Linear Programming

Linear Programming Alayssa Silva

July 4, 2012

What is Linear Programming (LP)?

A particular type of mathematical model in which relationships involving the variables are linear. This model use a mathematical technique called linear programming which determines the best or optimal decision even there are thousands of variables and relationships. It has been applied to a wide variety of decision problems in business and the public sector. The LP model is designed maximize (minimize) an objective function of the form.

Where f is same economic objective such as profit, production, cost, workweeks, or tons shipped.

f =C1X1 + C2X2 + . . . = CnXn

What is Linear Programming (LP)?

LP is composed of two main parts namely: objective function and constraints.The objective function is either to maximize or to minimize.The constraints are mathematical, relationships expressed in terms of linear equations or linear equalities.

Where: A coefficients are constraints B1 restricts f1, the objective function

A1X1 + A2X2 + . . . = AnXn < B1

Linear Programming Applications

1. Scheduling school buses to minimize total distance traveled 2. Allocating police patrol units to high crime areas in order to

minimize response time to 911 calls3. Scheduling tellers at banks so that needs are met during

each hour of the day while minimizing the total cost of labor4. Selecting the product mix in a factory to make best use of

machine- and labor-hours available while maximizing the firm’s profit

5. Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs

6. Determining the distribution system that will minimize total shipping cost

7. Developing a production schedule that will satisfy future demands for a firm’s product and at the same time minimize total production and inventory costs

8. Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company

Formulation of Linear Programming Models

Linear programming problems can be formulated in a systematic way by following these simple steps:

1.Define the specific decision variablesThis is assigning variables to the given products.

2.Identify the objective function which is either to maximize or to minimize. This is the objective that you are trying to achieve in solving the problem.

Example: Maximize: Contribution to profitMinimize: Cost

3.List down the constraints that affect the decision. There are different types of constraints which can be found on a given word problem solving.


Types of constraintsa.Capacity ConstraintsThere are limitations on the amount of equipment ,

space, or stuff availability. Example:

There are only two machines available, machine A and machine B. Machine A is available for 12 hours, while machine B is available for 20 hrs. b. Market Constraints

These are limitations (either lower or upper limits or both) on how much upper can be sold or used.

c. Availability ConstraintsThese are limitations on the available raw materials, labor, funds or other resources.


Types of constraintsd. Quality or Blending Constraints

These are limitations on the mixes of ingredients, usually defining the output of products.

e. Production Technology or Material Balance ConstraintsThese are limitations on the mixes of ingredients, usually defining the output of products.

f. Definitional ConstraintsThese are constraints that define a given variable, often, such constraints came from accountancy definitions.

4. Define the specific constraints using the decision variables.

Problem Example 1: A clock maker makes two types of wood clock to sell at various malls. It takes him three (3) hours to assemble a pine clock, which requires two (2) oz of vanish . It takes four (4) hours to assemble a molave clock, which takes four oz of varnish. He has eight oz of varnish available in stock and he can work 12 hours. If he makes P100 profit on each pine clock and P120 on each molave clock, how many of each type should he make to maximize his profits? Formulate the linear program.

Solution:Following the steps in formulating a linear program, we have,1. Let x = number of pine clock

y = number of molave clock2. Objective function

MAXIMIZE: Profit = P100x + P120y

Problem Example 1: (cont’n)3. Constraints

Raw materials and process requirements like- varnish requirement- processing time

4. Specific constraintsVarnishing 3x + 4y ≤ 12Processing time 2x + 4y ≤ 16

x ≥ 0 y ≥ 0

Thus, the linear program would≥ be:MAXIMIZE : Profit = P100x + P120ySUBJECT TO:Varnishing 3x + 4y ≤ 12Processing time 2x + 4y ≤ 16

x ≥ 0 y ≥ 0

Graphical Method of Linear Programming

The set of all points that satisfy all the constraints of the model is called a feasible region.However, it has been proven that the maximum or minimum value always occurs at a vertex of the feasible region.

Steps to be followed in LP Graphical Method1. Formulate the linear program.2. Graph the inequalities and shade the feasible region.3. Solve for the coordinates of the vertices of the feasible

region.4. Substitute the coordinates of the vertices of the feasible

region to the objective function. 5. Formulate your decision. If it is maximization, choose the

vertex that will give you the lowest value.

© 2008 Prentice Hall, Inc. B – 11

Formulating LP Problems

The product-mix problem at Shader Electronics

Two products

1. Shader X-pod, a portable music player

2. Shader BlueBerry, an internet-connected color telephone

Determine the mix of products that will produce the maximum profit



X-pods BlueBerrys Available HoursDepartment (X1) (X2) This Week

Hours Required to Produce 1 Unit

Electronic 4 3 240

Assembly 2 1 100

Profit per unit P7 P5

Decision Variables:X1 = number of X-pods to be producedX2 = number of BlueBerrys to be produced

Table B.1



Objective Function:

Maximize Profit = P7X1 + P5X2

There are three types of constraints Upper limits where the amount used is ≤ the

amount of a resource Lower limits where the amount used is ≥ the

amount of the resource Equalities where the amount used is = the

amount of the resource



Second Constraint:

2X1 + 1X2 ≤ 100 (hours of assembly time)

Assemblytime available

Assemblytime used is ≤

First Constraint:

4X1 + 3X2 ≤ 240 (hours of electronic time)

Electronictime available

Electronictime used is ≤


Graphical Solution

Can be used when there are two decision variables1. Plot the constraint equations at their

limits by converting each equation to an equality

2. Identify the feasible solution space

3. Create an iso-profit line based on the objective function

4. Move this line outwards until the optimal point is identified


Graphical Solution

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

–| | | | | | | | | | |

0 20 40 60 80 100

Num

ber

of B

lueB

erry

s

Number of X-pods

X1

X2

Assembly (constraint B)

Electronics (constraint A)Feasible region

Figure B.3


Graphical Solution

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

–| | | | | | | | | | |

0 20 40 60 80 100

Num

ber

of W

atch

TV

s

Number of X-pods

X1

X2

Assembly (constraint B)

Electronics (constraint A)Feasible region

Figure B.3

Iso-Profit Line Solution Method

Choose a possible value for the objective function

P210 = 7X1 + 5X2

Solve for the axis intercepts of the function and plot the line

X2 = 42 X1 = 30


Graphical Solution

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

–| | | | | | | | | | |

0 20 40 60 80 100

Num

ber

of B

lueB

erry

s

Number of X-pods

X1

X2

Figure B.4

(0, 42)

(30, 0)

P210 = P7X1 + P5X2


Graphical Solution

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

–| | | | | | | | | | |

0 20 40 60 80 100

Num

ber

of B

lueB

eryy

s

Number of X-pods

X1

X2

Figure B.5

P210 = P7X1 + P5X2

P350 = P7X1 + P5X2

P420 = P7X1 + P5X2

P280 = P7X1 + P5X2


Graphical Solution

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

–| | | | | | | | | | |

0 20 40 60 80 100

Num

ber

of B

lueB

erry

s

Number of X-pods

X1

X2

Figure B.6

P410 = P7X1 + P5X2

Maximum profit line

Optimal solution point(X1 = 30, X2 = 40)

Graphical Method of Linear Programming: Problem Example 1

To make one unit of product A requires 3 minutes in Dept. I and 1 minute in Dept II. One unit of product B requires 4 minutes in Dept I and 2 minutes in Dept II. Profit contribution is P5/unit of A and P8/unit of B. Find the number of units of A and B, which should be made to maximize profit if Dept I and II have 150 and 60 minutes available respectively. What is the maximum and minimum profit?

Solution: First, we formulate the linear programLet x= number of units of product A y = number of units of product B

P = incremental profit

Objective Function: To maximize: P = P5x + P8yConstraints: Subject to:

time in Dept I: 3x + 4y ≤ 150time in Dept II: x + 2y ≤ 60

x ≥ 0 y ≥ 0

Then, we graph the constraints by finding the x and y intercepts

Notice that the constraints x ≥ 0 and y ≥ 0 restrict the graph to Quadrant I.

x-intercept y-intercept

3x + 4y ≤ 150 3x + 4y = 150 (50, 0) (0, 37⅟2)

x + 2y ≤ 60 x + 2y = 60 (60,0) (0,30)

x ≥ 0 x = 0y ≥ 0 y = 0

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Linear Programming

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