Linear Programming 1 Objectives Requirements for a linear

Linear Programming 1

Objectives – Requirements for a linear programming model. – Graphical representation of linear models. – Linear programming results: • • Unique optimal solution Alternate optimal solutions Unbounded models Infeasible models – Extreme point principle. 2

Objectives - continued – Sensitivity analysis concepts: • • • Reduced costs Range of optimality--LIGHTLY Shadow prices Range of feasibility--LIGHTLY Complementary slackness Added constraints / variables – Computer solution of linear programming models • WINQSB • EXCEL 3

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 4

• The Importance of Linear Programming – Many real static problems lend themselves to linear programming formulations. – Many real problems can be approximated by linear models. – The output generated by linear programs provides 5 useful “what’s best” and “what-if” information.

Assumptions of Linear Programming • The decision variables are continuous or divisible, meaning that 3. 333 eggs or 4. 266 airplanes is an acceptable solution • The parameters are known with certainty • The objective function and constraints exhibit constant returns to scale (i. e. , linearity) • There are no interactions between decision variables 6

Methodology of Linear Programming Determine and define the decision variables Formulate an objective function verbal characterization Mathematical characterization Formulate each constraint 7

THE GALAXY INDUSTRY PRODUCTION PROBLEM - A Prototype Example • Galaxy manufactures two toy models: – Space Ray. – Zapper. • Purpose: to maximize profits • How: By choice of product mix – How many Space Rays? – How many Zappers? 8

Galaxy Resource Allocation Resources are limited to – 1200 pounds of special plastic available per week – 40 hours of production time per week. • All LP Models have to be formulated in the context of a production period – In this case, a week 9

• 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 input – Space Rays require 2 pounds of plastic and 10 3 minutes of labor per dozen.

• 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 profit per dozen). Win. QSB report is at the end. • The current production plan consists of: Space Rays = 550 dozens Zapper = 100 dozens Profit = 4900 dollars per week 11

MODEL FORMULATION • Decisions variables: – X 1 = Production level of Space Rays (in dozens per week). – X 2 = Production level of Zappers (in dozens per week). • Objective Function: – Weekly profit, to be maximized 12

The Objective Function Each dozen Space Rays realizes $8 in profit. Total profit from Space Rays is 8 X 1. Each dozen Zappers realizes $5 in profit. Total profit from Zappers is 5 X 2. The total profit contributions of both is 8 X 1 + 5 X 2 (The profit contributions are additive because of the linearity assumption) 13

• 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. 14

The Linear Programming Model Max 8 X 1 + 5 X 2 (Weekly profit) subject to 2 X 1 + 1 X 2 < = 1200 (Plastic) 3 X 1 + 4 X 2 < = 2400 (Production Time) X 1 + X 2 < = 800 (Total production) X 1 - X 2 < = 450 (Mix) Xj> = 0, j = 1, 2 (Nonnegativity) 15

The Set of Feasible Solutions for Linear Programs The set of all points that satisfy all the constraints of the model is called a FEASIBLE REGION 16

Using a graphical presentation we can represent all the constraints, the objective function, and the three 17

X 2 1200 The plastic constraint: The Plastic constraint 2 X 1+X 2<=1200 Total production constraint: X 1+X 2<=800 Infeasible 600 Production Feasible Time 3 X 1+4 X 2<=2400 Production mix constraint: X 1 -X 2<=450 600 800 Interior points. Boundary points. Extreme points. X 1 18

We now demonstrate the search for an optimal solu Start at some arbitrary profit, say profit = $2, 000. . . Then increase the profit, if possible. . . X 2 1200 . . . and continue until it becomes infeasibleio 800 Profit =$5040 4, Profit = 3, $ 2, 000 e l b si ea f e h t l al c e R 600 X 1 400 600 800 19 g e R

1200 X 2 Let’s take a closer look at the optimal point 800 Infeasible 600 Feasible region 400 X 1 600 800 20

Summary of the optimal solution Space Rays = 480 dozens Zappers = 240 dozens Profit = $5040 – This 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). 21

• Extreme points and optimal solutions – If a linear programming problem has an optimal solution, it will occur at an extreme point. • Multiple optimal solutions – For 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 22

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. 23

Sensitivity Analysis of Objective Function Coefficients. • Range of Optimality – The optimal solution will remain unchanged as long as • An objective function coefficient lies within its range of optimality • There are no changes in any other input parameters. – The value of the objective function will change 24 if

The effects of changes in an objective function coef on the optimal solution 1200 X 2 800 ax Ma 4 x 1 x 3 +. 755 x x 12 + ax 1 8 x 600 M M + 2 5 x 5 x 2 Max 2 x 1 +5 x 2 X 1 400 600 800 25

The effects of changes in an objective function coef on the optimal solution + 2 5 x 5 x 2 2 1+ 2 2 5 x 5 x 1+ + 5 x 1 8 x 3. 7 + x 1 5 x 5 x ax M Ma x . 7. 7 5 Range of optimality 100 1 x 33 800 x 1 8 x Ma 600 Ma X 2 ax M 1200 400 600 800 X 1 26

• Multiple changes – The range of optimality is valid only when a single objective function coefficient changes. – When more than one variable changes we turn to the 100% rule. This is beyond the scope of this course 27

• Reduced costs The reduced cost for a variable at its lower bound (usually zero) yields: • The amount the profit coefficient must change before the variable can take on a value above its lower bound. • Complementary slackness At the optimal solution, either a variable is at its lower bound or the reduced cost is 0. 28

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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. 30

• 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? 31

X 2 1200 1+ 2 x The Plastic constraint 2 x 1+ 0 35 =1 0 20 Maximum profit = 5040 Production mix constraint 2< 1 x =1 2< 1 x 600 The new Plastic constraint Feasible Production time constraint Infeasible extreme points X 1 600 800 32

• 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 33

• Range of feasibility – The 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 34

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. 35

Models Without Optimal Solutions • Infeasibility: Occurs when a model has no feasible point. • Unboundedness: Occurs when the objective can become infinitely large. 36

Infeasibilit y No point, simultaneously, 1 lies both above line and below lines 2 and 3 . 2 3 1 37

Unbounded solution th Ma xim bj ize ec tiv Th e. F ef un re ea ct gi sib io o e. O n n le 38

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 of Vitamin A, Vitamin D, and Iron. 39

• Decision variables – X 1 (X 2) -- The number of two-ounce portions of Texfoods (Calration) product used in a serving. • The Model Minimize 0. 60 X 1 + 0. 50 X 2 Subject to 20 X 1 + 50 X 2 % Vitamin A provided per 2 oz. 25 X 1 + 25 X 2 50 X 1 + 10 X 2 X 1, X 2 0 Cost per 2 oz. 100 Vitamin A 100 Vitamin D % required 100 Iron 40

The Graphical solution 5 4 The Iron constraint Feasible Region Vitamin “D” constraint 2 Vitamin “A” constraint 2 4 5 41

• 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. 42

Computer Solution of Linear Programs With Any Number of Decision Variables • Linear 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). 43 – The type of each constraint: .

• 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 44

Variable and constraint name can be changed here WINQSB Input Data for the Galaxy Industries Problem Variables are restricted to >= 0 Click to solve No upper bound 45

Basis and non-basis variables • The basis variable values are free to take on values other than their lower bounds • The non-basis variables are fixed at their lower bounds (0) • THERE ALWAYS AS MANY BASIS VARIABLES AS THERE ARE CONSTRAINTS, ALWAYS 46

Another problem with 10 products • max 10 x 1 + 12 x 2 + 15 x 3 + 5 x 4 + 8 x 5 + 17 x 6 + 3 x 7 + 9 x 8 + 11 x 10 • s. t. • 2 x 1 + x 2 + 3 x 3 + x 4 + 2 x 5 + 3 x 6 + x 7 + 3 x 8 + 2 x 9 + x 10 <= 100 • all xi >= 0 • How many basis variables? • How many products should we be 47 making?

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