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DECISION MAKING

Learn about decision-making tools such as linear programming and operations research to solve optimization problems. Maximize or minimize objective functions while considering various constraints.

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DECISION MAKING

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  1. DECISION MAKING Pongsa Pornchaiwiseskul Pongsa.P@chula.ac.th http://pioneer.chula.ac.th/~ppongsa Faculty of Economics Chulalongkorn University Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  2. DECISION MAKING TOOLS • Statistics • operations research (a mathematical optimization model of a complex system) • deterministic, e.g., Linear Programming, transportation problem • stochastic, e.g., decision tree, queuing theories Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  3. OPERATIONS RESEARCH • Multiple objective • Network system • Inter-temporal (dynamic) system • uncertainty (stochastic) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  4. Pongsa’s Sessions Linear Programming(LP) LINDO program exercise Transportation Problem CPM/PERT Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  5. MATHEMATICAL PROBLEMS Most decision making problems could be formulated as a mathematical problem. Decision making processes can be represented with mathematical functions and constraints. Some decision making problems involve optimization of an objective function. Others try to simulate the result of various decisions before the decision can be made. Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  6. OPTIMIZATION PROBLEMS • maximize or minimize an objective function • subject to various constraints. Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  7. EXAMPLES OF OBJECTIVE FUNCTIONS • net profit • output or throughput • cost or input • time • likeliness or probability • variance or uncertainty Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  8. EXAMPLES OF CONSTRAINTS • staffing • space • budget • time • technology Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  9. LINEAR PROGRAMMING(LP) PROBLEM • mathematical optimization problem • its objective function and constraints are all linear Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  10. EXAMPLE OF AN LP PROBLEM A hypothetical hospital of 30 beds has two service departments (OPD and IPD). The In-Patient Department (IPD) needs 2 doctor-hours and 4 nurse-hours for every patient-day. In Out-Patient Department(OPD), each doctor-hour must be supported by 4 nurse-hours. The hospital can offer no more than 50 doctor-hours of OPD service in a day. The hospital can earn $120 for every patient-day in IPD and $80 for every doctor-hour in OPD. The hospital has a limited resource of 80 doctor-hours and 240 nurse-hours per day. Determine the resource allocation between OPD and IPD so that income of the hospital is maximized. Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  11. STRUCTURE OF THE LP(for the example) Objective = maximize income Constraints 1) # of doctor-hours 2) # of nurse-hours 3) # of beds 4) # of OPD service hours Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  12. DEFINE THE VARIABLES X1 = # of in-patients per day X2 = # of OPD doctor-hours per day Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  13. LP REPRESENTATION max Z = 120X1 + 80X2 X1 ,X2 s.t. 2X1 + X2 <= 80 --------(1) 4X1 + 4X2 <= 240 --------(2) X1 <= 30 --------(3) X2 <= 50 --------(4) X1 ,X2 >= 0 Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  14. NOTE You will get a different LP representation if you define the variables differently. Anyway, all the LPs still represent the same decision-making problem Let’s define Y1 = # of IPD doctor-hours per day Y2 = # of OPD nurse-hours per day Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  15. ANOTHER LP REPRESENTATION max Z = 60Y1 + 20Y2 Y1 ,Y2 s.t. Y1 + 0.25Y2 <= 80 --------(1’) 2Y1 + Y2 <= 240 --------(2’) Y1 <= 60 --------(3’) Y2 <= 200 --------(4’) Y1 ,Y2 >= 0 Replace X1 with 0.5Y1 and X2 with 0.25Y2 Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  16. HOW TO SOLVE AN LP PROBLEM • Graphical Method • Simplex method • by hand (not discussed) • by computer Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  17. GRAPHICAL METHOD • good for LPs with two variables • make it easy to understand solving an LP Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  18. SIMPLEX METHOD • algebraic method developed by George Dantzig • need to understand it very well if you want to do it by hand or develop a computer program to do it • quite a few computer programs can do this,e.g., • LINDO (DOS and Windows versions) • LP83 (DOS) • MPSX (IBM mainframe) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  19. LP SOLUTIONS OR RESULTS • Unique Solution • no other values of LP variables will yield the same optimized value of objective function • Alternative Solution (non-unique) • infinitely many feasible solutions • Infeasible solution • No values of LP variables satisfying the constraints Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  20. LINDO Inputs MAX 120 x1 + 80 x2 SUBJECT TO 2) 2 x1 + x2 <= 80 3) 4 x1 + 4 x2 <=240 4) x1 <= 30 5) x2 <= 50 END Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  21. LINDO Report LP OPTIMUM FOUND AT STEP 3 OBJECTIVE FUNCTION VALUE 1) 5600.000 VARIABLE VALUE REDUCED COST X1 20.000000 0.000000 X2 40.000000 0.000000 Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  22. LINDO Report (cont’d) ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 40.000000 3) 0.000000 10.000000 4) 10.000000 0.000000 5) 10.000000 0.000000 NO. ITERATIONS= 3 Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  23. DUAL SOLUTIONS • For each constraint, there exist a dual variable sometimes called dual price • Define mi = dual variable for constraint i = change in Z / change in RHS value • Graphical method cannot give a dual solution (value of dual variables) but Simplex can Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  24. MEANING OF DUAL SOLUTION(First Representation) m1 = opportunity cost of a doctor-hour m2 = opportunity cost of a nurse-hour m3 = opportunity cost of a bed-day m4 = opportunity cost of the OPD limit Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  25. HEREWORK The Malaria Division has to allocate 20,000 kg of insecticide and 150,000 blood test kits among two provinces(say A and B) so that the number of malaria cases will be minimized. The division has two control options. A unit of Option I is the use of 1 kg of insecticide and 10 test kits while a unit of Option II is 2 kg of insecticide and 5 blood test kits. Input/case reduced keeps increasing when the incidence decreases. Schedules of # of unit of inputs per case reduced shown in the table below. Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  26. Province A Decrease in cases Option I Option II 0- 5,000 1.0 1.0 5,000-10,000 1.5 2.0 10,000+ 3.0 3.5 Province B Decrease in cases Option I Option II 0- 3,000 1.5 1.8 3,000-6,000 2.0 2.5 6,000+ 3.5 4.5 Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  27. STRUCTURE OF THE LP(for the herework) Objective = maximize change in number of cases in Provinces A and B Constraints 1) insecticide 2) blood test kits 3) incidence functions(stepwise) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

  28. QUESTIONS(for the herework) 1) Number of cases reduced in each province 2) Opportunity cost of insecticide and blood test kits in terms of malaria cases reduced Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University

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