Modeling and Optimization Tools for Solving PdProduc tion Plann ing an dShdlid Scheduling Pro blems Alkis Vazacopoulos CAPD Meeting Carnegie Mellon University Pittsburgh PA March 2011 1 Production Planning and Scheduling Problems » Strategic » Tactical » Operational 2 Software Companies/Academic Instit/Consulting Companies » SAP » Oracle » Aspentech » Honeywell » KBC » Consulting companies » IBM-CPLEX, Gurobi, FICO, GAMS, ….. 3 Technologies » LP » MIP » NLP » CP » Modeling » Platforms » Business Rules 4 LP Performance across releases (FICO Xpress) LP Performance 25000 800 795 20000 790 785 15000 780 Total Time 775 ber Solved ber Number Solved tal Time(s)tal mm oo T 10000 770 Nu 765 5000 760 755 0 750 2003B 2004B 2005B 2006B 2007B 2008A FICO 7.0 Re le as e Internal test set of 796 public and customer models 5 LP Performance across releases LP Performance 25000 800 795 20000 790 785 15000 780 Primal Dual Barrier Total Time 775 ber Solved ber Number Solved tal Time(s)tal mm oo T 10000 770 Nu 765 5000 760 755 0 750 2003B 2004B 2005B 2006B 2007B 2008A FICO 7.0 Re le as e Internal test set of 796 public and customer models 6 LP Performance across releases LP Performance 25000 800 795 20000 790 Good 785 15000 780 Primal Dual Barrier Total Time 775 ber Solved ber Number Solved tal Time(s)tal mm oo Is important T 10000 770 Nu 765 5000 760 755 0 750 2003B 2004B 2005B 2006B 2007B 2008A FICO 7.0 Re le as e Internal test set of 796 public and customer models 7 MIP Performance across releases 290 300000 270 250000 250 200000 Time ved ll 230 nn 150000 Numbers Solved 210 Total Time Number So 100000 Solutio Total 190 50000 170 150 0 2003B 2004B 2005B 2006B 2007B 2008A 7.0 7.1 Release Internal test set of 320 public and customer models 8 MIP Performance across releases MIP Performance 250000 290 270 200000 250 150000 Heuristics Cuts 230 Branch and General and Total Time Classic Number Solved Bound problem210 specific Total Time(s) 100000 ppproblem specific Number Solved parallelize 190 50000 170 0 150 2003B 2004B 2005B 2006B 2007B 2008A FICO 7.0 Re le as e 9 Feedback Loop MIP Performance 250000 290 270 200000 250 150000 Decomposition Perform Reengineer 230 Algorithms – OperationsTotal - Time B&B Number Solved mber Solved otal Time(s) more flexible Node210 uu TT 100000 N 190 50000 170 0 150 2003B 2004B 2005B 2006B 2007B 2008A FICO 7.0 Re le as e 10 Comparison GUROBI vs CPLEX » Gurobi: developpged to take advantage of multi-core machines » Growing very fast (150 commercial licenses, 25 academic sales, in addition to the academic program, source: jtoedm. com » LP 10-30% better than CPLEX (Mittelman website) » MIP 10% - 30% better on optimality » CPLEX better on Feasibility problems » New MIPLIB data soon 11 Open Source » SKIP – ZIB, Very competitive: used by Dynadec » CBC – used as an entryyyyp by many companies » Comparison from Hans Mittelman’s website 12 Benchmarking MIP » Optimality (what about problems won’t solve after 30 minutes, 60 minutes) » Best feasible solutions » Find k best solutions » Size of the tree » Presolve (reduction of the matrix) » Develop Sets of Problems » Etc…. » 13 Xpress 7 parallel speed ups Xppress 7 Xppress 7 1 thread 4 thread Improvement “Internal” deterministic 25,724 17,447 -47% “Internal” opportunistic 25,724 13,067 -96% Coral deterministic 24,484 16,129 -52% Coral opportunistic 24,484 11,137 -120% 14 mzzv42z: Single/Two Threads Xpress FICO IVE »1 thread »2 threads »15 15 Xpress 7 parallel speed ups Xppress 7 Xppress 7 1 thread 4 thread Improvement “Internal” 25,724 17,447 -47% deterministicParallel Parallel MIP “Internal” processors (same Deterministic opportunistic on a 25,724computer) 13,067P-MIP -96% Network Coral P-MIP deterministic 24,484 16,129 -52% Coral opportunistic 24,484 11,137 -120% 16 Concurrent LP solver » Efficient LP method parallelization Primal Dual Barrier Concurrent Overall Speedup 1 thread 19,940 13,459 - - 2 threads 19,940 13,459 6,193 (B+D) 23% 4 threads 19,940 13,459 7,540 7,491 70% Time(concurrent) ~ min(Time(dual), Time(primal), Time(barrier)) Controls: LPTHREADS and DETERMINISTIC (concurrent is nondeterministic!) 17 Concurrent LP solver » Efficient LP method parallelization Primal Dual Barrier Concurrent Overall Speedup 1 thread 19,940 13,459 - - P-D-B 2 threadsP-Barrier 19,940 13,459 6,193 (B+D) 23% 4 threads 19,940 13,459Combinations 7,540 7,491 70% Time(concurrent) ~ min(Time(dual), Time(primal), Time(barrier)) Controls: LPTHREADS and DETERMINISTIC (concurrent is nondeterministic!) 18 Customers want multiple alternative solutions Example: air04 » Set partitioning problem for airline crew scheduling » Available from MIPLIB 2003 Default Solver 5-Best Solution Solver 1st SltiSolution 56137 56137 2nd Solution 56166 56137 3rd Solution 56307 56137 4th Solution 56854 56137 5th Solution 57562 56137 Total B&B Nodes 141 15 981 Total Time 26 sec 29 sec 19 Customers want multiple alternative solutions Example: air04 » Set partitioning problem for airline crew scheduling » Available from MIPLIB 2003 Default Solver 5-Best Solution Solver 1st SltiSolution 56137 56137 Strategies- Strategy Robust - nd Business Refinenement 2 SolutionStrategies 56166 56137 Rules Rebalancing 3rd Solution 56307 56137 4th Solution 56854 56137 5th Solution 57562 56137 Total B&B Nodes 141 15 981 Total Time 26 sec 29 sec 20 Xpress-Tuner: How to Tune (Automatically) an Optimization Problem? 21 Xpress-Tuner: Tuning Process 22 Xpress-Tuner: Detailed Results 23 Xpress-Tuner: Comparing logs 24 Xpress-Tuner: Tuning MIP families 25 Xpress-Tuner: Tuning Methods 26 Benchmarking - Optimality Problems with less than 60 minutes Reduce time Problems Problems with less Problems Parallel MIP with less more than 2 than 30 Opportunisti hours seconds than 30 cMIPc MIP Solve in 1/10 mitinutes What is a of time – Reduce time examples good Google Parallel MIP solution? Matrix Gen Deterministic Multiple MIP Solutions Parallel Opportunisti Modeling c Languages Explain Solutions 27 MIP Research » More research on Deterministic Parallel » Predict Size of BB Tree/ Structure » UblUnbalance dTd Trees/h ow t o B Blalance » Solve Smaller problems faster (E Danna – Google) » New Cu ts /Feas ibility P ump » More CP in Presolve » HdCtit/SftCtitHard Constraints / Soft Constraints – OtiiBBOperations in BB nodes » Parallelize cuts/Heuristics in root node » How to scale better 4 to 8 cores » Data mining in Parameters – usinggp supervised learning » **** Adaptive search – change the parameters dynamically during the Search 28 Non Linear » Knitro » CONOPT » Xpress SLP : good for process industry » MINLP CMU-IBM , minlppg.org » BONMIN Basic Open Source Nonlinear Mixed Integer programming 29 Comparisons Mittelman » Knitro vs IPOPT » 27 better Knitro » IPOPT better in 14 cases 30 Global Optimization other technologies » Baron » Disjjpggunctive programming » Robust » Stochastic 31 Constraint Programming » ILOG-CP » CHIP » Artelys – Kalis » COMET – Dynadec » SIMPL (Yunes, Aron, Hooker) » Benchmarking CP codes???? 32 »Example of relaxation for alldifferent alldiff (x1,, xn ) x j 1,,n »Convex hull relaxation, which is the strongest possible linear relaxation (JNH, Williams & Yan): n x 1 n(n 1) j 2 j1 1 x | J | (| J | 1), all J 1, ,nwith | J | n j 2 jJ »For n = 4: x1 x 2 x3 x 4 10 x1 x 2 x3 6, x1 x 2 x 4 6, x1 x3 x 4 6, x 2 x3 x 4 6 x1 x 2 3, x1 x3 3, x1 x 4 3, x 2 x3 3, x 2 x 4 3, x3 x 4 3 x1 , x 2 , x3 , x 4 1 33 »Automatic relaxations »The following constraints can be relaxed automatically with Xpress‐Kalis : » linear constraints . alldifferent(X) . occurrence(X,v) opY . distribute(X,A,m,M) . X = Min/Max(Y) . absolute value X = | Y | . distance v op |X‐Y| . element(2D) X = A[I] | X = A[I][J] . cycle . logical constraints (,,or,and) 34 An integer knapsack problem with ssdecostaide constrain t min 5x1 8x2 4x3 subj.to 3x1 5x2 2x3 30 alldiff(x1, x2, x3) x j {1,2,3,4} »MILP needs additional xi jyij , all i constraints and 00--11 j variables to express the y 1, all i alldiff: ij j 35 Pure CP model in Xpress-KALIS 36 HYBRID Model in Xpress -KALIS 37 Modeling Platforms 38 Modeling Languages » OPL: Integrates MIP and CP » GAMS: All solvers available, NLP, MIP, Stochastic » Mosel : Programming and Modeling together 39 Modeling Languages » OPL: Integrates MIP and CP » GAMS: All solvers available, NLP, MIP, Stochastic » Mosel : Programming and Modeling together 40 More » IBM ODM: Optimization Decision Manager » Next ggpeneration : Eclipse » Simulation » Python link 41 Eclipse – Platform - Future 42 »CP, MIP and their combination » Oppgtimization Technologies » Mixed Integer Programming: MIP » Finite Domain Constraint Progggramming: CP » Planning & Scheduling »CP »CP+MIP» MIP » Long and mid-term planning: MIP » SSothort-term pl annin g, sc h eduling : C P » Time Horizon Supply chain optimization Reqq,,uires MIP, CP, and their combination 43 Solving Planning and Scheduling Models »INTEGRATED ENV IRONMENT »Common Plann ing and S ch ed uli ng Model »LP/MIP »CP 44 »Combining MIP & CP »Master problem »MIP »Sub problem »CP »Feasible »Infeasible »(OK) »Add Cut to MIP » Mixed l ogica l/ Linea r Prog ra mming fra me w ork (Hooker et al) 45 Project » LISCOS Project » European Union 5th Framework » BASF - Chem ica ls » P&G - Food » PSA - Cars » Barbot - Paint 46 Production Scheduling Problem » Schedules Crude-Oil Blend-Shops in Oil-Refineries. » In the Categgyory of “Closed-Shop” Scheduling. » Involves Quantity, Logic and Quality Decisions. » Quantity = Flow, Rate, Inventory » Logic = Mixing-Delay, Up-Time, One-Flow-In » Quality = Octane, Sulphur, Yield, Temperature » Intended to Reduce Quality Variation. Key Absolute Maximum Limit Planning Proxy Target Set Closer to Limit Target Low Variability Reduced Manual Crude-Oil Blend Time Scheduling Scheduling Installed » 47 Production Scheduling Problem » Finite-Capacity Scheduling.