1. *
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Wiederholung
Operations Research *

2. *
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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 Real world problem Real world solution Model Model solution*
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OR Process
Assessment
Real world problem
Real world solution
Abstraction
Interpretation
Analysis
Model
Model solution *

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

5. *
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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*
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Mathematical Programming
Optimization Models *

7. *
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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. *
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Mortimer Middleman *

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

10. *
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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. *
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Main constraints
Main constraints of optimization models specify the restriction and interactions, other than variable type, that limit decision variables. *

12. *
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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. *
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Mortimer Middleman *

15. Parameters – Output Variables*
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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*
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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*
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Two Crude Petroleum Case
gasoline
jet fuel
lubricant
Saudi
Venezuelan *

18. *
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Fence Excercise *

19. Howie’s Hot Tub Problem*
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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*
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Howie’s Decision Problem
Let
X1 = # of Aqua-spas produced
X2 = # of Hydro-Luxs produced
Maximize Z = 350 X X2
s.t.
X1 + X2 <= (pumps)
9 X1 + 6 X2 <= 1, (labor hours)
12 X X2 <= (feet of tubing)
X1, X2 >= (non-negativity)
Constraints, from upper left to lower right are tubing, pumps, labor. So tubing is non-binding at the optimal solution. *

21. *
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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. *
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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*
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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. *
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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.)*
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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. Mathematical Programming*
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Mathematical Programming
Graphical Solution *

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

28. An optimization model can have only one optimal value.*
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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*
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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. *
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Fence Excercise *

31. Mathematical Programming*
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Mathematical Programming
Large Scale Optimisation *

32. *
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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, *

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

34. Pi Hybrids Example The producing cost($/bag)*
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Pi Hybrids Example
The producing cost($/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. *
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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 Indexes: f = 1...l (facilities)*
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Pi Hybrids Example
Indexes:
f = 1...l (facilities)
h = 1...m (hybrid variety)
r = 1...n (sales region) *

37. *
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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 Variables: xf,h f = 1,...,l; h = 1,...,m*
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Pi Hybrids Example
Variables:
xf,h f = 1,...,l; h = 1,...,m
bags h at facility f
yf,h,r f = 1,...,l; h = 1,...m, r = 1,...,n
bags h from facility f to region r *

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

40. Pi Hybrids Example Parameters (continued): uf f = 1,...,l;*
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Pi Hybrids Example
Parameters (continued):
uf f = 1,...,l;
capacity (bushel)
ah h = 1,...,m;
(bushel/bag)
dh,r h = 1,...,m; r = 1,...,n
demand (bag) *

41. Indexed families of Constraints*
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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. *
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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. Mathematical Programming*
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Mathematical Programming
Linear or Nonlinear *

45. *
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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. *
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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 Non-Proportional regular hourly wage rates*
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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*
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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 x
Not linear
f(x) = sin(x)
f(x1, x2) = x1/x2
f(x) = ex *

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

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

51. *
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E-Mart *

52. Mathematical Programming*
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Mathematical Programming
Discrete (Integer) vs. Continuous *

53. *
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Discrete decision var.
A variable is discrete if it is limited to a fixed countable set of values. Often, the choices are integer or only binary (0 and 1).
A variable is continuous if it can take on any value in a specified interval. *

54. *
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Integer Program
An optimization model is an integer program (IP) if any one if its decision variables is discrete. If all variables are discrete, the model is a (pure) integer program; otherwise, it is a mixed-integer program (MIP). *

55. *
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Bethlehem Ingot Mold
F.J. Vasko, F.E. Wolf, K.S. Stott, J.W. Scheirer (1989),
„Selecting Optimal Ingot Sizes for Bethlehem Steel“,
Interfaces, 19:1, 68-84 *

56. *
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Bethlehem Ingot Mold *

57. *
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Integer Program
A discrete or integer programming model is an integer linear program (ILP) if its (single) objective function and all main constraints are linear.
A discrete or integer programming model is an integer nonlinear program (INLP) if its (single) objective function or any of its main constraints is nonlinear. *

58. Exam Scheduling C.J. Horan and W.D. Coates (1990)*
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Exam Scheduling
C.J. Horan and W.D. Coates (1990)
„Using More Than ESP to Schedule Final Exams: Purdue‘s Examination Scheduling Procedure II (ESP II)“
College and University Computer Users Conference Proceedings, 35, *

59. *
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Exam Scheduling *

60. LP Models are preferred*
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LP Models are preferred
When there is an option, such as when optimal variable magnitudes are likely to be large enough that fractions have no practical importance, modeling with continuous variables is preferred. *

61. Mathematical Programming*
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Mathematical Programming
Multiobjective Optimization Models *

62. DuPage Land Use Deepak Bammi and Dalip Bammi (1979)*
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DuPage Land Use
Deepak Bammi and Dalip Bammi (1979)
„Development of a Comprehensive Land Use Plans by means of a Multiple Objective Mathematical Progamming Model,“
Interfaces, 9:2, part 2, 50-63 *

63. DuPage Land Use Single-family residential Multiple-family residential*
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DuPage Land Use
Single-family residential
Multiple-family residential
Commerical
Offices
Manufacturing
Schools and other institutions
Open space *

64. *
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DuPage Land Use *

65. LP Model Linear programming requires a single objective function*
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LP Model
Linear programming requires a single objective function
If not:
including objectives as constraints in the model + Sensitivity Analysis
Goal Programming
MCDM *

66. Single objectives are preferred*
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Single objectives are preferred
When there is an option, single-objective optimization models are preferred to multiobjective ones because conflicts among objectives usually make multiobjective models less tractable. *

67. Beispiel 1
Production Allocation: The Acme Axle Company produces both car and track axles for national and international markets. Each axle must complete two manufacturing processes: molding and finishing. Each car axle requires 16 units of molding and 10 units of finishing, whereas a truck axle requires 24 units of molding and 20 units of finishing. Weekly, 480 units of molding and 360 units of finishing are available. The demand for Acme‘s axles is such that the firm may sell all it produces. Acme achieves a profit of $50 per car axle and $60 per truck axle. Acme also has an agreement with the Spitz Motor Company to supply 12 car axles and 8 truck axles weekly. Given the above constraints and requirements, Acme desires to know what amounts of car and truck axles to produce weekly in order to maximize profit. Formulate an LP Model to gain insights on the optimal production mix.

68. Beispiel 2
Large scale: Suppose that the decision variables of a mathematical programming model are
xi l t … amount of product i produced on manufacturing line l during week t
where i=1,…,17; l=1,…,5; t=1,…,7. Use summation and indexed notation to write expressions for each of the following systems of constraints in terms of these decision variables, and determine how many constraints belong to each system:
Total production on any line in any week should not exceed 200.
The total 7-week production of product i=5 should not exceed 4000.
At least 100 units of each product should be produced each week.

69. Bsp - Evaluierung
Evaluation: Five car salespeople had the following sales for the past two months:
The general manager believes that total dollar sales doesn’t adequately capture performance and would like to use a weighted average of luxury car, SUV, and mid-sized sales instead. The manager asks each salesperson to come up with a (positive) weight for each car category, but stipulates that weights cannot allow anyone’s total weighted score to exceed For example, defining w1 = luxury weight, w2 = SUV weight, and w3 = mid-sized weight, Fred’s weighted score would be: 3 w1 + 6 w w3. Develop an LP model that will find a set of weights that will make John’s weighted score as large as possible.

70. *
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Break *