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.
Formulation of Linear Programming Models
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.
Formulation of Linear Programming Models
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
© 2008 Prentice Hall, Inc. B – 12
Formulating LP Problems
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
© 2008 Prentice Hall, Inc. B – 13
Formulating LP Problems
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
© 2008 Prentice Hall, Inc. B – 14
Formulating LP Problems
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 ≤
© 2008 Prentice Hall, Inc. B – 15
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
© 2008 Prentice Hall, Inc. B – 16
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
© 2008 Prentice Hall, Inc. B – 17
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
© 2008 Prentice Hall, Inc. B – 18
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
© 2008 Prentice Hall, Inc. B – 19
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
© 2008 Prentice Hall, Inc. B – 20
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
0 50 1000
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Y-Values
Y-Values
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