QMT437
Pn Pa’ezah Hamzah
Linear Programming
Part 2
‐
The Simplex Method (for LP problems in
≥
2 variables)
Learning Outcomes
Students should be able to:
1.
Solve LP problems by the simplex method (focusing on maximization
problems)
2.
Interpret the optimal simplex tableau.
I.
Introduction
The simplex method can be used to solve LP problems in two or more decision
variables
.
For illustration, we will solve an LP problem in 2 variables using both the
graphical and simplex procedures.
Example:
XYZ Company produces two types of toys:
tricycles and scooters.
Tricycles need to
be assembled at Workstation 1, and scooters at Workstation 2.
However, both
products require processing time at Workstation 3. The following table summarizes
the data of the problem.
Hours per batch
Workstation
tricycles
scooters
1
1
0
2
0
2
3
3
2
The available time for Workstation 1, Workstation 2, and Workstation 3 are 4, 12,
and 18 hours, respectively. The profits from one batch of tricycles and one batch of
scooters are RM3,000 and RM5,000, respectively.
The LP model for XYZ problem is
Let X
1
= number of batches of tricycles, X
2
= number of batches of scooters
Maximize profit (Z) = 3X
1
+ 5X
2
(RM’000)
subject to:
X
1
≤
4
;Workstation 1 (hours)
2X
2
≤
12
;Workstation 2 (hours)
3X
1
+
2X
2
≤
18
;Workstation 3 (hours)
X
1
,X
2
≥
0
Corner Point solution
Corner
points
Z=3X
1
+5X
2
(RM000)
(0,6)
(0,0)
0
(0,6)
30
(2,6)
36
(4,3)
27
(4,0)
12
(4,0)
6
(2, 6)
•
Feasible
region
3X
1
+2X
2
=18
•
(4,3)
X
1
=4
2X
2
=12
X
2
•
9
X
1
(0,0)
•
•
Optimal solution
:
produce 2 batches
of tricycles and 6 batches of scooters.
Maximum profit = RM36,000
1