LINEAR PROGRAMMING: THE GRAPHICAL METHOD

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LINEAR PROGRAMMING: THE LINEAR PROGRAMMING: THE GRAPHICAL METHODGRAPHICAL METHOD

Linear Programming ProblemLinear Programming Problem Properties of LPsProperties of LPs LP SolutionsLP Solutions Graphical SolutionGraphical Solution Introduction to Sensitivity AnalysisIntroduction to Sensitivity Analysis

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Linear Programming (LP) ProblemLinear Programming (LP) Problem

A A mathematical programming problemmathematical programming problem is one is one that seeks to maximize or minimize an that seeks to maximize or minimize an objective function subject to constraints. objective function subject to constraints.

If both the objective function and the If both the objective function and the constraints are linear, the problem is referred to constraints are linear, the problem is referred to as a as a linear programming problemlinear programming problem..

Linear functionsLinear functions are functions in which each are functions in which each variable appears in a separate term raised to variable appears in a separate term raised to the first power and is multiplied by a constant the first power and is multiplied by a constant (which could be 0).(which could be 0).

Linear constraintsLinear constraints are linear functions that are are linear functions that are restricted to be "less than or equal to", "equal restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant.to", or "greater than or equal to" a constant.

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Building Linear Programming ModelsBuilding Linear Programming Models

1.1. What are you trying to decide - Identify the What are you trying to decide - Identify the decision variable to solve the problem and define decision variable to solve the problem and define appropriate variables that represent them. For appropriate variables that represent them. For instance, in a simple maximization problem, RMC, Inc. instance, in a simple maximization problem, RMC, Inc. interested in producing two products: fuel additive interested in producing two products: fuel additive and a solvent base. The decision variables will be X1 and a solvent base. The decision variables will be X1 = tons of fuel additive to produce, and X2 = tons of = tons of fuel additive to produce, and X2 = tons of solvent base to produce. solvent base to produce.

2.2. What is the objective to be maximized or What is the objective to be maximized or minimized? Determine the objective and express it as minimized? Determine the objective and express it as a linear function. When building a linear programming a linear function. When building a linear programming model, only relevant costs should be included, sunk model, only relevant costs should be included, sunk costs are not included. In our example, the objective costs are not included. In our example, the objective function is: function is: z = 40X1 + 30X2; z = 40X1 + 30X2; where 40 and 30 are the objective function where 40 and 30 are the objective function coefficients. coefficients.

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3. What limitations or requirements restrict the 3. What limitations or requirements restrict the values of the decision variables? Identify and write values of the decision variables? Identify and write the constraints as linear functions of the decision the constraints as linear functions of the decision variables. Constraints generally fall into one of the variables. Constraints generally fall into one of the following categories:following categories:• a.a. Limitations Limitations - The amount of material used - The amount of material used

in the production process cannot exceed the in the production process cannot exceed the amount available in inventory. In our example, the amount available in inventory. In our example, the limitations are:limitations are:

• Material 1 = 20 tonsMaterial 1 = 20 tons• Material 2 = 5 tonsMaterial 2 = 5 tons• Material 3 = 21 tons available.Material 3 = 21 tons available.• The material used in the production of X1 and X2 The material used in the production of X1 and X2

are also known. are also known.

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To produce one ton of fuel additive uses .4 ton of To produce one ton of fuel additive uses .4 ton of material 1, and .60 ton of material 3. To produce material 1, and .60 ton of material 3. To produce one ton of solvent base it takes .50 ton of material 1, one ton of solvent base it takes .50 ton of material 1, .20 ton of material 2, and .30 ton of material 3. .20 ton of material 2, and .30 ton of material 3. Therefore, we can set the constraints as follows:Therefore, we can set the constraints as follows:

.4X1 + .50 X2 .4X1 + .50 X2 <= 20<= 20 .20X2 <= 5.20X2 <= 5

.6X1 + .3X2 .6X1 + .3X2 <=21, where .4, .50, .20, .6, and .3 are called <=21, where .4, .50, .20, .6, and .3 are called constraint coefficients. The limitations (20, 5, and constraint coefficients. The limitations (20, 5, and 21) are called Right Hand Side (RHS).21) are called Right Hand Side (RHS).

b.b. Requirements - specifying a minimum Requirements - specifying a minimum levels of performance. For instance, levels of performance. For instance, production must be sufficient to satisfy production must be sufficient to satisfy customers’ demand. customers’ demand.

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Properties of LPsProperties of LPs

ProportionalityProportionality

The profit contribution and the amount The profit contribution and the amount of the resources used by a decision variable is of the resources used by a decision variable is directly proportional to its value.directly proportional to its value.

Additivity Additivity

The value of the objective function and The value of the objective function and the amount of the resources used can be the amount of the resources used can be calculated by summing the individual calculated by summing the individual contributions of the decision variables.contributions of the decision variables.

DivisibilityDivisibility

Fractional values of the decision Fractional values of the decision variables are permitted.variables are permitted.

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LP SolutionsLP Solutions

The The maximizationmaximization or or minimizationminimization of some quantity of some quantity is the is the objectiveobjective in all linear programming problems. in all linear programming problems.

A A feasible solutionfeasible solution satisfies all the problem's satisfies all the problem's constraints.constraints.

Changes to the objective function coefficients do Changes to the objective function coefficients do not affect the feasibility of the problem.not affect the feasibility of the problem.

An An optimal solutionoptimal solution is a feasible solution that results is a feasible solution that results in the largest possible objective function value, in the largest possible objective function value, zz, , when maximizing or smallest when maximizing or smallest zz when minimizing. when minimizing.

In the graphical method, if the objective function In the graphical method, if the objective function line is parallel to a boundary constraint in the line is parallel to a boundary constraint in the direction of optimization, there are direction of optimization, there are alternate alternate optimal solutionsoptimal solutions, with all points on this line , with all points on this line segment being optimal.segment being optimal.

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A graphical solution method can be used to A graphical solution method can be used to solve a linear program with two variables.solve a linear program with two variables.

If a linear program possesses an optimal If a linear program possesses an optimal solution, then an solution, then an extreme pointextreme point will be optimal. will be optimal.

If a constraint can be removed without If a constraint can be removed without affecting the shape of the feasible region, the affecting the shape of the feasible region, the constraint is said to be constraint is said to be redundantredundant. .

A A nonbinding constraintnonbinding constraint is one in which there is is one in which there is positive slack or surplus when evaluated at the positive slack or surplus when evaluated at the optimal solution.optimal solution.

A linear program which is overconstrained so A linear program which is overconstrained so that no point satisfies all the constraints is said that no point satisfies all the constraints is said to be to be infeasibleinfeasible..

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A A feasible regionfeasible region may be unbounded and yet there may be unbounded and yet there may be optimal solutions. This is common in may be optimal solutions. This is common in minimization problems and is possible in minimization problems and is possible in maximization problems.maximization problems.

The feasible region for a two-variable linear The feasible region for a two-variable linear programming problem can be nonexistent, a single programming problem can be nonexistent, a single point, a line, a polygon, or an unbounded area.point, a line, a polygon, or an unbounded area.

Any linear program falls in one of three categories:Any linear program falls in one of three categories:• is infeasible is infeasible • has a unique optimal solution or alternate optimal has a unique optimal solution or alternate optimal

solutionssolutions• has an objective function that can be increased has an objective function that can be increased

without boundwithout bound

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Slack and Surplus VariablesSlack and Surplus Variables

A linear program in which all the variables are A linear program in which all the variables are non-negative and all the constraints are non-negative and all the constraints are equalities is said to be in equalities is said to be in standard formstandard form. .

Standard form is attained by adding Standard form is attained by adding slack slack variables variables to "less than or equal to" constraints, to "less than or equal to" constraints, and by subtracting and by subtracting surplus variablessurplus variables from from "greater than or equal to" constraints. "greater than or equal to" constraints.

Slack and surplus variables represent the Slack and surplus variables represent the difference between the left and right sides of difference between the left and right sides of the constraints.the constraints.

Slack and surplus variables have objective Slack and surplus variables have objective function coefficients equal to 0. function coefficients equal to 0.

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Example: Graphical SolutionExample: Graphical Solution

Solve graphically for the optimal solution:Solve graphically for the optimal solution:

Min Min zz = 5 = 5xx11 + 2 + 2xx22

s.t. 2s.t. 2xx11 + 5 + 5xx22 >> 10 10

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

xx11, , xx22 >> 0 0

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Graph the ConstraintsGraph the Constraints

Constraint 1Constraint 1: When : When xx11 = 0, then = 0, then xx22 = 2; when = 2; when xx22 = = 0, 0, then then xx11 = 5. Connect (5,0) and (0,2). The ">" = 5. Connect (5,0) and (0,2). The ">" side is side is above this line.above this line.

Constraint 2Constraint 2: When : When xx22 = 0, then = 0, then xx11 = 3. But setting = 3. But setting xx11 to to 0 will yield 0 will yield xx22 = -12, which is not on the graph. = -12, which is not on the graph.

Thus, to get a second point on this line, set Thus, to get a second point on this line, set xx11 to any to any number larger than 3 and solve for number larger than 3 and solve for xx22: : when when xx11 = 5, = 5, then then xx22 = 8. Connect (3,0) and = 8. Connect (3,0) and (5,8). The ">" side is (5,8). The ">" side is to the right.to the right.

Constraint 3Constraint 3: When : When xx11 = 0, then = 0, then xx22 = 4; when = 4; when xx22 = = 0, 0, then then xx11 = 4. Connect (4,0) and (0,4). The ">" = 4. Connect (4,0) and (0,4). The ">" side is side is above this line.above this line.

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Constraints GraphedConstraints Graphed

55

44

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22

11

55

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1 2 3 4 5 61 2 3 4 5 6 1 2 3 4 5 61 2 3 4 5 6

xx22xx22

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

22xx11 + 5 + 5xx22 >> 10 1022xx11 + 5 + 5xx22 >> 10 10

xx11xx11

Feasible RegionFeasible Region

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Graph the Objective FunctionGraph the Objective Function

Set the objective function equal to an arbitrary Set the objective function equal to an arbitrary constant (say 20) and graph it. For 5constant (say 20) and graph it. For 5xx11 + 2 + 2xx22 = 20, = 20, when when xx11 = 0, then = 0, then xx22 = 10; when = 10; when xx22= 0, then = 0, then xx11 = 4. = 4. Connect (4,0) and (0,10).Connect (4,0) and (0,10).

Move the Objective Function Line Toward OptimalityMove the Objective Function Line Toward Optimality

Move it in the direction which lowers its value Move it in the direction which lowers its value (down), since we are minimizing, until it touches the (down), since we are minimizing, until it touches the last point of the feasible region, determined by the last point of the feasible region, determined by the last two constraints. This is called the Iso-Value Line last two constraints. This is called the Iso-Value Line Method.Method.

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Objective Function GraphedObjective Function Graphed

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5

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1 2 3 4 5 6 1 2 3 4 5 6

xx22xx22Min Min zz = 5 = 5xx11 + 2 + 2xx22

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

Min Min zz = 5 = 5xx11 + 2 + 2xx22

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

22xx11 + 5 + 5xx22 >> 10 1022xx11 + 5 + 5xx22 >> 10 10

xx11xx11

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Solve for the Extreme Point at the Intersection of the Solve for the Extreme Point at the Intersection of the Two Binding ConstraintsTwo Binding Constraints

44xx11 - - xx22 = 12 = 12

xx11+ + xx22 = 4 = 4

Adding these two equations gives: Adding these two equations gives:

55xx11 = 16 or = 16 or xx11 = 16/5. = 16/5.

Substituting this into Substituting this into xx11 + + xx22 = 4 gives: = 4 gives: xx22 = 4/5 = 4/5 Solve for the Optimal Value of the Objective FunctionSolve for the Optimal Value of the Objective Function

Solve for Solve for zz = 5 = 5xx11 + 2 + 2xx22 = 5(16/5) + 2(4/5) = 88/5. = 5(16/5) + 2(4/5) = 88/5.

Thus the optimal solution is Thus the optimal solution is

xx11 = 16/5; = 16/5; xx22 = 4/5; = 4/5; z z = 88/5= 88/5

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1 2 3 4 5 6 1 2 3 4 5 6

xx22xx22Min Min zz = 5 = 5xx11 + 2 + 2xx22

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

Min Min zz = 5 = 5xx11 + 2 + 2xx22

44xx11 - - xx22 >> 12 12

xx11 + + xx22 >> 4 4

22xx11 + 5 + 5xx22 >> 10 10

Optimal: Optimal: xx11 = 16/5 = 16/5

xx22 = 4/5 = 4/5

22xx11 + 5 + 5xx22 >> 10 10

Optimal: Optimal: xx11 = 16/5 = 16/5

xx22 = 4/5 = 4/5xx11xx11

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Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis is used to determine effects is used to determine effects on the optimal solution within specified ranges on the optimal solution within specified ranges for the objective function coefficients, constraint for the objective function coefficients, constraint coefficients, and right hand side values. coefficients, and right hand side values.

Sensitivity analysis provides answers to certain Sensitivity analysis provides answers to certain what-if questions.what-if questions.

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Range of OptimalityRange of Optimality

A A range of optimalityrange of optimality of an objective function of an objective function coefficient is found by determining an interval for the coefficient is found by determining an interval for the objective function coefficient in which the original objective function coefficient in which the original optimal solution remains optimal while keeping all optimal solution remains optimal while keeping all other data of the problem constant. The value of the other data of the problem constant. The value of the objective function may change in this range.objective function may change in this range.

Graphically, the limits of a range of optimality are Graphically, the limits of a range of optimality are found by changing the slope of the objective function found by changing the slope of the objective function line within the limits of the slopes of the binding line within the limits of the slopes of the binding constraint lines. (This would also apply to constraint lines. (This would also apply to simultaneous changes in the objective coefficients.)simultaneous changes in the objective coefficients.)

The slope of an objective function line, Max The slope of an objective function line, Max cc11xx11 + + cc22xx22, is -, is -cc11//cc22, and the slope of a constraint, , and the slope of a constraint, aa11xx11 + + aa22xx22 = = bb, is -, is -aa11//aa22..

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Shadow PriceShadow Price

A A shadow priceshadow price for a right hand side value (or for a right hand side value (or resource limit) is the amount the objective resource limit) is the amount the objective function will change per unit increase in the right function will change per unit increase in the right hand side value of a constraint. Mathematically, hand side value of a constraint. Mathematically, the shadow price is the rate of improvement in the shadow price is the rate of improvement in the objective value per unit increase in a the objective value per unit increase in a constraint right hand side(RHS). Economically, constraint right hand side(RHS). Economically, the shadow price measures the marginal benefit the shadow price measures the marginal benefit of having one additional unit of a scarce of having one additional unit of a scarce resources. Therefore, depending on the cost per resources. Therefore, depending on the cost per unit of the limited resource, you use the shadow unit of the limited resource, you use the shadow price to decide whether to buy one additional price to decide whether to buy one additional unit of that resource.unit of that resource.

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Shadow PriceShadow Price

Graphically, a shadow price is determined by Graphically, a shadow price is determined by adding +1 to the right hand side value in adding +1 to the right hand side value in question and then resolving for the optimal question and then resolving for the optimal solution in terms of the same two binding solution in terms of the same two binding constraints. constraints.

The shadow price is equal to the difference in The shadow price is equal to the difference in the values of the objective functions between the values of the objective functions between the new and original problems.the new and original problems.

The shadow price for a nonbinding constraint is The shadow price for a nonbinding constraint is 0. A constraint is nonbinding if its constraint 0. A constraint is nonbinding if its constraint limit is not reached (we have more of that limit is not reached (we have more of that resource than required). resource than required).

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Dual PriceDual Price

A A dual pricedual price for a right hand side value (or for a right hand side value (or resource limit) is the amount the objective resource limit) is the amount the objective function will improve per unit increase in the function will improve per unit increase in the right hand side value of a constraint. right hand side value of a constraint.

For maximization problems dual prices and For maximization problems dual prices and shadow prices are the same.shadow prices are the same.

For minimization problems, shadow prices are For minimization problems, shadow prices are the negative of dual prices.the negative of dual prices.

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Range of FeasibilityRange of Feasibility

The The range of feasibilityrange of feasibility for a change in the right for a change in the right hand side value is the range of values for this hand side value is the range of values for this coefficient in which the original shadow price coefficient in which the original shadow price remains constant.remains constant.

Graphically, the range of feasibility is Graphically, the range of feasibility is determined by finding the values of a right hand determined by finding the values of a right hand side coefficient such that the same two lines that side coefficient such that the same two lines that determined the original optimal solution determined the original optimal solution continue to determine the optimal solution for continue to determine the optimal solution for the problem.the problem.

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Example: Sensitivity AnalysisExample: Sensitivity Analysis


Max Max zz = 5 = 5xx11 + 7 + 7xx22

s.t. s.t. xx11 << 6 6

22xx11 + 3 + 3xx22 << 19 19

xx11 + + xx22 << 8 8

xx11, , xx22 >> 0 0

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Graphical SolutionGraphical Solution

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1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10

22xx11 + 3 + 3xx22 << 19 19

xx22

xx11

xx11 + + xx22 << 8 8Max 5x1 + 7x2Max 5x1 + 7x2

xx11 << 6 6

Optimal Optimal xx11 = 5, = 5, xx22 = 3 = 3 zz = 46 = 46

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Range of Optimality for Range of Optimality for cc11

The slope of the objective function line is The slope of the objective function line is --cc11//cc22. The slope of the first binding constraint, . The slope of the first binding constraint, xx11 + + xx22 = 8, is -1 and the slope of the second binding = 8, is -1 and the slope of the second binding constraint, constraint, xx11 + 3 + 3xx22 = 19, is -2/3. = 19, is -2/3.

Find the range of values for Find the range of values for cc11 (with (with cc22 staying staying 7) such that the objective function line slope lies 7) such that the objective function line slope lies between that of the two binding constraints:between that of the two binding constraints:

-1 -1 << - -cc11/7 /7 << -2/3 -2/3

Multiplying through by -7 (and Multiplying through by -7 (and reversing the inequalities):reversing the inequalities):

14/3 14/3 << cc11 << 7 7

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Range of Optimality for Range of Optimality for cc22

Find the range of values for Find the range of values for cc22 ( with ( with cc11 staying 5) such that the objective function line staying 5) such that the objective function line slope lies between that of the two binding slope lies between that of the two binding constraints:constraints:

-1 -1 << -5/ -5/cc22 << -2/3 -2/3

Multiplying by -1: 1 Multiplying by -1: 1 >> 5/ 5/cc22 >> 2/3 2/3

Inverting, Inverting, 1 1 << cc22/5 /5 << 3/2 3/2

Multiplying by 5: Multiplying by 5: 5 5 << cc22 << 15/2 15/2

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Shadow PricesShadow Prices

Constraint 1Constraint 1: Since : Since xx11 << 6 is not a binding 6 is not a binding constraint, constraint, its shadow price is 0.its shadow price is 0.

Constraint 2Constraint 2: Change the RHS value of the : Change the RHS value of the second second constraint to 20 and resolve for constraint to 20 and resolve for the optimal point the optimal point determined by the last two determined by the last two constraints: constraints:

22xx11 + 3 + 3xx22 = 20 and = 20 and xx11 + + xx22 = 8. = 8.

The solution is The solution is xx11 = 4, = 4, xx22 = 4, = 4, zz = 48. = 48. Hence, the Hence, the

shadow price = shadow price = zznewnew - - zzoldold = 48 - 46 = 2. = 48 - 46 = 2.

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Shadow Prices (continued)Shadow Prices (continued)

Constraint 3Constraint 3: Change the RHS value of the third : Change the RHS value of the third constraint to 9 and resolve for the constraint to 9 and resolve for the

optimal point optimal point determined by the last two determined by the last two constraints: constraints:

22xx11 + 3 + 3xx22 = 19 and = 19 and xx11 + + xx22 = 9. = 9.

The solution is: The solution is: xx11 = 8, = 8, xx22 = 1, = 1, zz = 47. = 47. Hence, the Hence, the shadow price is shadow price is zznewnew - - zzoldold = 47 - 46 = = 47 - 46 = 1.1.

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Example: Infeasible ProblemExample: Infeasible Problem


Max Max zz = 2 = 2xx11 + 6 + 6xx22

s.t. 4s.t. 4xx11 + 3 + 3xx22 << 1212

22xx11 + + xx22 >> 8 8

xx11, , xx22 >> 0 0

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Example: Infeasible ProblemExample: Infeasible Problem

There are no points that satisfy both constraints, There are no points that satisfy both constraints, hence this problem has no feasible region, and hence this problem has no feasible region, and no optimal solution. no optimal solution.

xx22

xx11

44xx11 + 3 + 3xx22 << 12 12

22xx11 + + xx22 >> 8 8

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Example: Unbounded ProblemExample: Unbounded Problem


Max Max z z = 3 = 3xx11 + 4 + 4xx22

s.t. s.t. xx11 + + xx22 >> 5 5

33xx11 + + xx22 >> 8 8

xx11, , xx22 >> 0 0

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Example: Unbounded ProblemExample: Unbounded Problem

The feasible region is unbounded and the The feasible region is unbounded and the objective function line can be moved parallel to objective function line can be moved parallel to itself without bound so that itself without bound so that zz can be increased can be increased infinitely. infinitely.

xx22

xx11

33xx11 + + xx22 >> 8 8

xx11 + + xx22 >> 5 5

Max 3Max 3xx11 + 4 + 4xx22

55

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88

2.672.67

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The End of Chapter 7The End of Chapter 7

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LINEAR PROGRAMMING: THE GRAPHICAL METHOD