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### Wiederholung

### Mathematical Programming

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

Real world solution

Abstraction

Interpretation

Analysis

Model

Model solution

Operations Research

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

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.

Optimization Models

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.

Mathematical Programming Deterministic Optimization

- Maximise/Minimise
- a single real function
- of real or integer variables

- subject to constraints on the variables

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.

Main constraints

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

Objective Functions

- Objective functions in optimization models quantify the decision consequences to be maximized or minimized.

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

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)

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

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

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)

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.

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.

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.

Optimal Solution

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

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

Mathematical Programming

Howie’s hot tube problem

Excel Workbook

Lawrence W. Robinson

Johnson Grad. School of Mgmt, Cornell University

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.

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

Mathematical Programming

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

Pi Hybrids Example

- l = 20 facilities
- m = 25 hybrid corn
- n = 30 sales region

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)

Indexing

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

Pi Hybrids Example

- f = 1...l (facilities)
- h = 1...m (hybrid variety)
- r = 1...n (sales region)

Indexes:

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

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:

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:

Pi Hybrids Example

Parameters (continued):

- uff = 1,...,l;
- capacity (bushel)

- ahh = 1,...,m;
- (bushel/bag)

- dh,rh = 1,...,m; r = 1,...,n
- demand (bag)

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.

Pi Hybrids Example

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.

Mathematical Programming

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.

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.

Linearity

- Proportional
- regular hourly wage rates
- machine output per hour
- …

- Non-Proportional
- wage rates for over time
- freight rates
- quantity purchasing discout

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

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, ...)

E-Mart

P. Doyle and J. Saunders (1990),

„Multiproduct Advertising Budgeting“,

Marketing Science, 9, 97-113

Discrete (Integer) vs. Continuous

Mathematical ProgrammingDiscrete 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.

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).

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

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.

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, 133-142

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.

Multiobjective Optimization Models

Mathematical ProgrammingDuPage 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

DuPage Land Use

- Single-family residential
- Multiple-family residential
- Commerical
- Offices
- Manufacturing
- Schools and other institutions
- Open space

LP Model

- Linear programming requires a single objective function
- If not:
- including objectives as constraints in the model + Sensitivity Analysis
- Goal Programming
- MCDM

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.

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.

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.

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 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.

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