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Short Tail Keywords linear programming constraint |
long Tail Keywords (2 words) linear programming c 0 feasible region flair furniture objective function |
long Tail Keywords (3 words) painting and varnishing 0 20 40 20 40 60 40 60 80 80 100 t c 0 20 40 20 c |
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slideshare scribd company logo rezaul karim chapter linear programming modelsformulation graphicalpresentation introductionmany management decisions involve trying makethe most effective use limited resources machinery labor money time warehouse space rawmaterialslinear programminglinear lplp widely usedmathematical modeling technique designed helpmanagers planning decision making relative toresource allocationbelongs broader field mathematicalmathematicalprogrammingprogrammingin sense programmingprogramming refers andsolving problem mathematically requirements linearprogramming has been applied many areas over thepast years all problems have properties common seek maximizemaximize minimizeminimize somequantity objective functionobjective function presence restrictions constraintsconstraints thatlimit degree can pursue ourobjective there must alternative courses action tochoose from constraints beexpressed terms linearlinear equations orinequalitiesinequalities examples successfullp applicationsdevelopment production schedule will satisfy future demands firms minimizingminimizing total inventory costsdetermination grades petroleum products yield themaximummaximum profitselection different blends raw materials feed millsto produce finished combinations minimumminimum costdetermination distribution system minimizeminimizetotal shipping cost several warehouses variousmarket locations assumptionsproperties programs one more are linearassumptions certainty proportionality additivity divisibility nonnegative variablestable basic assumptions assume conditions certaintycertainty exist numbers inthe known anddo not change during period being studied proportionalityproportionality exists andconstraints constancy between increases resourceutilization unit needs hours then require additivityadditivity activities equalsthe sum individual divisibilitydivisibility solutions need wholenumbers answers variables nonnegativenonnegative aredealing real physical quantities formulating program involves developinga mathematical model represent themanagerial steps completely understand managerialproblem faced identify define writemathematical expressions objectivefunction applications theproduct mix problemproduct two produced using limitedresources such personnel machines rawmaterials profit firm seeks maximize basedon contribution per product would like determine how manyunits should tomaximize overall given its flair furniture produces inexpensivetables chairs processes similar both certainamount carpentry work paintingand varnishing department table takes ofpainting chair requires hour available painting yields aprofit wants bestcombination tables reachthe maximum profithours required toproduce unitdepartment tablescchairsavailable hoursthis weekcarpentry used cannotexceed week usedcannot exceed representing actualdecisions make aret number weekc create andcmaximize develop relationships twoconstraints tablenumber chairnumber know thatcarpentry similarlypainting timethis means producedrequires andvarnishing these restrict capacityand affect values nonnegativet greater than orequal thanor equal complete stated mathematicallymaximize csubject constraint constraintt nonnegativity graphical solution easiest way solve small problemsis approach method only works thereare just amore complex needed notpossible plot twodimensional graph provides valuableinsight into other approaches representation tnumberofchairsnumber tablesthis axis represents theconstraint figure first step solving toidentify set region feasible equationon start graphing equality portion ofthe intercepts draw theline carpentryconstraint similarly line shown following tablest equation tablesfigure any point below plotwill violate restriction above therestriction lies exactlysatisfies andsatisfies anddoes varnishingconstraint departmentsmust find satisfies bothconstraints simultaneouslysimultaneously new shows plots regionfeasible area solutionsarea iswhere satisfied inside feasiblefeasible outside infeasibleinfeasiblesolution furniturepaintingvarnishing constraintcarpentry constraintfeasibleregion usedpaintingconstraint isoprofit once graphed tofind optimal possible speediest linemethod starting but value graphthe move direction ofincreasing maintaining slope last touches theoptimal choose obviously best further graphs created larger profits origin maintainingthe staying within boundaries thefeasible highest generated theisoprofit passes through four lines cisoprofit theflair problemoptimal pointt second problemsemploys corner methodcorner looking every cornerpoint theory behind lie cornercornerpointspoints extreme pointextreme afoursided polygon pointslabeled graphcorner points because returns coordinates accurately wehave intersection constraintlines details slide simultaneous methodsimultaneous multiplythe add carpentryequation linec substituting either original equationsallows summary methodsisoprofit select specific increasing decreasingcost isthe compute profitor costcorner reason found solutiontable minimization minimizing anobjective instead maximizing solvedgraphically setting feasiblesolution thecorner isocost lineapproach analogous isoprofitapproach maximization findthe minimum holiday meal turkey ranch considering buyingtwo brands blending themto provide good lowcost diet turkeysminimize cents subject ounces ingredient constraintx constraintholiday ranchx pounds brand purchasedx purchasedlet ranchingredientcomposition poundof ozminimum monthlyrequirement perturkey ozbrand feeda pound datatable constructthe solutionregion solutionwill thecorners wouldin maximizationproblemholiday poundsofbrand constraintingredient constraintfeasible regionabcfigure three cand algebra back findcost lowest purchase ofbrand atotal isocostapproach choosing initialcost itis clearimprovement ispossibleholiday linedirection decreasing special cases difficulties arise attimes tosolving infeasibility unboundedness redundancy alternate problemthat occurrence world generally relaxeduntil satisfying constraintsregion constraintsfigure satisfyingthird sometimes finitesolution solutionvariables made infinitelylarge without violating beopen ended usually beenformulated improperly unbounded right regionx redundant does affectthe may binding very realworld causes particular eliminatingredundant simplifies aredundantconstraint redundantconstraintfeasibleregionx occasionally mayexist graphically occurs objectivefunctions runs perfectlyparallel actually allows great flexibilityin deciding combination theprofit same example ofalternate optimalsolutions feasibleregionisoprofit consists allcombinations along theab segmentisoprofit overlays segment abbamaximize subj sensitivity analysis thus far beenfound under called deterministicdeterministicassumptionsassumptions thedata dynamic andchanging analyze sensitivesensitive deterministic solutionis changes analysissensitivity postoptimalitypostoptimalityanalysisanalysis parametric programmingparametric optimalityoptimalityanalysisanalysis often series whatifquestions concerning variablecoefficients less advertising trialanderror wherevalues changed entire resolved preferred analytic postoptimalityanalysis after solved rangeof parameters solutionwithout resolving deal onlywith errors estimating input thelp also managementsexperiments thefirm simplex feasibility study proposals service design tqm presentation jollibee integer transportation formulation queuing duality marketing aspects technical location process selection facility layout operations research game operation rsh qam liner science senstivity market projections manual sustainability supply chain cellular goal slides bba methodlppqarender pptx lpp ppt ppts pomppt class quantitativetechniqueformanagerialdecisionlinearprogramming phpa revisedpptx linearprogramingproblemlpp operational truth about dinesh bafnas situationpdf understanding uae labour law key employers employees fexle salesforce lightning 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