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Lecture 6 Simplex method for linear programming

Simplex method for linear programmingslides credit: Weinan EExamples and standard formFundamental theoremSimplex algorithmOutlineExamples and standard formFundamental theoremSimplex algorithmExamples and standard formFundamental theoremSimplex algorithmExample: Transportation problemSchematics of transportation problemaaabbbb12m12n 1nCijDestinationOriginExamples and standard formFundamental theoremSimplex algorithmExample: Transportation problemIFormulation:mins=m i=1n j=1cijxijsubject to theconstraintm i=1xij bj, j= 1, .. , nn j=1xij ai, i= 1, .. , mxij 0, i= 1, .. , m;j= 1, .. , the supply of thei-th origin,bjis the demand of thej-thdestinations,xijis the amount of the shipment from sourceitodestinationjandcijis the unit transportation cost problem ( Simplex method )Examples and standard formFundamental theoremSimplex algorithmLinear programmingIDefinition:If the minimized (or maximized) function and the constraints are all inlinear forma1x1+a2x2+ +anxn+ type of optimization is called li

I Linear programming maxw = 10x 1 + 11x 2 3x 1 + 4x 2 ≤ 17 2x 1 + 5x 2 ≤ 16 x i ≥ 0, i = 1,2 I The set of all the feasible solutions are called feasible region. feasible region I 5 3 Thisfeasible region is a colorredconvex polyhedron spanned bypoints …

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