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Graduate Program in Business Information Systems

Graduate Program in Business Information Systems. Linear Programming and Applications. Aslı Sencer. OPERATIONS RESEARCH. What is Operations Research? Collection of techniques used to allocate the scarce resources in the “best” –OPTIMAL – way! Best of what?

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Graduate Program in Business Information Systems

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  1. Graduate Program in Business Information Systems Linear Programming and Applications Aslı Sencer

  2. OPERATIONS RESEARCH • What is Operations Research? • Collection of techniques used to • allocate the scarce resources • in the “best” –OPTIMAL – way! • Best of what? • We need an “objective” to be minimized or maximized! • Profit, Cost, Utility, Delay, Distance, Flow, etc. BIS 517-Aslı Sencer

  3. Some applications • Resource allocation • Production and inventory planning • Capacity Planning • Workers and machine scheduling • Investment planning • Formulating marketing and military strategies BIS 517-Aslı Sencer

  4. Some news about OR: • A Wall Street Journal Article lists the use of LP as one of the greatest technological innovations of the past 1000 years. • 1975 Nobel Prize for economics: T.C. Koopmans and L.V. Kantoprovich for their contributions in the field • 1992 Nobel Prize: Harry Markowitz for his LP based research. BIS 517-Aslı Sencer

  5. Basic Optimization Techniques • Linear Programming • Integer and Goal Programming • Transportation, Assignment Models • Network Models • Nonlinear Programming • Stochastic Programming • Also Simulation BIS 517-Aslı Sencer

  6. LINEAR PROGRAMMING • Most successful of all modern quantitative methods. • Program here is not a computer code! It is a plan that efficiently allocate limited resources to achive a goal. • Involves linear relationships, i.e. relations are in the form of lines, planes! BIS 517-Aslı Sencer

  7. Basic LP Models: Product Mix • Redwood Furniture Co. BIS 517-Aslı Sencer

  8. What is the optimal plan to max. Profit? • Option 1: Allocate all resources to the more profitable item. Total quantity, profit? Any resource left? • Option 2: Is it more profitable to produce less chairs and more tables? Linear Programming BIS 517-Aslı Sencer

  9. Formulating a Linear Problem • Define variables: : number of tables produced in a period : number of chairs produced in a period • Define constraints: • Define Objective Function BIS 517-Aslı Sencer

  10. How is an LP solved? • Graphical Method: Applicable to a maximum of two decision variables. • Simplex Method: Applicable to all LP. Takes long to implement manually. • Use softwares based on simplex and other techniques. BIS 517-Aslı Sencer

  11. Graphical Solution Xc 15 Optimal Solution: Xt=4 tables, Xc=9 Chairs Profit*=$96 Constraint 1 11 (4,9) Constraint 2 Xt 10 22 BIS 517-Aslı Sencer

  12. Basic LP Models: Feed Mix • Two types of seeds are mixed to formulate the wheat of wild birdseed. BIS 517-Aslı Sencer

  13. LP Formulation BIS 517-Aslı Sencer

  14. Graphical Solution to Feed Mix Problem Xs Optimal Solution: Xb*=3750 lb, Xs*=7500 lb Cost*=$1425 (3750,7500) (15000,0) (10000,0) Xb Roughage Fat Protein BIS 517-Aslı Sencer

  15. Types of Feasible Regions Bounded Feasible Region Unbounded Feasible Region BIS 517-Aslı Sencer

  16. Cases in an LP: Infeasible Solution Do all LP has an optimal Solution? No feasible region If an LP has no feasible region, then the solution is INFEASIBLE! BIS 517-Aslı Sencer

  17. Cases in an LP: Multiple Optima Optimal Solutions: Point(1): X1*=4 2/7, X2*=6 3/7 Point(2): X1*=6 6/7, X2*=4 2/7 P*=$120 Infinite number of optimal solutions exist in the form (1) (2) BIS 517-Aslı Sencer

  18. Cases in an LP: Unbounded Optimal Solution Optimal Solution: X1*=, X2*= P*= BIS 517-Aslı Sencer

  19. Solving Linear Programs with a Spreadsheet • Write out the formulation table • Put the formulation table into a spreadsheet • Use Excel’s Solver to obtain a solution BIS 517-Aslı Sencer

  20. Solution in the Excel Solver BIS 517-Aslı Sencer

  21. Applications of LP:Transportation Models • Sporting goods company Warehouses Demand Capacity Plants Frankfurt 100 300 200 Juarez 150 100 200 150 NY Seoul Phoenx Tel Aviv Yokohama BIS 517-Aslı Sencer

  22. LP:Transportation Models (cont’d.) • What are the optimal shipping quantities from the • plants to the warehouses, if the demand has to be met • by limited capacities while the shipping cost is minimized? Shipping Costs per pair of skis BIS 517-Aslı Sencer

  23. LP:Transportation Models (cont’d.) Xij: Number of units shipped from plant i to warehouse j. i=1,2,3 and j=1,2,3,4. Minimize shipping costs=19X11+7X12+3X13+21X14 +15X21+21X22+18X23+6X24 +11X31+14X32+15X33+22X34 BIS 517-Aslı Sencer

  24. LP:Transportation Models (cont’d.) subject to #shipped from a plant can not exceed the capacity: X11+X12+X13+X14≤100 (Juarez Plant) X21+X22+X23+X24≤300 (Seoul Plant) X31+X32+X33+X34≤200 (Tel Aviv Plant) #shipped to a warehouse can not be less than the demand: X11+X21+X31≥150 (Frankfurt) X12+X22+X32≥100 (NY) X13+X23+X33≥200 (Phoenix) X14+X24+X34≥150 (Yokohama) Nonnegativity Xij ≥0 for all i,j. BIS 517-Aslı Sencer

  25. LP:Transportation Models (cont’d.) Optimal Solution: Optimal cost=$6,250 Warehouses Demand Capacity Plants Frankfurt 100 300 200 Juarez 150 100 200 150 50 100 NY Seoul 100 100 Phoenx Tel Aviv 100 Yokohama 150 BIS 517-Aslı Sencer

  26. LP: Marketing Applications • How to allocate advertising budget between mediums such as TV, radio, billboard or magazines? Ex: Real Reels Co. Allocated ad. Budget=$100,000 • No more than 5 ads in True and at least two ads in Playboy and Esquire BIS 517-Aslı Sencer

  27. LP: Marketing Applications (cont’d.) Not integer? BIS 517-Aslı Sencer

  28. LP: Assignment Models • Assignment of a set of workers to a set of jobs BIS 517-Aslı Sencer

  29. LP: Assignment Models (cont’d.) BIS 517-Aslı Sencer

  30. LP: Diet Problem • How much to use of each ingredient so that the nutritional requirements are met in the cheapest way? • Ex: Feed Mix problem given at the beginning of the lecture BIS 517-Aslı Sencer

  31. LP:Labor Planning • Addresses staffing needs over a specific time period. Hong Kong Bank of Commerce: • 12 Full time workers available, but may fire some. • Use part time workers who has to work for 4 consequtive hours in a day. • Luch time is one hour between 11a.m. and 1p.m. shared by full time workers. • Total part time hours is less than 50% of the day’s total requirement. • Part-timers earn $4/hr (=$16/day) and full timers earn $50/day. BIS 517-Aslı Sencer

  32. LP:Labor Planning (Cont’d.) BIS 517-Aslı Sencer

  33. LP:Labor Planning (cont’d.) BIS 517-Aslı Sencer

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