JW LP Formulation and Extra Examples

JW LP Formulation and Extra Examples - ITIS 1P97 Walker...

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Fall 2010 CHAPTER 4: LINEAR PROGRAMMING MODELS A Standard Form of LP: Maximize Z = c 1 x 1 + c 2 x 2 + . . . . . + c n x n subject to the constraints 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 where c j = contribution into profit from one unit of product j a ij = amount of resource type i used to make one unit of product j b i = amount of resource type i available Some Terminology in LP: Objective function - maximization/minimization Inequality (equality) constraints Nonnegativity constraints Decision variables Parameters LP formulation - translation (or transformation) of a real world problem into a linear programming model Feasible point (region) Graphical method Optimal point (region) The simplex method Corner (or extreme) points Slack - amount of unused resources Surplus - amount exceed minimum requirement ASSUMPTIONS (LIMITATIONS) OF LINEAR PROGRAMMING 1. Proportionality: Individual activities are considered independently of the others. These quantities are directly proportional to the level of each activity. i) the measure of effectiveness : z = c k x k in O.F. ii) the usage of each resource : i = a ik x k in constraints 2. Additivity: The total usage of each resource and the resulting total measure of effectiveness equal the sum of the corresponding quantities generated by each activity. No interactions between any of the activities (no cross-product terms) 3. Divisibility: Activity units can be divided into any fractional levels (noninteger values for decision variables are permissible). 4. Certainty: All parameter values (a ij , b i , and c j ) are known constant. Use of ‘sensitivity analysis’
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JW LP Formulation and Extra Examples - ITIS 1P97 Walker...

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