Chapter4+LP+Formulation[1]

# Chapter4+LP+Formulation[1] - ITIS 1P97 Canbolat Fall 2009...

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ITIS 1P97– Canbolat Fall 2009 1 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: No interactions between any of the activities (no cross-product terms) The total usage of each resource and the resulting total measure of effectiveness equal the sum of the corresponding quantities generated by each activity. 3.

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Chapter4+LP+Formulation[1] - ITIS 1P97 Canbolat Fall 2009...

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