs Earlier: We can reach opt. By traversing % of tree

21. 10. Computational Results • The effect of Increasing N and M on running time of B&B Big Gap btwn «Min» and «Max» for each lower bounding method due to different K levels Calculating LBR is faster . As N and M increase, the CPU time required by LBRapproaches to LBLP, so we may expect tosee that LBLPwill have shorter CPU times for larger problem sizes.. If we check the CPU time ratio LBR /LBIP, we observe that as N isincreased, performanceof the LBRgets closer to the performanceof LBIP, but as M is increased, we observe the opposite.This is due to the fact that computing LBIP isitself an NP-hard problem and requires much moretime when M is increased.

22. 10. Computational Results • Size of the eliminated and traversed nodes with different K • Soln. Quality Results of IS, BS, RBS: (RBS is the best quality and time)RA : : % deviation from the opt. with algo. A • RA = (cost achieved by A) – (opt cost:B&B) / (opt cost: B&B) As K is increased, the size of traversed tree increases since fewer number of nodes will be eliminated due to feasibility. Hence CPU increases too.

23. 10. Computational Results • ISH achieves a significant improvement for all 3 cases RA+ISH = starting soln with algo A and improve through ISH Comparing Table 5 and 6

24. 10. Computational Results • Soln quality of BS and ISH improves as K is increased • Hence, if B&B: not efficient, BS and ISH can achieve soln closer to optimum. • This is due to the shape of manuf. Cost func. • When K is increased, we have higher process. Time: flatter manuf. Cost func.

25. 10. Computational Results • Testing IS, RBS, ISH for 50-100 jobs and 2,3,4 machines IRBS= % deviation of RBS from initial soln. achieved by IS • We cannot solve these instances to the optimum due to theCPU time requirements. • Therefore, we comparedthe results achieved by the RBS algorithm withrespect to the initial results given by the IS algorithm • The required CPU times are still reasonableeven for the large problem instances.

26. 10. Computational Results • Our computational results show that B&Bcan solve the problems byjust traversing the 5% of the maximal possible (Slide 20)-Table-2 • B&B tree size and the proposed lower boundingmethods can eliminate up to 80% of the search tree. (Table 2) • For the cases where B&B is not computationallyefficient, our BSand improvement searchalgorithms achieved solutions within 1% of the optimumon the average in a very short computation time.