Direct Programming

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2. Goals. Prerequisites for a straight programming model.Graphical representation of direct models.Linear programming results:Unique ideal solutionAlternate ideal solutionsUnbounded modelsInfeasible modelsExtreme point standard.. 3. Destinations - proceeded. Affectability examination concepts:Reduced costsRange of optimality- - LIGHTLYShadow pricesRange of possibility - LIGHTLYComplementary slackne

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﻿Direct Programming

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Objectives Requirements for a straight programming model. Graphical portrayal of direct models. Direct programming outcomes: Unique ideal arrangement Alternate ideal arrangements Unbounded models Infeasible models Extreme point standard.

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Objectives - proceeded with Sensitivity investigation ideas: Reduced costs Range of optimality- - LIGHTLY Shadow costs Range of achievability - LIGHTLY Complementary slackness Added requirements/factors Computer arrangement of direct programming models WINQSB EXCEL LINDO

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3.1 Introduction to Linear Programming A Linear Programming model tries to boost or limit a straight capacity, subject to an arrangement of direct imperatives. The straight model comprises of the accompanying parts: An arrangement of choice factors. A goal work. An arrangement of limitations. Demonstrate FORMAT

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The Importance of Linear Programming Many genuine static issues loan themselves to direct programming definitions. Numerous genuine issues can be approximated by direct models. The yield created by direct projects gives useful "what's ideal" and "consider the possibility that" data.

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Assumptions of Linear Programming (p. 48) The choice factors are persistent or detachable, implying that 3.333 eggs or 4.266 planes is a worthy arrangement The parameters are known with assurance The target capacity and limitations show consistent comes back to scale (i.e., linearity) There are no collaborations between choice factors

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Methodology of Linear Programming Determine and characterize the choice factors Formulate a target work verbal portrayal Mathematical portrayal Formulate every imperative

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3.2 THE GALAXY INDUSTRY PRODUCTION PROBLEM - A Prototype Example Galaxy makes two toy models: Space Ray. Critic. Reason: to expand benefits How: By decision of item blend what number Space Rays? What number of Zappers? A RESOURCE ALLOCATION PROBLEM

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Galaxy Resource Allocation Resources are constrained to 1200 pounds of unique plastic accessible every week 40 hours of creation time every week. All LP Models must be figured with regards to a generation period For this situation, seven days

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Marketing prerequisite Total creation can't surpass 800 handfuls. Number of many Space Rays can't surpass number of many Zappers by more than 450. Innovative information Space Rays require 2 pounds of plastic and 3 minutes of work for every dozen. Critics require 1 pound of plastic and 4 minutes of work for every dozen.

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Current creation arrange calls for: Producing however much as could reasonably be expected of the more beneficial item, Space Ray (\$8 benefit per dozen). Utilize assets left over to deliver Zappers (\$5 benefit per dozen). The present generation arrange comprises of: Space Rays = 550 handfuls Zapper = 100 handfuls Profit = 4900 dollars for each week

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Management is looking for a creation calendar that will build the organization's benefit.

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MODEL FORMULATION Decisions factors : X1 = Production level of Space Rays (in handfuls every week). X2 = Production level of Zappers (in handfuls every week). Target Function: Weekly benefit, to be boosted

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The Objective Function Each dozen Space Rays acknowledges \$8 in benefit. Add up to benefit from Space Rays is 8X1. Every dozen Zappers acknowledges \$5 in benefit. Add up to benefit from Zappers is 5X2. The aggregate benefit commitments of both is 8X1 + 5X2 (The benefit commitments are added substance as a result of the linearity suspicion)

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we have a plastics asset imperative, a creation time requirement, and two promoting limitations. PLASTIC: every dozen units of Space Rays requires 2 lbs of plastic; every dozen units of Zapper requires 1 lb of plastic and inside any given week, our plastic provider can give 1200 lbs.

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The Linear Programming Model Max 8X1 + 5X2 (Weekly benefit) subject to 2X1 + 1X2 < = 1200 (Plastic) 3X1 + 4X2 < = 2400 (Production Time) X1 + X2 < = 800 (Total creation) X1 - X2 < = 450 (Mix) X j > = 0, j = 1,2 (Nonnegativity)

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3.4 The Set of Feasible Solutions for Linear Programs The arrangement of all focuses that fulfill every one of the imperatives of the model is known as a FEASIBLE REGION

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Using a graphical introduction we can speak to every one of the requirements, the goal work, and the three sorts of achievable focuses.

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The plastic limitation: 2X1+X2<=1200 The Plastic requirement Production blend imperative: X1-X2<=450 X2 1200 Total creation imperative: X1+X2<=800 Infeasible 600 Feasible Production Time 3X1+4X2<=2400 X1 600 800 Interior focuses. There are three sorts of attainable focuses Boundary focuses. Extraordinary focuses.

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3.5 Solving Graphically for a Optimal Solution

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Profit = \$ 000 Recall the achievable Region We now show the scan for an ideal arrangement Start at some self-assertive benefit, say benefit = \$2,000... At that point increment the benefit, if conceivable... X2 1200 ...and proceed until it gets to be distinctly infeasible Profit =\$5040 4, 2 , 3, 800 600 X1 400 600 800

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X2 1200 Let's investigate the ideal point 800 Infeasible 600 Feasible area Feasible locale X1 400 600 800

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Summary of the ideal arrangement Space Rays = 480 handfuls Zappers = 240 handfuls Profit = \$5040 This arrangement uses all the plastic and all the creation hours. Add up to creation is just 720 (not 800). Space Rays generation surpasses Zapper by just 240 handfuls (not 450).

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Extreme focuses and ideal arrangements If a straight programming issue has an ideal arrangement, it will happen at an extraordinary point. Numerous ideal answers For different ideal answers for exist, the target work must be parallel to an imperative that characterizes the limit of the practical district. Any weighted normal of ideal arrangements is additionally an ideal arrangement.

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3.6 The Role of Sensitivity Analysis of the Optimal Solution Is the ideal arrangement delicate to changes in info parameters? Conceivable explanations behind posing this question: Parameter values utilized were just best gauges. Dynamic condition may bring about changes. "Imagine a scenario where" examination may give temperate and operational data.

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3.7 Sensitivity Analysis of Objective Function Coefficients. Range of Optimality The ideal arrangement will stay unaltered the length of A target work coefficient exists in its scope of optimality There are no adjustments in some other info parameters. The estimation of the target capacity will change if the coefficient increases a variable whose esteem is nonzero.

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The impacts of changes in a target work coefficient on the ideal arrangement X2 1200 800 Max 8x1 + 5x2 600 Max 4x1 + 5x2 Max 3.75x1 + 5x2 Max 2x1 + 5x2 X1 400 600 800

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Range of optimality The impacts of changes in a target work coefficient on the ideal arrangement X2 1200 Max8x1 + 5x2 10 Max 10 x1 + 5x2 Max 3.75 x1 + 5x2 3.75 800 600 Max8x1 + 5x2 Max 3.75x1 + 5x2 X1 400 600 800

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Multiple progressions The scope of optimality is legitimate just when a solitary target work coefficient changes. At the point when more than one variable changes we swing to the 100% run the show. This is past the extent of this course

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Reduced costs The lessened cost for a variable at its lower bound (generally zero) yields: The sum the benefit coefficient must change before the variable can go up against an incentive over its lower bound. Integral slackness At the ideal arrangement, either a variable is at its lower bound or the lessened cost is 0.

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3.8 Sensitivity Analysis of Right-Hand Side Values Any adjustment in a correct hand side of a coupling limitation will change the ideal arrangement. Any adjustment in a right-hand side of a non-restricting requirement that is not as much as its slack or overflow, will bring about no adjustment in the ideal arrangement.

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In affectability examination of right-hand sides of imperatives we are keen on the accompanying inquiries: Keeping every single other variable the same, what amount would the ideal estimation of the goal work (for instance, the benefit) change if the right-hand side of a requirement changed by one unit? For what number of extra UNITS is this per unit change substantial? For what number of less UNITS is this per unit change legitimate?

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The Plastic requirement The new Plastic limitation Maximum benefit = 5040 Production blend imperative Production time imperative X2 1200 2x1 + 1x2 <=1200 2x1 + 1x2 <=1350 600 Feasible Infeasible extraordinary focuses X1 600 800

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Skip this detail Correct Interpretation of shadow costs Sunk costs: The shadow cost is the estimation of an additional unit of the asset, since the cost of the asset is excluded in the figuring of the target work coefficient. Included costs: The shadow cost is the exceptional incentive over the current unit esteem for the asset, since the cost of the asset is incorporated into the computation of the target work coefficient.

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Range of practicality The arrangement of right - hand side qualities for which a similar arrangement of imperatives decides the ideal extraordinary point. The range over-which similar factors stay in arrangement (which is another method for saying that a similar outrageous point is the ideal extraordinary point) Within the scope of attainability, shadow costs stay consistent; nonetheless, the ideal target work esteem and choice variable qualities will change if the comparing imperative is official

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3.9 Other Post Optimality Changes SKIP THIS Addition of a limitation. Erasure of a requirement. Expansion of