1. Discovering Combinatorial Optimization with the ILOG Optimization Suite Gregory Glockner, Ph.D. Optimization Product Manager ILOG, Inc. gglockner@ilog.com

2. 2 Agenda q About ILOG q Brief introduction to constraint programming q Exploring combinatorial optimization q Exploring iterative optimization q Discrete scheduling Overview

3. 3 q Founded 1987 q 590 employees q 2,000+ customers q Selling in 30 countries q NASDAQ/Euronext 300 200 120 70 98990001 ISV/OEM Partners Most influential IT companies for World Leader in Software Components About ILOG

4. 4 Views Component Suite JViews Component Suite JTGO CPLEX Solver JRules Rules ILOG Core Technologies About ILOG

5. 5 ILOG CPLEX CPLEX ILOG Solver & ILOG Solver & ILOG JSolver ILOG Concert Technology (C++ & ) ILOG Concert Technology (C++ & Java) Hybrid ILOG Optimization Suite ILOG OPL Studio ILOG OPL Studio ILOGSchedulerILOGDispatcherILOGConfigurator ILOG JConfigurator About ILOG

6. 6 ILOG Optimization Suite – 2 q Core Engines q ILOG Solver - Constraint Programming Engine q ILOG CPLEX - Math Programming Engine q Vertical Engine Extensions q ILOG Scheduler - Constraint-Based Scheduling q ILOG Dispatcher - Vehicle Routing, Technician Dispatching q ILOG Configurator - Product and Service Configuration q Modeling Tools q OPL Studio - Rapid Development of Optimization Apps q AMPL - Modeling Support for CPLEX About ILOG We use OPL Studio here since its high level language makes it an easy starting point

7. 7 Inside ILOG OPL Studio q OPL: Optimization Programming Language q High-level language specialized for optimization modeling q OPLScript q Procedural language to control optimization process q OPL Studio q IDE to edit and solve models and to view results q OPL Component Libraries q APIs to integrate models into standalone applications via C++, Java, and Microsoft COM and.NET (C#, VB, VBA) q Direct links to the ILOG Optimization Suite q ILOG CPLEX, ILOG Solver, ILOG Scheduler About ILOG

8. 8 Month ç Week ç Day ç Hour Time steps Scope Strategic ç Tactical ç Operational Application Long-term planning Published Schedule Operational Scheduling Drivers Economics ç Feasibility Technology LP ç MIP/Hybrid ç CP Range of Optimization Applications Only ILOG has optimization technology for the entire planning horizon About ILOG

9. 9 Agenda q About ILOG q Brief introduction to constraint programming q Exploring combinatorial optimization q Exploring iterative optimization q Discrete scheduling Overview

10. 10 Constraint Programming Example Find a solution to: x + y < 5, x  y, x, y  {1, …, 5} 21543 Branching on y 21543 Branching on x Inconsistent values removed by constraint propagation Constraint Programming

11. 11 CP in a Nutshell q Essential problem: To satisfy a given set of constraints by assigning values to variables q Typical problems involve discrete variables q Enumerate values via branch and bound q Try all values in domain q Use general and custom searches q Constraint propagation reduces domain at each node q Eliminate inconsistent values from current partial solution Constraint Programming

12. 12 Comparing CP and IP q Computation at nodes q CP: Constraint propagation is fast but naïve q IP: LP relaxation is slow, powerful algorithm q Constraint representation q IP uses linear inequalities q CP uses logical constraints q Search methods q IP has built-in branch-and-bound algorithm q User supplies CP search Constraint Programming

13. 13 Some CP Constraints – 1 q Logical and high-order q (x > 0)  (y = 0) q (Q = 0)  (Q  100) q C = 5  Q + 1000  (Q > 0) q Element and table q T =  i C[x[ i ]] q (x, y)  {(0,0), (2,3), (3,2), (5,0), (0,5)} Constraint Programming

14. 14 Some CP Constraints – 2 q Global q AllDifferent(x) q Cardinality: | { x  S : x = 3 } | = 5 q Path: x defines a Hamiltonian cycle q Custom constraints q Define your own by extending constraint classes Constraint Programming

15. 15 Search Algorithms q IP uses priorities and economic pseudocosts to direct branch-and-bouond tree exploration scheme q CP requires explicit search algorithm q Ex 1: Branch on variable with smallest domain q Ex 2: Branch on variable with lowest costs q Exploiting problem structure leads to fastest solutions Constraint Programming

16. 16 Comparing IP and CP modeling q Example: Pick two different items q Each has a natural formulation q IPs have more variables and constraints q CPs have large variable domains Constraint Programming IP: Use binary variables and linear constraints CP: Use general variables with logical constraints  i x[ i ] = 1  i y[ i ] = 1 x[ i ] + y[ i ] <= 1 for all i x  { 1, …, n } x <> y

17. 17 Typical Applications q LP/MIP q Planning models with complex economics where optimal solution is required q CP q Operational models with complex operations where many good feasible solutions are required Constraint Programming

18. 18 Agenda q About ILOG q Brief introduction to constraint programming q Exploring combinatorial optimization q Exploring iterative optimization q Discrete scheduling Overview

19. 19 Exercise 1: Solving a set covering IP q Set of tasks q enum Tasks...; q Set of workers q range Workers 1..nbWorkers; q A subset of workers are qualified to perform each task q {Workers} qualified[Tasks] =...; q Find cheapest set of workers who are qualified to cover all jobs Combinatorial Optimization

20. 20 Completing exercise 1 q First, setup tutorial files and launch OPL Studio q ~gglockner/tutorial/setup q cd ~/ilog q opl q Select File > Open > Project and open setcvr.prj q Write objective function and covering constraint q Run by selecting Execution > Run q Stop by selecting Execution > Abort  Note: Your personal ilog directory (e.g. ~/ilog ) contains all sample files and a shell script for OPL Studio Combinatorial Optimization

21. 21 Exercise 2: Map coloring CP q Set of countries q enum Country...; q Set of colors q enum Colors...; q Set of bordering pairs q {adj} Border =...; q Assign a color to each country so that no bordering countries have same color Combinatorial Optimization

22. 22 Completing exercise 2 q Load map.prj q Add logical constraint to require bordering countries to have different colors q Run and use continue to find all solutions q Optional: Rewrite the problem to see if this map can be colored using fewer colors! q Create binary indicator variable colorUsed[i] q Add objective and constraint on colorUsed indicators m NB: Unlike MP, you can index a CP decision variable by another q Change “solve” to “subject to” q Save as map2 since we will reuse old version for exercise 3 Combinatorial Optimization

23. 23 Exercise 3: Tracing the algorithm q The CP search procedure q As ILOG Solver tries assignments, it propagates the effects to eliminate particular combinations q Once it finds a solution, it backtracks through the search tree and generate alternate solutions q OPL Studio can animate the search procedure q A graphical tree represents the search tree q A table shows the variable domains Combinatorial Optimization

24. 24 Completing exercise 3 q Load completed map coloring model (map.prj) q Select Execution > Browse Active Model q Right click on Variables > color and select display domain q Select Debug > Stop at Choice Point and Debug > Display Search Tree q Select Execution > Next to step through search procedure q Make sure to uncheck Debug > Stop at Choice Point and Debug > Display Search Tree when finished Combinatorial Optimization

25. 25 Demo 1: Advanced graphics q OPL’s graphical drawing board lets you develop custom graphics for a search procedure q Lines, ellipses, rectangles, and polygons q Various colors are available q Text labels q Graphics are undone when backtracking in the search tree q Try it! Run mapgr.prj Combinatorial Optimization

26. 26 Exercise 4: Factory sequencing q There are three car colors: red, blue, and gray q Day is divided into periods q You can make one car each period q Changing colors requires one idle period q You must meet daily production requirements q A simplified supply chain application Combinatorial Optimization

27. 27 Completing exercise 4 q Load factory.prj q Add constraint to satisfy daily demands q Careful! Should be cumulative q Add logical constraint for color changes  Useful logical operators: = (equals), <> (not equals), \/ (or), & (and), ! (not), => (implies) Combinatorial Optimization

28. 28 Improving performance q CP models benefit from redundant constraints that reduce search space q Like IP cuts, these are implied by model structure q Ex 1: No sequence of idle periods q For any two consecutive periods, at least one should not be idle q Ex 2: Color change q If a period is idle, then the surrounding periods should have different colors Combinatorial Optimization

29. 29 Improving performance – 2 q Alternate formulation q For each batch of cars of the same color, determine start time, duration, and the color q Focuses on runs rather than slots q Can use both formulations simultaneously! q More complicated, but more efficient q Easier to constrain run length, etc. q Fewer variables with larger domains q Take advantage of scheduling algorithms in ILOG Scheduler Combinatorial Optimization

30. 30 Agenda q About ILOG q Brief introduction to constraint programming q Exploring combinatorial optimization q Exploring iterative optimization q Discrete scheduling Overview

31. 31 What is iterative optimization? q Solving a sequence of related models q Try various simplified models q Decompose a large problem into subproblems q View model results under different scenarios Iterative Optimization

32. 32 Exercise 5: Resolving q Chicken nuggets q Sold in packages of 4, 6, and 9 q You can buy 10 by combining 4 & 6 q You cannot buy 7 by combining any of these boxes q What is the largest number you cannot make? q Determine this iteratively, not by number theory! Iterative Optimization

33. 33 Completing exercise 5 q Load nuggets.mod q Add constraints to determine if a total of nuggets can be created q Load and run nuggets.osc OPLScript q Optional q Try nuggets in sizes of 4,5,6 or 6,9,20 q Can you change the script to be more efficient? Iterative Optimization

34. 34 Traveling Salesman Problem (TSP) q Canonical combinatorial optimization problem q Create a tour that visits every city exactly once q Make the tour as cheap as possible q Common iterative solution method q Solve an integer programming problem q If you receive sub-cycles, add constraints to break the cycles (subtours) q Repeat until you find a single tour Iterative Optimization

35. 35 Demo 2: Solving the TSP q Open OPLScript in tsp.osc q Solves the TSP as a series of integer programs q Animates subtours via the drawing board q Try it! q You can change the data sets in the OPLScript q Note: Some data do not have graphics Iterative Optimization

36. 36 Agenda q About ILOG q Brief introduction to constraint programming q Exploring combinatorial optimization q Exploring iterative optimization q Discrete scheduling Overview

37. 37 Describing a scheduling problem q Modeling entities for scheduling problems q Activity variables m Defined by start time, end time, and duration m Can be breakable m Various precendence constraints for activities q Resource constraints m Unary Resources m Discrete Resources m Reservoirs m State Resources with Transition Times q Combine scheduling with generic constraint programming elements Discrete scheduling

38. 38 Want to decide the sequence of tasks to build a house Each task requires a certain amount of time to be completed Each task requires certain other tasks to be completed enum Tasks { masonry, carpentry, plumbing, ceiling, roofing, painting, windows, facade, garden, moving }; int duration[Tasks] = [7,3,8,3,1,2,1,2,1,1]; Activity a[t in Tasks](duration[t]);... a[masonry] precedes a[ceiling]; a[carpentry] precedes a[roofing]; a[ceiling] precedes a[painting];... Activity scheduling: Construction Discrete scheduling

39. 39 Each task requires one worker from a set Want to balance the workload enum Workers { Alan, Beth, Carl }; UnaryResource worker[Workers]; var int assign[Tasks,Workers] in 0..1; forall(t in Tasks, w in Workers) a[t] requires (assign[t,w]) worker[w]; forall(t in Tasks) sum(w in Workers) assign[t,w] = 1; Activity onsite[w in Workers]; minimize max(w in Workers) onsite[w].duration forall(t in Tasks, w in Workers) assign[t,w]=1 => (onsite[w].start <= a[t].start & onsite[w].end >= a[t].end); sum(w in Workers) onsite[w].duration >= sum(t in Tasks) duration[t]; Activity scheduling (2) Redundant Constraints Discrete scheduling

40. 40 enum Tasks { masonry, carpentry, plumbing, ceiling, roofing, painting, windows, facade, garden, moving }; enum Workers {Alan, Beth, Carl}; int duration[Tasks] = [7,3,8,3,1,2,1,2,1,1]; scheduleHorizon = sum(t in Tasks) duration[t]; Activity a[t in Tasks](duration[t]); Activity onsite[Workers]; var int assign[Tasks,Workers] in 0..1; UnaryResource worker[Workers]; minimize max (w in Workers) onsite[w].duration subject to { a[masonry] precedes a[carpentry]; a[masonry] precedes a[plumbing]; a[masonry] precedes a[ceiling]; a[carpentry] precedes a[roofing];... forall(t in Tasks, w in Workers) a[t] requires(assign[t,w]) worker[w]; forall(t in Tasks) sum(w in Workers) assign[t,w] = 1; forall(t in Tasks, w in Workers) assign[t,w]=1 => (onsite[w].start <= a[t].start & onsite[w].end >= a[t].end); sum (w in Workers) onsite[w].duration >= sum (t in Tasks) duration[t]; }; Decision Variables Objective Function Constraints Data OPL scheduling model Discrete scheduling

41. 41 Exercise 6: Scheduling q Open and run the model balhouse.mod q Look at the Gantt and resource charts q Double-click on the activities and resources q Try adding a constraint that Beth cannot do the plumbing q Try adding a constraint that Carl must be finished no later than day 13 m Hint: Every activity has a start, end, and duration time Discrete scheduling