Presentation Transcript

1. Mathematical Modeling

   Tran, Van HoaiFaculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai
2. What is it ? MAXIMIZE 50D + 30C + 6M SUBJECT TO 7D + 3C + 1.5M ≤ 2000 D ≥ 100 C ≤ 500 D, C, M ≥ 0 D, C integers (Total profit) (Raw steel) (Contract) (Cushions) (Nonnegativity) (Discrete) Mathematical Modeling = process to translate observed or desired phenomena into mathematical expressions Tran Van Hoai
3. Modeling profit NetOffice: a company to produce Desk (D = number of desks) Chair (C = number of chairs) Molded steel (M = pounds of molded steel) Profit (net) $50/a desk $30/a chair $6/a pound molded steel 50D + 30C + 6M Tran Van Hoai
4. Modeling functional constraints Raw steel 7 pounds for a desk 3 pounds for a chair 1.5 pounds for a pound of molded steel 7D + 3C + 1.5M Functional constraint 7D + 3C + 1.5M ≤ 2000 NewOffice only has 2000 pounds of raw steel Tran Van Hoai
5. Modeling variable constraints Limited number of cushions (lótnệm) C ≤ 500 Contract commitments C ≥ 100 Trivial constraints D, C, M ≥ 0 D, C integers Tran Van Hoai
6. Solving the model is quite simple MAXIMIZE 50D + 30C + 6M SUBJECT TO 7D + 3C + 1.5M ≤ 2000 D ≥ 100 C ≤ 500 D, C, M ≥ 0 D, C integers Spreadsheet, WinQSB, Gurubi, COIN, ILOG,… D = 100 (desks) C = 433 (chairs) M = 2/3 (pound) Tran Van Hoai
7. Mathematical models Optimization model is to maximize/minimize a quantity that maybe restricted by a set of constraints Prediction model is to describe/predict events given a certain conditions Deterministic model is in which profit, cost,…assumed to be known with certainty Stochastic model is in which (at least) one values of parameters determined by probability distributions Tran Van Hoai
8. MS process – step 1:Defining the problem General situation to apply MS/OR Designing/implementing new operations Evaluating ongoing set of operations Determining/recommending corrective action for operations which producing unsatisfactory results Good principle wrong answer to right question is not fatal Right question to wrong answer is disastrous (thảmkhốc) Tran Van Hoai
9. Factors to be faced “Fuzzy” (incomplete, conflicting) “Soft” constraints (goals or restrictions) Different opinions (worker/manager/owner) Limited budget for analyses Limited time for analyses/recommendations Political “turf wars” No idea on what is wanted (ask consultant to tell) Tran Van Hoai
10. Suggested approach Observe operations Understanding at least as well as those directly involved Ease into complexity Recognize political realities Decide what is really wanted Making company be sure of its objective Identify constraints Seek continuous feedback Relate closely to models Tran Van Hoai
11. Delta Hardware StoreProblem statement 3 warehouses 1 production plant Do not expand production capacity Subcontract other manufacturer (label product s by Delta) To find least cost distribution scheme (from its plant, shipments from subcontractor) To meet demands its warehouses Google.com Tran Van Hoai
12. MS process – step 2:Building mathematical model “Put scattered thoughts, ideas, conflicting objectives/constraints into logical coherent decision framework” “Mathematical modeling is an art” Tran Van Hoai
13. Suggested approach Identify decision variables Quantify the objectives/constraints Construct a model shell Gather data – Consider time/cost issues Tran Van Hoai
14. Decision variables & decision makers “Controllable” or “uncontrollable” depend on who has control Inputs Manager PRODUCTION PROCESS $ Owner Tran Van Hoai
15. Quick guide Controllable input = decision variable Uncontrollable input = parameter Hardest part to build mathematical model Ask “Does the decision maker have the authority to decide the numerical value of the item?” If answer = “yes”, it is decision variable Be very precise in the units (& time frame) of each decision variable Tran Van Hoai
16. Delta Hardware StoreVariable definition Decision maker has no control over demand, production capacities, unit costs Tran Van Hoai
17. Quantify objective/constraints Often, there is single objective function ≥2 objective functions → multicriteria decision problem Constraints can be definitional in nature Artificial constraints can be added to strengthen model Total profit = Total revenues – Total cost Tran Van Hoai
18. Quick guide Create limiting condition in words as follows (amount of resource required) (Has some relation to) (Availability of the resource) Translate to math expressions, using known, parameters, and variables Move variables to left side, constants to right side Construct model shell Use generic symbols for parameters (until actual data determined) Tran Van Hoai
19. Delta Hardware StoreAdditional observation Additional information Finite production capacity at Phoenix plant Limited amount of paint available from subcontractor Different requirements for 3 warehouses Orders in unit of 1000 gallons of paints (=a truck delivery), cost = f( time, distance ) Subcontractor charges fixed fee for a 1000-gallon order, a delivery charge for each city Tran Van Hoai
20. Delta Hardware StoreInformal model Create a model in words Minimize overall monthly cost (manufacturing, transporting, subcontracting) Subject to Phoenix plant cannot operate beyond its capacity Amount order to subcontractor is not over a maximum limit Orders at each warehouse will be fulfilled Tran Van Hoai
21. Objective function MINIMIZE (M+T1)X1+ (M+T2)X2+ (M+T3)X3+ (C+S1)X4+ (C+S2)X5+ (C+S3)X6 Tran Van Hoai
22. Constraints (1) 1. Number of truckloads shipped out from Phoenix cannot exceed plant capacity X1 + X2 + X3 ≤ Q1 2. Number of gallons ordered from subcontractor cannot exceed order limit X4 + X5 + X6 ≤ Q2 Tran Van Hoai
23. Constraints (2) 3. Number of gallons received at each warehouse equals to its total order X1 + X4 = R1 X2 + X5 = R2 X3 + X6 = R3 4. All shipments are nonnegative and integers X1, X2, X3, X4, X5, X6 ≥ 0 X1, X2, X3, X4, X5, X6 integer Need gathering (or approximating) data for parameters Tran Van Hoai
24. Data gathering- time/cost issues Time/cost of collecting, organizing, sorting relevant “Hard” data >< “soft” data Harder the data, more costly/time consuming to obtaint Time/cost of generating solution approach Simplifying solution technique can lead to unrealistic Time/cost of using the model Management must respond rapidly to dynamic business → impact on model selected RULE OF THUMB “Pareto principle” or “80/20 rule” A business client settles for 80% of optimal solution at 20% of cost to obtain it Tran Van Hoai
25. Delta Hardware StoreData gathering Simplify the problem Transportation problem with only cost for manufacturing, ordering, transportation Partial truckload, wholesale pricing, time-dependent cost,…are ignored Tran Van Hoai
26. Production limit No plant runs continuously at full capacity due to machine failure, partial staffing, limited resource Two possibilities Theoretical production limit * reduction factor Ask plant manager “what is best estimations?” Make a forecast E.g., compute an average production (except outlier) Q1 = AVG(production)past months = 7.9 (~8) Tran Van Hoai
27. Plant product/transportation costs Production cost Direct: $2.25 Indirect: $6000/8000 Transportation cost Loading (at Phoenex): $100 Unloading: (San Jose) $150, (Fresno) $100, (Azusa) $120 Mileage: (to San Jose) $800, (to Fresno) $550, (to Azusa) $430 M = $3.00 * 1000 = $3000 Q1 = $100 + $150 + $800 = $1050 Q2 = $100 + $100 + $555 = $750 Q3 = $100 + $120 + $430 = $650 Tran Van Hoai
28. Final model Minimize 4050X1 + 3750X2 + 3650X3 + 6200X4 + 6400X5 + 6100X6 S.t. X1 + X2 + X3 ≤ 8 X4 + X5 + X6 ≤ 5 X1 + X4 = 4 X2 + X5 = 2 X3 + X6 = 4 Xi ≥ 0, integer i=1,…,6 Tran Van Hoai
29. MS process – step 3:Solving mathematical model Choose an appropriate solution techniques Generate model solutions Test/Validate model results Return to modeling step if unacceptable results Perform “what-if” analyses Cost/time must be considered Large classes of problems have efficient solution techniques Tran Van Hoai
30. How to choose solution techniques? Can apply observation of experts Woolsey’s Laws Managers would rather live with a problem they can’t solve than use a technique they don’t trust Managers don’t want the best solution, they simply want a better one If the solution technique will cost you more than you will save, don’t use it Tran Van Hoai
31. Test/Validate model results Due to simplification, optimal/heuristical, simulated solutions Good solutions are not for real-life situation We need test/validate to answer Do the results make sense ? Intuitive ? Can solution be integrated in current conditions ? Changes needed ? Does solution modify plans of the organization ? Testing/Validating is time-consuming process Historical/Simulated (hypothetical) data can be used Tran Van Hoai
32. Iterative development If one team not successful, other team comes with fresh mind Manager Analysist MODEL – SOLVE – VERIFY Tran Van Hoai
33. What-if analyses Computer solution to a model is “an answer” for the model Managers need anticipating more Management concerns Potential new opportunities Possible changes What-if Tran Van Hoai
34. Report Tran Van Hoai
35. MS process – step 4:Communicating/Implementing results Prepare a business report/presentation Monitor the progress of the implementation HOMEWORK Read textbook 1.5. Writing business report/memos 1.6 . Using speadsheets in management science models 2.5. Using Excel Solver to find an optimal solution and analyze results Tran Van Hoai
36. Next Linear Programming Models Integer Linear Programming Models Tran Van Hoai