Chapter 6 Supplement
Linear Programming
6S-1
Learning Objectives
▪ Describe the type of problem that would lend
itself to solution using linear programming
▪ Formulate a linear programming model from
a description of a problem
▪ Solve linear programming problems using
the graphical method
▪ Interpret computer solutions of linear
programming problem
6S-2
Linear Programming
▪ Used to obtain optimal solutions to
problems that involve restrictions or
limitations, such as:
▪
▪
▪
▪
Materials
Budgets
Labor
Machine time
6S-3
Linear Programming
▪ Linear programming (LP) techniques
consist of a sequence of steps that will
lead to an optimal solution to problems,
in cases where an optimum exists
6S-4
Linear Programming Model
▪ Objective Function: mathematical statement
of profit or cost for a given solution
▪ Decision variables: amounts of either inputs
or outputs
▪ Feasible solution space: the set of all
feasible combinations of decision variables as
defined by the constraints
▪ Constraints: limitations that restrict the
available alternatives
▪ Parameters: numerical values
6S-5
Linear Programming
Assumptions
▪ Linearity: the impact of decision variables is
linear in constraints and objective function
▪ Divisibility: noninteger values of decision
variables are acceptable
▪ Certainty: values of parameters are known and
constant
▪ Nonnegativity: negative values of decision
variables are unacceptable
6S-6
Linear Programming
A company is manufacturing two products A and B. The
manufacturing time required to make them, the profit and capacity
available at each work centre are given as follows:
Product Matching
Fabrication
Assembly
A
1 Hour
5 Hours
3 Hours
B
2 Hours
4 Hours
1 Hour
Profit per unit
80
100
Total Capacity 720 Hours 1800 Hours 900 Hours
6S-7
Linear Programming
Maximize, Z = 80x1 + 100x2 subject to the constraints:
x1 + 2x2 ≤
720
5x1 + 4x2 ≤ 1800
3x1 + x2 ≤
x1 , x2
900
≥ 0.
6S-8
Linear Programming
A company produces three products P, Q, and R
from three raw materials: A, B and C. One unit of
product P requires 2 units of A and 3 units of B. One
unit of product Q requires 2 units of B and 5 units of C
and one unit of product R requires 3 units A, 2 units of
B and 4 units of C. The company has 8 units of
material A, 10 units of material B and 15 units of
material C available to it. Profits per unit of products P,
Q and R are Tk3, Tk.5, and Tk4 respectively.
Formulate the program mathematically.
6S-9
Linear Programming
Decision
variable
Product Types of raw material Profit per
A B C Unit (Tk)
x1
P
2 3 -
3
x2
Q
- 2 5
5
x3
R
3 2 4
4
Max 8
10 15
Unit of materials
available
6S-10
Linear Programming
Maximize, Z = 3x1 + 5x2 + 4x3 subject to the constraints:
2x1 + 3x3 ≤ 8
3x1 + 2x2 + 2x3 ≤
10
5x2 + 4x3 ≤ 15
x1 , x2, x3 ≥ 0
6S-11
Linear Programming
A diet conscious housewife wishes to ensure certain
minimum intake of vitamins A, B, and C for the family.
The minimum daily (quantity) needs of the vitamins A, B,
C for the family are respectively 30, 20, and 16 units. For
the supply of these minimum vitamin requirements, the
housewife relies on two fresh foods. The first one
provides 7, 5, 2 units of the three vitamins per gram
respectively and the second one provides 2, 4, 8 units of
the same three vitamins per gram respectively. The first
foodstuff costs Tk.3 per gram and the second Tk.2 per
gram. How many grams of each foodstuff should the
housewife buy everyday to keep her food bill as low as
possible ?
6S-12
Linear Programming
Minimize, Z = 3x1 + 2x2 subject to the constraints:
7x1 + 2x2
≥ 30
5x1 + 4x2 ≥
2x1 + 8x2 ≥
20
16
x1 , x2 ≥ 0.
6S-13
Linear Programming
A factory manufactures two articles A and B. To
manufacture the article A, a certain machine has to be
worked for 1.5 hours and in addition a craftsman has to
work for 2 hours. To manufacture the article B, the
machine has to be worked for 2.5 hours and in addition a
craftsman has to work for 1.5 hours. In a week the
factory can avail of 80 hours of machine time and 70
hours of craftsman’s time. The profit on each article A is
Tk.5 and that on each article B is Tk.4. If all the articles
produced can be sold away, find how many of each kind
should be produced to earn the maximum profit per
week.
6S-14
Linear Programming
Decision Article
Hours on Hours on
Profit per
variable
Machine Craftsman
(Tk)
x1
A
1.5
2
5
x2
B
2.5
1.5
4
Hours Available
(per week)
80
Maximum
70
Maximum
x1 = number of units of Article A
x2 = number of units of Article B
6S-15
Linear Programming
Maximize, Z = 5x1 + 4x2 subject to the constraints:
1.5x1 + 2.5x2 ≤ 80
2x1 + 1.5x2 ≤ 70
x1 , x2 ≥
0.
6S-16
Graphical Linear Programming
Graphical method for finding optimal
solutions to two-variable problems
1. Set up objective function and
constraints in mathematical format
2. Plot the constraints
3. Identify the feasible solution space
4. Plot the objective function
5. Determine the optimum solution
6S-17
Linear Programming Example
▪ Objective - profit
Maximize, Z = 60X1 + 50X2
Subject to:
Assembly 4X1 + 10X2 ≤ 100 hours
Inspection 2X1 + 1X2 ≤ 22 hours
Storage 3X1 + 3X2 ≤ 39 cubic feet
X1, X2 ≥ 0
6S-18
Linear Programming Example
▪ Objective
Maximize, Z = 2X1 + 10X2
Subject to:
Durability 10X1 + 4X2 ≥ 40 wk
StrengthX1 + 6X2 ≥ 24 psi
Time
X1 + 2X2 ≤ 14 hr
X1, X2 ≥ 0
6S-19
Solutions and Corner Points
▪ Feasible solution space is usually a polygon
▪ Solution will be at one of the corner points
▪ Enumeration approach: Substituting the
coordinates of each corner point into the
objective function to determine which corner
point is optimal.
6S-20
Slack and Surplus
▪ Surplus: when the optimal values of
decision variables are substituted into a
greater than or equal to constraint and the
resulting value exceeds the right side value
▪ Slack: when the optimal values of decision
variables are substituted into a less than or
equal to constraint and the resulting value is
less than the right side value
6S-21
Simplex Method
▪ Simplex: a linear-programming
algorithm that can solve problems
having more than two decision
variables
6S-22
MS Excel Worksheet for
Microcomputer
Problem
Figure 6S.15
6S-23
MS Excel Worksheet Solution
Figure 6S.17
6S-24