Transcription of Lecture 6 Simplex method for linear programming
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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 linear and standard formFundamental theoremSimplex algorithmGeneral form of constraints of linear programmingIThe minimized function will always beminxw=cTx(ormax)wher
3x + 10y ≤ 300 x ≥ 0,y ≥ 0 I Example 3: (type II) minw = x 1 + 3x 2 + 4x 3 s.t. x 1 + 2x 2 + x 3 = 5 2x 1 + 3x 2 + x 3 = 6 x 2 ≥ 0,x 3 ≥ 0. Examples and standard form Fundamental theorem Simplex algorithm Standard form of constraints
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