2 Objectives Requirements for a linear programming model. Graphical representation of linear models.Linear programming results:Unique optimal solutionAlternate optimal solutionsUnbounded modelsInfeasible modelsExtreme point principle.
3 Objectives - continued Sensitivity analysis concepts:Reduced costsRange of optimality--LIGHTLYShadow pricesRange of feasibility--LIGHTLYComplementary slacknessAdded constraints / variablesComputer solution of linear programming modelsWINQSBEXCELLINDO
4 Introduction to Linear Programming A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints.The linear model consists of the following components:A set of decision variables.An objective function.A set of constraints.SHOW FORMAT
5 The Importance of Linear Programming Many real static problems lend themselves to linearprogramming formulations.Many real problems can be approximated by linear models.The output generated by linear programs providesuseful “what’s best” and “what-if” information.
6 Assumptions of Linear Programming The decision variables are continuous or divisible, meaning that eggs or airplanes is an acceptable solutionThe parameters are known with certaintyThe objective function and constraints exhibit constant returns to scale (i.e., linearity)There are no interactions between decision variables
7 Methodology of Linear Programming Determine and define the decision variablesFormulate an objective functionverbal characterizationMathematical characterizationFormulate each constraint
8 THE GALAXY INDUSTRY PRODUCTION PROBLEM - A Prototype Example Galaxy manufactures two toy models:Space Ray.Zapper.Purpose: to maximize profitsHow: By choice of product mixHow many Space Rays?How many Zappers?A RESOURCE ALLOCATION PROBLEM
9 Galaxy Resource Allocation Resources are limited to1200 pounds of special plastic available per week40 hours of production time per week.All LP Models have to be formulated in the context of a production periodIn this case, a week
10 Marketing requirement Total production cannot exceed 800 dozens.Number of dozens of Space Rays cannot exceed number of dozens of Zappers by more than 450.Technological inputSpace Rays require 2 pounds of plastic and3 minutes of labor per dozen.Zappers require 1 pound of plastic and4 minutes of labor per dozen.
11 Current production plan calls for: Producing as much as possible of the more profitable product, Space Ray ($8 profit per dozen).Use resources left over to produce Zappers ($5 profitper dozen).WinQSB report is at the end.The current production plan consists of:Space Rays = 550 dozensZapper = 100 dozensProfit = 4900 dollars per week
12 MODEL FORMULATION Decisions variables: Objective Function: X1 = Production level of Space Rays (in dozens per week).X2 = Production level of Zappers (in dozens per week).Objective Function:Weekly profit, to be maximized
13 The Objective Function Each dozen Space Rays realizes $8 in profit.Total profit from Space Rays is 8X1.Each dozen Zappers realizes $5 in profit.Total profit from Zappers is 5X2.The total profit contributions of both is8X1 + 5X2(The profit contributions are additive because of the linearity assumption)
14 we have a plastics resource constraint, a production time constraint, and two marketing constraints. PLASTIC: each dozen units of Space Rays requires 2 lbs of plastic; each dozen units of Zapper requires 1 lb of plastic and within any given week, our plastic supplier can provide 1200 lbs.
15 The Linear Programming Model Max 8X1 + 5X2 (Weekly profit)subject to2X1 + 1X2 < = (Plastic)3X1 + 4X2 < = (Production Time)X1 + X2 < = (Total production)X X2 < = (Mix)Xj> = 0, j = 1,2 (Nonnegativity)
16 The Set of Feasible Solutions for Linear Programs The set of all points that satisfy all the constraints of the model is calledaFEASIBLE REGION
17 Using a graphical presentation we can represent all the constraints,the objective function, and the threetypes of feasible points.
19 We now demonstrate the search for an optimal solution Start at some arbitrary profit, say profit = $2,000...Then increase the profit, if possible...X21200...and continue until it becomes infeasibleProfit =$5040Profit = $2,3,4,800Recall the feasible Region600X1400600800
20 Let’s take a closer look at the optimal point X21200Let’s take a closer look atthe optimal point800Infeasible600FeasibleregionFeasibleregionX1400600800
21 Summary of the optimal solution Space Rays = 480 dozensZappers = 240 dozensProfit = $5040This solution utilizes all the plastic and all the production hours.Total production is only 720 (not 800).Space Rays production exceeds Zapper by only 240 dozens (not 450).
22 Extreme points and optimal solutions If a linear programming problem has an optimal solution, it will occur at an extreme point.Multiple optimal solutionsFor multiple optimal solutions to exist, the objective function must be parallel to a constraint that defines the boundary of the feasible region.Any weighted average of optimal solutions is also an optimal solution.
23 The Role of Sensitivity Analysis of the Optimal Solution Is the optimal solution sensitive to changes in input parameters?Possible reasons for asking this question:Parameter values used were only best estimates.Dynamic environment may cause changes.“What-if” analysis may provide economical and operational information.
24 Sensitivity Analysis of Objective Function Coefficients. Range of OptimalityThe optimal solution will remain unchanged as long asAn objective function coefficient lies within its range of optimalityThere are no changes in any other input parameters.The value of the objective function will change if the coefficient multiplies a variable whose value is nonzero.
25 The effects of changes in an objective function coefficient on the optimal solution X21200800600Max 8x1 + 5x2Max 4x1 + 5x2Max 3.75x1 + 5x2Max 2x1 + 5x2X1400600800
26 The effects of changes in an objective function coefficient on the optimal solution X21200Range of optimalityMax8x1 + 5x210Max 10 x1 + 5x2Max 3.75 x1 + 5x23.75800600Max8x1 + 5x2Max 3.75x1 + 5x2X1400600800
27 Multiple changesThe range of optimality is valid only when a single objective function coefficient changes.When more than one variable changes we turn to the100% rule.This is beyond the scope of this course
28 Complementary slackness Reduced costsThe reduced cost for a variable at its lower bound (usually zero) yields:The amount the profit coefficient must change beforethe variable can take on a value above its lower bound.Complementary slacknessAt the optimal solution, either a variable is at its lower bound or the reduced cost is 0.
30 Sensitivity Analysis of Right-Hand Side Values Any change in a right hand side of a binding constraint will change the optimal solution.Small change in a right-hand side of a non-binding constraint that is less than its slack or surplus, will cause no change in the optimal solution.
31 In sensitivity analysis of right-hand sides of constraints we are interested in the following questions:Keeping all other factors the same, how much would the optimal value of the objective function (for example, the profit) change if the right-hand side of a constraint changed by one unit?For how many additional UNITS is this per unit change valid?For how many fewer UNITS is this per unit change valid?
32 Feasible The Plastic constraint The new Plastic constraint X21200The Plastic constraintThe new Plastic constraint2x1 + 1x2 <=1200Maximum profit = 50402x1 + 1x2 <=1350600Production mix constraintFeasibleProduction timeconstraintInfeasible extreme pointsX1600800
33 Correct Interpretation of shadow prices Sunk costs: The shadow price is the value of an extra unit of the resource, since the cost of the resource is not included in the calculation of the objective function coefficient.Included costs: The shadow price is the premium value above the existing unit value for the resource, since the cost of the resource is included in the calculation of the objective function coefficient.
34 Range of feasibilityThe set of right - hand side values for which the same set of constraints determines the optimal extreme point.The range over-which the same variables remain in solution (which is another way of saying that the same extreme point is the optimal extreme point)Within the range of feasibility, shadow prices remain constant; however, the optimal objective function value and decision variable values will change if the corresponding constraint is binding
35 Other Post Optimality Changes Addition of a constraint.Deletion of a constraint.Addition of a variable.Deletion of a variable.Changes in the left - hand side technology coefficients.
36 Models Without Optimal Solutions Infeasibility: Occurs when a model has no feasible point.Unboundedness: Occurs when the objective can become infinitely large.
37 Infeasibility No point, simultaneously, lies both above line and below lines and123213
38 Unbounded solutionMaximizethe Objective FunctionThe feasible region
39 Navy Sea Ration A cost minimization diet problem Mix two sea ration products: Texfoods, Calration.Minimize the total cost of the mix.Meet the minimum requirements ofVitamin A, Vitamin D, and Iron.
40 Decision variables The Model X1 (X2) -- The number of two-ounce portions of Texfoods (Calration) product used in a serving.The ModelMinimize 0.60X X2Subject to20X X Vitamin A25X X Vitamin D50X X IronX1, XCost per 2 oz.% Vitamin Aprovided per 2 oz.% required
41 The Graphical solution 5The Iron constraintFeasible Region4Vitamin “D” constraint2Vitamin “A” constraint245
42 Summary of the optimal solution Texfood product = 1.5 portions (= 3 ounces)Calration product = 2.5 portions (= 5 ounces)Cost =$ 2.15 per serving.The minimum requirements for Vitamin D and iron are met with no surplus.The mixture provides 155% of the requirement for Vitamin A.
43 Computer Solution of Linear Computer Solution of Linear Programs With Any Number of Decision VariablesLinear programming software packages solve large linear models.Most of the software packages use the algebraic technique called the Simplex algorithm.The input to any package includes:The objective function criterion (Max or Min).The type of each constraint:The actual coefficients for the problem.
44 The typical output generated from linear programming software includes: Optimal value of the objective function.Optimal values of the decision variables.Reduced cost for each objective function coefficient.Ranges of optimality for objective function coefficients.The amount of slack or surplus in each constraint.Shadow (or dual) prices for the constraints.Ranges of feasibility for right-hand side values.
45 WINQSB Input Data for the Galaxy Industries Problem Variable andconstraint name can bechanged hereWINQSB Input Data for the Galaxy Industries ProblemClick to solveVariables arerestricted to >= 0No upper bound
46 Basis and non-basis variables The basis variable values are free to take on values other than their lower boundsThe non-basis variables are fixed at their lower bounds (0)THERE ARE ALWAYS AS MANY BASIS VARIABLES AS THERE ARE CONSTRAINTS, ALWAYS
47 Another problem with10 products max 10x x x3 + 5 x4 + 8 x5 + 17x6 + 3 x7 + 9x8 + 11x10s.t.2x1 + x2 + 3x3 + x4 + 2x5 + 3x6 + x7 + 3x8 + 2x9 + x10 <= 100all xi >= 0How many basis variables?How many products should we be making?