Wiederholung. Operations Research. 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. OR Process. Assessment. Real world problem.
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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.
Real world problem
Real world solution
Operations Research deals with decision problems by formulating and analyzing mathematical models – mathematical representations of pertinent problem features.
The model-based OR approach to problem solving works best on problems important enough to warrant the time and resources for a careful study.
The three fundamental concerns of forming operations research models are
subject to g(x) <= 0
x >= 0
Taken from Ragsdale’s Book
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)
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.
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.
Optimal solutions show graphically as points lying on the best objective function contour that intersects the feasible region.
Lawrence W. Robinson
Johnson Grad. School of Mgmt, Cornell University
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
The first step in formulating a large optimization model is to choose appropriate indexes for the different dimensions of the problem.
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.
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.
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.
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.
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, ...)
P. Doyle and J. Saunders (1990),
„Multiproduct Advertising Budgeting“,
Marketing Science, 9, 97-113
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).
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
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, 133-142
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.
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
When there is an option, single-objective optimization models are preferred to multiobjective ones because conflicts among objectives usually make multiobjective models less tractable.
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.
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:
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 100. For example, defining w1 = luxury weight, w2 = SUV weight, and w3 = mid-sized weight, Fred’s weighted score would be: 3 w1 + 6 w2 + 12 w3. Develop an LP model that will find a set of weights that will make John’s weighted score as large as possible.