1. Mathematical Modeling Tran, Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology 2012-20131Tran Van Hoai

2. What is it ? MAXIMIZE50D+ 30C+ 6M SUBJECT TO 7D+ 3C+ 1.5M≤ 2000 D≥ 100 C≤ 500 D,C,M≥ 0 D,C integers 2012-2013Tran Van Hoai2 Mathematical Modeling = process to translate observed or desired phenomena into mathematical expressions (Total profit) (Raw steel) (Contract) (Cushions) (Nonnegativity) (Discrete)

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 2012-2013Tran Van Hoai3

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 2012-2013Tran Van Hoai4

5. Modeling variable constraints Limited number of cushions (lót nệm) C ≤ 500 Contract commitments C ≥ 100 Trivial constraints D, C, M ≥ 0 D, C integers 2012-2013Tran Van Hoai5

6. Solving the model is quite simple 2012-2013Tran Van Hoai6 MAXIMIZE50D+ 30C+ 6M SUBJECT TO 7D+ 3C+ 1.5M≤ 2000 D≥ 100 C≤ 500 D,C,M≥ 0 D,C integers MAXIMIZE50D+ 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) D = 100 (desks) C = 433 (chairs) M = 2/3 (pound)

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 2012-2013Tran Van Hoai7

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 2012-2013Tran Van Hoai8 Good principle wrong answer to right question is not fatal Right question to wrong answer is disastrous (thảm khốc) Good principle wrong answer to right question is not fatal Right question to wrong answer is disastrous (thảm khốc)

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) 2012-2013Tran Van Hoai9

10. Suggested approach 1.Observe operations – Understanding at least as well as those directly involved 2.Ease into complexity 3.Recognize political realities 4.Decide what is really wanted – Making company be sure of its objective 5.Identify constraints 6.Seek continuous feedback 2012-2013Tran Van Hoai10 Relate closely to models

11. Delta Hardware Store Problem statement 2012-2013Tran Van Hoai11 Google.com 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 To find least cost distribution scheme (from its plant, shipments from subcontractor) To meet demands its warehouses

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” 2012-2013Tran Van Hoai12

13. Suggested approach 1.Identify decision variables 2.Quantify the objectives/constraints 3.Construct a model shell 4.Gather data – Consider time/cost issues 2012-2013Tran Van Hoai13

14. Decision variables & decision makers “Controllable” or “uncontrollable” depend on who has control 2012-2013Tran Van Hoai14 PRODUCTION PROCESS Inputs Manager Owner $

15. Quick guide 1.Ask “Does the decision maker have the authority to decide the numerical value of the item?” – If answer = “yes”, it is decision variable 2.Be very precise in the units (& time frame) of each decision variable 1.Ask “Does the decision maker have the authority to decide the numerical value of the item?” – If answer = “yes”, it is decision variable 2.Be very precise in the units (& time frame) of each decision variable 2012-2013Tran Van Hoai15 Controllable input = decision variable Uncontrollable input = parameter Hardest part to build mathematical model

16. Delta Hardware Store Variable definition X1X1 Amount of paint shipped from Phoenix to San Jose X2X2 Amount of paint shipped from Phoenix to Fresno X3X3 Amount of paint shipped from Phoenix to Azusa X4X4 Amount of paint subcontracted for San Jose X5X5 Amount of paint subcontracted for Fresno X6X6 Amount of paint subcontracted for Azusa 2012-2013Tran Van Hoai16 Decision maker has no control over demand, production capacities, unit costs

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 2012-2013Tran Van Hoai17 Total profit = Total revenues – Total cost

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) 2012-2013Tran Van Hoai18

19. Delta Hardware Store Additional 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 2012-2013Tran Van Hoai19

20. Create a model in words Minimize overall monthly cost (manufacturing, transporting, subcontracting) Subject to 1.Phoenix plant cannot operate beyond its capacity 2.Amount order to subcontractor is not over a maximum limit 3.Orders at each warehouse will be fulfilled 2012-2013Tran Van Hoai20 Delta Hardware Store Informal model

21. Objective function MManufacturing cost at Phoenix plant T 1, T 2 T 3 Shipping cost from Phoenix to San Jose, Fresno, Azusa CFixed cost per 1000 gallons from subcontractor S 1, S 2 S 3 Shipping charge by subcontractor to San Jose, Fresno, Azusa 2012-2013Tran Van Hoai21 MINIMIZE(M+T 1 )X 1 + (M+T 2 )X 2 + (M+T 3 )X 3 + (C+S 1 )X 4 + (C+S 2 )X 5 + (C+S 3 )X 6

22. Constraints (1) 2012-2013Tran Van Hoai22 Q1Q1 Capacity of the Phoenix plant Q2Q2 Maximum number of gallons available from subcontractor R 1 R 2 R 3 Respective orders at warehouses in San Jose, Fresno, Azusa 1. Number of truckloads shipped out from Phoenix cannot exceed plant capacity X 1 + X 2 + X 3 ≤ Q 1 2. Number of gallons ordered from subcontractor cannot exceed order limit X 4 + X 5 + X 6 ≤ Q 2 1. Number of truckloads shipped out from Phoenix cannot exceed plant capacity X 1 + X 2 + X 3 ≤ Q 1 2. Number of gallons ordered from subcontractor cannot exceed order limit X 4 + X 5 + X 6 ≤ Q 2

23. Constraints (2) 2012-2013Tran Van Hoai23 3. Number of gallons received at each warehouse equals to its total order X 1 + X 4 = R 1 X 2 + X 5 = R 2 X 3 + X 6 = R 3 4. All shipments are nonnegative and integers X 1, X 2, X 3, X 4, X 5, X 6 ≥ 0 X 1, X 2, X 3, X 4, X 5, X 6 integer 3. Number of gallons received at each warehouse equals to its total order X 1 + X 4 = R 1 X 2 + X 5 = R 2 X 3 + X 6 = R 3 4. All shipments are nonnegative and integers X 1, X 2, X 3, X 4, X 5, X 6 ≥ 0 X 1, X 2, X 3, X 4, X 5, X 6 integer Need gathering (or approximating) data for parameters

24. 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 A business client settles for 80% of optimal solution at 20% of cost to obtain it RULE OF THUMB “Pareto principle” or “80/20 rule” RULE OF THUMB “Pareto principle” or “80/20 rule” Data gathering- time/cost issues 2012-2013Tran Van Hoai24

25. Simplify the problem – Transportation problem with only cost for manufacturing, ordering, transportation – Partial truckload, wholesale pricing, time- dependent cost,…are ignored 2012-2013Tran Van Hoai25 Delta Hardware Store Data gathering R1R1 4S1S1 $1200 R2R2 2S2S2 $1400 R3R3 5S3S3 $1100 Q2Q2 5C$5000

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) 2012-2013Tran Van Hoai26 Q 1 = AVG(production) past months = 7.9 (~8)

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 2012-2013Tran Van Hoai27 M = $3.00 * 1000 = $3000 Q 1 = $100 + $150 + $800 = $1050 Q 2 = $100 + $100 + $555 = $750 Q 3 = $100 + $120 + $430 = $650 M = $3.00 * 1000 = $3000 Q 1 = $100 + $150 + $800 = $1050 Q 2 = $100 + $100 + $555 = $750 Q 3 = $100 + $120 + $430 = $650

28. Final model Minimize 4050X 1 + 3750X 2 + 3650X 3 + 6200X 4 + 6400X 5 + 6100X 6 S.t. X 1 + X 2 + X 3 ≤ 8 X 4 + X 5 + X 6 ≤ 5 X 1 + X 4 = 4 X 2 + X 5 = 2 X 3 + X 6 = 4 X i ≥ 0, integer  i=1,…,6 2012-2013Tran Van Hoai28

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 2012-2013Tran Van Hoai29 Cost/time must be considered Large classes of problems have efficient solution techniques Cost/time must be considered Large classes of problems have efficient solution techniques

30. How to choose solution techniques? Can apply observation of experts 2012-2013Tran Van Hoai30 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 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

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 ? 2012-2013Tran Van Hoai31 Testing/Validating is time-consuming process Historical/Simulated (hypothetical) data can be used Testing/Validating is time-consuming process Historical/Simulated (hypothetical) data can be used

32. Iterative development If one team not successful, other team comes with fresh mind 2012-2013Tran Van Hoai32 MODEL – SOLVE – VERIFY ManagerAnalysist

33. What-if 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 2012-2013Tran Van Hoai33

34. Report Adjustable Cells FinalReducedObjectiveAllowable CellNameValueCost Coefficie ntIncreaseDecrease $B$13 PHOENIX PLANT SAN JOSE1040502150300 $C$13 PHOENIX PLANT FRESNO2037505001E+30 $D$13 PHOENIX PLANT AZUSA5036503001E+30 $B$14 SUBCONTRACTOR SAN JOSE3062003002150 $C$14 SUBCONTRACTOR FRESNO050064001E+30500 $D$14 SUBCONTRACTOR AZUSA030061001E+30300 2012-2013Tran Van Hoai34

35. MS process – step 4: Communicating/Implementing results Prepare a business report/presentation Monitor the progress of the implementation 2012-2013Tran Van Hoai35 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 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

36. Next Linear Programming Models Integer Linear Programming Models 2012-2013Tran Van Hoai36