2. Graphical Solution

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ADM2302 ~ Rim Jaber 1

Linear Programming:The Graphical Method 1. Problem description

2. Formulate the problem as a linearprogram

3. Plot the constraints 4. Identify the feasible region

5. Find the optimal solution

A. Isoprofit line solution method B. Corner point solution method

Interpretation of the solution


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RMC, Inc. is a firm that produces chemical based products. In a particularprocess three raw materials are used to produce two products. TheMaterial requirements per ton are:

Product Material 1 Material 2 Material 3Fuel additive 2/5 0 3/5

Solvent base 1/2 1/5 3/10

For the current production period RMC has available the followingquantities of each raw material. Because of spoilage, any materials not

used for current production must be discarded.Number of Tons

Material Available for Production

Material 1 20

Material 2 5

Material 3 21

If the contribution to the profit is $40 for each ton of fuel additive and $30for each ton of solvent base, How many tons of each product should beproduced in order to maximize the total contribution profit?


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2. RMC Problem Formulation

Max Z = $ 40 x1 + 30 x2Subject to

2/5 x1 + 1/2 x2 20 (Material 1)

1/5 x2 5 (Material 2)

3/5 x1 + 3/10 x2 21 (Material 3)

x1 0, x2 0

x1: number of tons of fuel additive that RMCproduces

x2: number of tons of solvent base thatRMC produces


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4. The Feasible Region

x1

10

20

30

40

500 10 20 30 40

50x2

2/5x1 + 1/2 x2 = 20

1/5 x2 = 5

3/5 x1 + 3/10 x2= 21


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5. The Optimal Solution:A.The Isoprofit or Isocost Line Solution

Method1. Select value for profit or cost, and draw isoprofit / isocostline to reveal its slope

Z = $240

2. With a maximization problem, maintain same slope andmove line up and right until it touches feasible region atone point. With minimization, move down and left until ittouches only one point in feasible region.

3. Identify optimal solution as coordinates of point

touched by highest possible isoprofit line or lowestpossible isocost line.

4. Read optimal coordinates and compute optimal profit orcost.


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x1

10

20

30

40

500 10 20 30 40

50x2

Maximum isoprofit line

Optimal Solution

Point

A. The Optimal Solution


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Interpretation of theSolution

The solution indicates that if RMC, Inc.produces 25 tons of fuel additive and20 tons of solvent base, it will receive

$1600, the maximum profit possiblegiven the Material constraints(resources constraints)

The solution tells management that the

optimal solution will require allavailable material 1 and material 3, butonly 4 of 5 tons of material 2. (the 1 tonof material 2 is referred as slack


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Corner Point PropertyVery important property of LinearProgramming problems:

Corner Point Property.

Property states optimal solution to LP problem willalwaysoccur at a corner point.

The solution point will be on the boundary of thefeasible solution area and at one of the corners ofthe boundary where two constraint lines intersect


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5. The Optimal Solution:B. The Corner Point Solution Methods

Solve simultaneous equations ateach corner point to find the

solution values at each point. Substitute these values into the

objective function (Z) to find the setof values that results in the

maximum Z value


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x1

10

20

30

40

500 10 20 30 40

50x2

Optimal Solution

Point(25,20)

(18.75,25)(0, 25)

(0,0) (35,0)

B. The Optimal Solution


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A Minimization LP ProblemMany LP problems involve minimizingobjective such as

costinstead of maximizing profit function.Examples:

Restaurant may wish to develop work schedule to meet

staffing needs whileminimizing total number ofemployees.

Manufacturer may seek to distribute its products from

several factories to its many regional warehouses in such a

way as tominimize total shipping costs.

Hospital may want to provide its patients with a daily meal

plan that meets certain nutritional standards while

minimizing food purchase costs.


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Example of a Two Variable MinimizationLP Problem

Holiday Meal Turkey Ranch

Buy two brands of feed for good, low-cost diet forturkeys.

Each feed may contain three nutritional ingredients(protein, vitamin, and iron).

One pound ofBrand A contains:

5 units of protein,

4 units of vitamin, and

0.5 units of iron.

One pound ofBrand B contains:

10 units of protein, 3 units of vitamins, and

0 units of iron.


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Example of Two Variable MinimizationLinear Programming Problem

Holiday Meal Turkey Ranch

Brand A feed costs ranch $0.02 per pound, while Brand

B feed costs $0.03 per pound.

Ranch owner would like lowest-cost diet that meets

minimum monthly intake requirementsfor each

nutritionalingredient.


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Summary of Holiday Meal TurkeyRanch Data


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Formulation of LP Problem:

Minimize cost (in cents) = 2A + 3BSubject to:

5A + 10B 90 (protein constraint)4A + 3B 48 (vitamin constraint)A 1 (iron constraint)A 0, B 0 (nonnegativity constraint)

Where:A denotes number of pounds ofBrand A feed, and Bdenote number of pounds ofBrand B feed.


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Graphical Solution of HolidayMeal Turkey Ranch Problem

Drawing Constraints:

5A + 10B 90

4A + 3B 48

A 1

Nonnegativity ConstraintA 0, B 0


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Isocost Line Method.

One can start by

drawing a 54-cent cost line

2A + 3B. = 54


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Isocost Line Method

Isocost line is moved parallel to 54-cent solution line toward lower leftorigin.

Last point to touch isocost line while still in contact with feasibleregion is corner point 2.


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19

Solving for corner point 2 with two equations with values 8.4 for A and

4.8 for B, minimum optimal cost solution is:

2A + 3B = (2)(8.4) + (3)(4.8) = 31.2

Isocost Line Method


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Corner Point Solution Method Point 1 - coordinates (A = 3, B = 12)

cost of 2(3) + 3(12) = 42 cents. Point 2 - coordinates (A = 8.4, b = 4.8)

cost of 2(8.4) + 3(4.8) = 31.2 cents

Point 3 - coordinates (A = 18, B = 0)

cost of (2)(18) + (3)(0) = 36 cents.

Optimal minimal cost solution:

Corner Point 2, cost = 31.2 cents


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Summary of Graphical Solution Methods

1. Graph each constraint equation.

2. Identify feasible solution region, that is, area that

satisfies all constraints simultaneously.

3. Select one of two following graphical solution

techniques and proceed to solve problem.

1. Corner Point Method.

2. Isoprofit or Isocost Method.

S f G hi l S l ti


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Summary of Graphical SolutionMethods (Continued)

Corner Point Method Determine coordinates of each of corner points

of feasible region by solving equations.

Compute profit or cost at each point bysubstitution of values of coordinates intoobjective function and solving for result.

Identify an optimal solution as a corner point

with highest profit (maximization problem), orlowest cost (minimization).

S f G hi l S l ti


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Summary of Graphical SolutionMethods (continued)

Isoprofit or Isocost Method Select value for profit or cost, and draw isoprofit /

isocost line to reveal its slope.

With a maximization problem, maintain same slopeand move line up and right until it touches feasible

region at one point. With minimization, move downand left until it touches only one point in feasibleregion.

Identify optimal solution as coordinates of point

touched by highest possible isoprofit line or lowestpossible isocost line.

Read optimal coordinates and compute optimalprofit or cost.


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Special Cases in LP

Infeasibility

Unbounded Solutions

(Unboundedness) Redundancy

More Than One

Optimal Solution(Alternate OptimalSolutions)


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Infeasibility

A problem with no feasible regionbecause of the presence of

conflicting constraints Consider the following 3 constraints:

X1 + 2X2


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A Problem with No Feasible Solution

X2

X1

8

6

4

2

02 4 6 8

Region Satisfying3rd Constraint

Region Satisfying First 2 Constraints


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Unboundedness

LP model does not have a finitesolution

Consider the following example:

Max 3X1 + 5X2

Subject to

X1 >= 5

X2 = 10

X1, X2 >= 0


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X2

X1

15

10

5

05 10 15

Feasible Region

X1 > 5X2 < 10

X1 +2X2 > 10

A Solution Region That isUnbounded to the Right


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Redundancy

A redundant constraint is constraint thatdoes not affect feasible region in anyway.

Consider the following example:

Max X1 + 2X2

Subject to

X1 + X2


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X2

X1

30

25

20

15

10

5

0

5 10 15 20 25 30

FeasibleRegion

2X1 + X2 < 30

X1 < 25

X1 +X2 < 20

RedundantConstraint

A Problem with aRedundant Constraint


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Alternate Optimal Solutions An LP problem may have more than one

optimal solution.

Graphically, when the isoprofit (or isocost)

line runsparallelto a constraintin problem

which lies in direction in which isoprofit (or

isocost) line is located.

In other words, when they have same slope.


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Alternate Optimal

Solutions (Contd) Consider the following example:

Max 3X1 + 2X2

Subject to6X1 + 4X2


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8

7

6

5

4

3

21

01 2 3 4 5 6 7 8

Optimal Solution Consists of AllCombinations ofX1 and X2 Alongthe AB Segment

Isoprofit Line for $12Overlays Line Segment

Isoprofit Line for $8

A

B AB

An Example of Alternate

Optimal Solutions

E l Alt t O ti l S l ti


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Example: Alternate Optimal Solutions

At profit level of $12, isoprofit line will rest directly on top of firstconstraint line.

This means that any point along line between corner points 1 and2 provides an optimal X andY combination.

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