Wiederholung

1. Wiederholung Operations Research

2. Operations Research Operations Research (OR) is the field of how to form mathematical models of complex management decision problems and how to analyze the models to gain insight about possible solutions.

3. OR Process Assessment Real world problem Real world solution Abstraction Interpretation Analysis Model Model solution

4. Operations Research Operations Research deals with decision problems by formulating and analyzing mathematical models – mathematical representations of pertinent problem features.

5. Operations Research The model-based OR approach to problem solving works best on problems important enough to warrant the time and resources for a careful study.

6. Mathematical Programming Optimization Models

7. OR models The three fundamental concerns of forming operations research models are • decisions open to decision makers, • the constraints limiting decision choices, and • the objectives making some decisions preferred to others.

8. Mortimer Middleman

9. Mathematical Programming Deterministic Optimization • Maximise/Minimise • a single real function • of real or integer variables • subject to constraints on the variables

10. Variables • Variables in optimization models represent the decisions to be taken. • Variable-type constraints specify the domain of definition for decision variables: the set of values for which the variables have meaning.

11. Main constraints • Main constraints of optimization models specify the restriction and interactions, other than variable type, that limit decision variables.

12. Objective Functions • Objective functions in optimization models quantify the decision consequences to be maximized or minimized.

13. Mortimer Middleman • d ... weekly demand • f ... fixed cost of replenishment • h ... cost per carat per week holding • s ... cost per carat lost sales • l ... lead time • m ... minimum order size

14. Mortimer Middleman

15. Parameters – Output Variables • Parameters – quantities taken as given • Weekly demand, fixed cost of replenishment, cost for holding inventory, cost per carat lost sales, lead time, minimum order size. • Parameters and decision variables determine results measured as output variables • c(r,q ; d,f,h,s,l,m)

16. Canonical Form of a (Non-Linear) Optimization Problem • Maximize f(x) subject to g(x) <= 0 x >= 0 • Key Components of Optimization Pb. • Objective Function • Decision Variables • Constraints

17. Two Crude Petroleum Case gasoline jet fuel lubricant Saudi Venezuelan

18. Fence Excercise

19. Howie’s Hot Tub Problem • Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Acqua-Spa and the Hydro-Lux. Howie Jones, the owner and manager of the company, needs to decide how many of each type of hot tub to produce during his next production cycle. Howie buys prefabricated fiberglass hot tub shells from a local supplier and adds the pump and tubing to the shells to create his hot tubs. (This supplier has the capacity to deliver as many hot tub shells as Howie needs.) Howie installs the same type of pump into both hot tubs. He will have only 200 pumps available during his next production cycle. From a manufacturing standpoint, the main difference between the two models of hot tubs is the amount of tubing and labor required. Each Acqua-Spa requires 9 hours of labor and 12 feet of tubing. Each Hydro-Lux requires 6 hours of labor and 16 feet of tubing. Howie expects to have 1,566 production labor hours and 2,880 feet of tubing available during the next production cycle. Howie earns a profit of $350 on each Aqua-Spa he sells and $300 on each Hydro-Luc he sells. He is confident that he can sell all the hot tubs he produces. The question is, how many Acqua-Spas and Hydro-Luxes should Howie produce if he wants to maximize his profits during the next production cycle? Taken from Ragsdale’s Book

20. Howie’s Decision Problem • Let • X1 = # of Aqua-spas produced • X2 = # of Hydro-Luxs produced • Maximize Z = 350 X1 + 300 X2 s.t. X1 + X2 <= 200 (pumps) 9 X1 + 6 X2 <= 1,566 (labor hours) 12 X1 + 16 X2 <= 2880 (feet of tubing) X1, X2 >= 0 (non-negativity)

21. Feasible • The feasible set (or region) of an optimization model is the collection of choices for decision variables satisfying all model constraints. • The feasible set for an optimization model is plotted by introducing constraints one by one, keeping track of the region satisfying all at the same time.

22. Optimal Solution An optimal solution is a feasible choice for decision variables with objective function value at least equal to that of any other solution satisfying all constraints.

23. Graphing Objective Functions Objective functions are normally plotted in the same coordinate system as the feasible set of optimization model by introducing contours – lines or curves through points having equal objective function values.

24. Optimal Solution Optimal solutions show graphically as points lying on the best objective function contour that intersects the feasible region.

25. Graphical Solution(Only practical for 2D Pbs.) • Plot the constraints • Identify the feasible region • Draw contours (level curves; iso-value lines) of objective function • Most desirable level curve will intersect feasible region

26. Graphical Solution Mathematical Programming

27. Howie’s hot tube problem Excel Workbook Lawrence W. Robinson Johnson Grad. School of Mgmt, Cornell University

28. Optimal Value • The optimal value in an optimization model is the objective function value of any optimal solution. • An optimization model can have only one optimal value.

29. Use Graphical Solution to Develop Some Intuition • Alternate optimal solutions • If obj. fn. is parallel to a binding constraint • Redundant constraints • Plays no role in determining feasible region • Unbounded solution • Can occur if feasible region is unbounded • Infeasible problem • There is no feasible region; constraints are inconsistent

30. Fence Excercise

31. Large Scale Optimisation Mathematical Programming

32. Pi Hybrids Example Mingjian Zuo, Way Kuo, and Keith L. McRoberts (1991), „Application of Mathematical programming to a Large-Scale Agricultural Production and Distribution System“, Journal of Operational Research Society, 42, 639-648

33. Pi Hybrids Example • l = 20 facilities • m = 25 hybrid corn • n = 30 sales region

34. Pi Hybrids Example • The producing cost($/bag) • The corn processing capacity (bushels) • The corn needed to produce a bag (bushels/bag) • Hybrid corn demanded (bag) • The cost per bag shipping ($/bag)

35. Indexing The first step in formulating a large optimization model is to choose appropriate indexes for the different dimensions of the problem.

36. Pi Hybrids Example • f = 1...l (facilities) • h = 1...m (hybrid variety) • r = 1...n (sales region) Indexes:

37. Indexing parameters To describe large-scale optimization models compactly it is usually necessary to assign indexed symbolic names to variables and to most input parameters, even though they are being treated as constant. Summation Notation

38. Pi Hybrids Example • xf,hf = 1,...,l; h = 1,...,m • bags h at facility f • yf,h,rf = 1,...,l; h = 1,...m, r = 1,...,n • bags h from facility f to region r Variables:

39. Pi Hybrids Example • pf,hf = 1,...,l; h = 1,...,m • production cost ($/bag) • sf,h,rf = 1,...,l; h = 1,...m, r = 1,...,n • shipping cost ($/bag) Parameters:

40. Pi Hybrids Example Parameters (continued): • uff = 1,...,l; • capacity (bushel) • ahh = 1,...,m; • (bushel/bag) • dh,rh = 1,...,m; r = 1,...,n • demand (bag)

41. Indexed families of Constraints Families of similar constraints distinguished by indexes may be expressed in a single-line format (constraint for fixed indexes) (ranges of indexes) which implies one constraint for each combination of indexes in the ranges specified.

42. Pi Hybrids Example

43. Large-scale Optimization models become large mainly by a relatively small number of objective function and constraint elements being repeated many times for different periods, locations, products, and so on.

44. Linear or Nonlinear Mathematical Programming

45. LP Model An optimization model is a linear program (or LP) if it has continuous variables, a single linear objective function, and all constraints are linear equalities or inequalities.

46. Linear functions • A function is linear if it is a constant-weighted sum of decision variables. Otherwise, it is nonlinear. • Linear functions implicitly assume that each unit increase in a decision variable has the same effect as the preceding increase: equal returns to scales.

47. Linearity • Proportional • regular hourly wage rates • machine output per hour • … • Non-Proportional • wage rates for over time • freight rates • quantity purchasing discout

48. f(x) is linear if it is a sum of constants times the components of x • Linear • y = f(x) = a x + b • f(x) = c0 + c1 x1 + c2 x2 + c3 x3 + ... • Not linear • f(x) = sin(x) • f(x1, x2) = x1/x2 • f(x) = ex

49. Linear Programming: A Special Kind of NLP • Suppose • Objective function is linear • Constraints are linear • Decision variables are continuous • Max cTx (i.e., c1 x1 + c2 x2 + ...) st A x <= b (a1,1 x1 + a1,2 x2 + ... <= b1 a2,1 x1 + a2,2 x2 + ... <= b2) x >= 0 (i.e., x1 >= 0, x2 >= 0, ...)

50. E-Mart P. Doyle and J. Saunders (1990), „Multiproduct Advertising Budgeting“, Marketing Science, 9, 97-113