[PDF] [PDF] LINEAR PROGRAMMING MODELS

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11 mai 2008 · For instance, several assumptions are implicit in linear programing The example of a canonical linear programming problem from the 

[PDF] LINEAR PROGRAMMING MODELS

A Linear programming problem can be expressed in the following standard form: max z= c1x1+ Assumptions of Linear Programming 1 Examples of LP

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11 mai 1998 · Linear programming is a mathematical technique for finding optimal solutions to problems cases, where, for example, the variables might represent the levels of a set The Fundamental Assumptions of Linear Programming

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tion Models B5 Assumptions of Linear Programming Models B6 The following example shows how an operational problem can be represented and analyzed 

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We now present examples of four general linear programming problems Under this assumption, the inequalities in constraints (2) and (3) must be satisfied

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example, let us assume that the revenue per table is 120 and revenue per chair is 70 define a Linear Programming problem, state the assumptions, introduce  

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For example, the objective function coefficient for x1 is 3, and the objec- Does the Dorian model meet the four assumptions of linear programming outlined in

[PDF] 101 Types of Constraints and Variables in Linear Programming

10 2 "Violations" of the Algorithmic Assumptions as well as LP assumptions 10 1 Types of Constraints and For example, a LP formulation of an automobile

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too many wild assumptions, in the form of an LP then you know you can get a The presentation below is given in terms of the following constraint types:

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INDR 262 Optimization Models and Mathematical Programming

LINEAR PROGRAMMING MODELS

Common terminology for linear programming:

- linear programming models involve . resources denoted by i, there are m resources . activities denoted by j, there are n acitivities . performance measure denoted by z

An LP Model:

1 n jj j zcx max s.t. 1 1 n ijji j axbi,...,m z : value of overall performance measure x j : level of activity j (j=1,...,n) c j : performance measure coefficient for activity j b j : amount of resource i available (i=1,...,m) a ij : amount of resource i consumed by each unit of activity j

Decision Variables: x

j

Parameters: c

j ,a ij ,b j

Standard Form of the LP Model

A Linear programming problem can be expressed in the following standard form: max z= c 1 x 1 + c 2 x 2 + ...+ c n x n s.t. a 11 x 1 + a 12 x 2 + ...+ a 1n x n b 1 a 21
x 1 + a 22
x 2 + ...+ a 2n x n b 2 a m1 x 1+ a m2 x 2 + ...+ a mn x n b m x 1 0 x 2 0 x n 0

Objective functions: overall performance measure

c 1 x 1 + c 2 x 2 + ...+ c n x n

Constraints: a

i1 x 1 +a i2 x 2 + ...+ a in x n b i i=1,...,m (Functional constraints) x j

0 j=1,...,n (Nonnegativity constraints)

Metin Turkay

1 INDR 262 Optimization Models and Mathematical Programming

Variations in LP Model

An LP model can have the following variations:

1. Objective Function: minimization or maximization problem.

2. Direction of constraints

a i1 x 1 +a i2 x 2 + ...+ a in x n b i i=1,...,m less than or equal to a i1 x 1 +a i2 x 2 + ...+ a in x n b i i=1,...,m greater than or equal to a i1 x 1 +a i2 x 2 + ...+ a in x n = b i i=1,...,m equality

3. Non-negativity constraints

-x j

Terminology for solutions of the LP Model

Solution: any specification of values for the decision variables, x j, is called a solution Infeasible Solution: a solution for which at least one constraint is violated. Feasible Solution: a solution for which all of the constraints are satisfied Corner - Point Feasible (CPF) solution: a solution that lies at the corner of the feasible region. Optimal Solution: a feasible solution that has the most favorable value of the objective function. maximization ĺ largest z minimization ĺ smallest z Multiple Optimal Solutions: infinite number of solutions with the most favorable value of the objective function

The best CPF = optimal solution

Metin Turkay

2 INDR 262 Optimization Models and Mathematical Programming

Assumptions of Linear Programming

1. Proportionality:

- contribution of each activity to the objective function, z, is proportional to its level. c j x j - contribution of each activity to each functional constraint is proportional to its level. a i x j

2. Additivity:

- every function is the sum of the individual contribution of the respective activities. 1 n jj j zc x 1 1 n ijji j axbi,...,m

3. Divisibility:

- decision variables are allowed to have any real values that satisfy the functional and non-negativity constraints.

4.Certanity:

- the parameter values are assumed to be known constants.

Examples of LP

- Radiation Therapy Design - Regional Planning - Controlling Air Pollution - Reclaiming Solid Water - Personnel Scheduling - Distribution Network - Product Mix - Planning read pp.44-67

Metin Turkay

3 INDR 262 Optimization Models and Mathematical Programming

SOLVING LP PROBLEMS

Consider the Wyndor Glass Co. problem:

max z= 3x 1 + 5x 2 s.t. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1 0 x 2 0

2x2 = 12

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