Solving Linear Programming
Problems Using Excel
Ken S. Li
Southeastern Louisiana University
Linear Programming
 The Problem
An optimization model is a linear program if it has
continuous variables, a single objective function, and
all constraints are linear equalities or inequalities
History
Linear programming was conceptually developed before
World War II by the outstanding Russian mathematician
A.N. Kolmogorov.
Who and When
Major contributors:
Kantorovich
Stigler
Dantzig
Karmarkar
Gonzaga
Wright
Terlaky
Todd
1945
1947
1984
1992
1996
1995
1991
Standard Formulation
Mathematical Formulation
minimize cx
subject to
Ax = b
x >= 0
Standard Formulation
where x is the vector of variables to be solved for, A
is a matrix of known coefficients,
and c and b are vectors of known coefficients.
The expression "cx" is called the objective
function,
and the equations "Ax=b" are called the
constraints.
Standard Formulation
Usually A has more columns than rows, and Ax=b
is therefore quite likely to be under-determined,
leaving great latitude in the choice of x with
which to minimize cx. The word "Programming"
is used here in the sense of "planning"; the
necessary relationship to computer programming
was incidental to the choice of name. Hence the
phrase "LP program" to refer to a piece of
software is not a redundancy, although I tend to
use the term "code" instead of "program" to avoid
the possible ambiguity.
Applications
 Applications
Linear and integer programming have proved
valuable for modeling many and diverse types of
problems in planning, routing, scheduling,
assignment, and design. Industries that make use
of LP and its extensions include transportation,
energy, telecommunications, and manufacturing
of many kinds.
Applications
 Specific Applications
1. Development of a production schedule that will satisfy
future demands for a firm’s production and at the
same time minimize total production and inventory
costs
2. Selection of the product mix in a factory to make best
use of machine-hours and labor-hours available while
maximizing the firm’s products
Application - Contd
3. Determination of grades of petroleum products to
yield the maximum profit
4. Selection of different blends of raw materials to feed
mills to produce finished feed combinations at
minimum cost
5. Determination of a distribution system that will
minimize total shipping cost from several warehouses
to various market locations
Computational Method
 Simplex Methods
Simplex methods, introduced by Dantzig about 50
years ago, visit "basic" solutions computed by
fixing enough of the variables at their bounds to
reduce the constraints Ax = b to a square system,
which can be solved for unique values of the
remaining variables. Basic solutions represent
extreme boundary points of the feasible region
defined by Ax = b, x >= 0, and the simplex
method can be viewed as moving from one such
point to another along the edges of the boundary.
Computational Method
 Interior Point Methods
Barrier or interior-point methods, by contrast,
visit points within the interior of the feasible
region. These methods derive from techniques for
nonlinear programming that were developed and
popularized in the 1960s by Fiacco and
McCormick, but their application to linear
programming dates back only to Karmarkar's
innovative analysis in 1984.
Interior Point Method
Step 1: Choose any feasible interior point solution,
x (0)  0 and set solution index t=0.
Step 2: If any component of x (t) is 0, or if recent
steps have made no significant change in the
solution value, stop. Current point is either
optimal or very nearly so.
Step 3: Construct the next move direction
x
( t 1)
  X t Pt c
(t )
Interior Point Method -contd
Where
 x1( t )

0

Xt 


 0
0
x2( t )
0
0 


0 

(t ) 
xn  ,
Pt  I  ATt (At ATt ) 1 A t , ct  X t c.
Interior Point Method - contd
Step 4: If there is no limit on feasible moves in the
direction x (t 1) (all components are
nonnegative), stop ; the given model is
unbounded. Otherwise, construct the step size

1
x
( t 1)
X
1
t
.
Interior Point Method - contd
Step 4: compute the new solution
x (t 1)  x (t )  x (t 1)
Then let t  t  1, and return to Step 2.
A Simple Example
The Marriott Tub Company manufactures two lines of
bathtubs, called Model A and model B. Every tub
requires a certain amount of steel and zinc; the company
has available a total of 25,000 pounds of steel and 6,000
pounds of zinc. Each model A bathtub requires a total of
125 pounds of steel and 20 pounds of zinc, and each
yields a profit of $90. Each model B bathtub can be sold
for a profit of $70; it in turn requires 100 pounds of steel
and 30 pounds of zinc. Find the best production mix of
the bathtubs.
The Formulation
Maximize
Subject to
P  90 x  70 y
125 x  100 y  25000
20 x  30 y  6000
x, y  0
Where x and y are the numbers of model A
and model B bathtubs that the company
will make, respectively.
Solving by Interior Point
Method
x1
x2
x3
x4
Obj
Initial
1st
2nd
3 rd
100
100
2500
1000
16000
129.4 86.4
178.8
818.8
17698
152.8 58.7
3.3
1184
17861
157
53.6
5.7
1250
17882
4 th
5 th
6 th
7 th
8 th
189
13.60
8.3
1810
17962
199.6 0.4
6.5
1995
17992
199.7 0.3
0.1
1994
17994
199.9 0.1
0.1
1999
17998
200
0
2000
18000
0
Solving by Excel
http://www.selu.edu/Academics/Faculty/kli
/linearprog.xls