The Big M-Method
Earlier, we have only treated problems where the constraints
were of ≤ type with non-negative RHS. But the constraints
can also have an = or ≥ signs. Example:
Maximize Z= 4x1 + 3x2+6x3
Subject to 2x1 + 3x2+2x3=440
4x1 +3x3≤ 470
2x1 +5x2 ≥ 430
x1, x2, x3 ≥ 0
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• In a case where at least one of the constraints is of = or
≥ type, we have to introduce another type of variables
called artificial variables.
• They are merely a device to get the starting basic
feasible solution so that simplex algorithm is applied
as usual to get optimal solution.
•
One of the techniques available to solve such type of
problems is a Big M-Method (also known as penalty
method)
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• This method consists of the following basic steps:
Step 1: Express the linear programming problem in standard form
by introducing slack and surplus variables.
– Slack variables are added to the left-hand sides of the
constraints of (≤) type and surplus variables are subtracted
from the constraints of (≥) type.
Step 2: Add non-negative artificial variables to the left-hand sides
of all the constraints of initially (≥) or (=) type.
– The purpose of introducing the artificial variables is just to
obtain an initial basic feasible solution. These artificial
variables have the following characteristics:
• They are fictitious, have no physical meaning or economic
significance and have no relevance to the problem.
• The artificial variables are a computational device.
• They keep the starting equations in balance and provide a
mathematical trick for getting a starting solution.
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• Their introduction (addition) violates the
equality of constraints.
• They are, therefore, rightly termed as artificial
variables as opposed to other real decision
variables in the problem.
• Therefore, we must get rid of these variables
and must not allow them to appear in the final
solution.
• To achieve this, these variables are assigned a
very large per unit penalty in the objective
function.
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This penalty is designated by - M for
maximization problems and + M for minimization
problems, where M > O.
Step 3: Solve the modified linear programming
problem by the simplex method.
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While making iterations, using the simplex method, one of
the following three cases may arise:
1. If no artificial variable remains in the basis and the
optimality condition is satisfied, then the solution is an
optimal feasible solution to the given problem.
2. If at least one artificial variable appears in the basis at
zero level (with zero value in b column) and the
optimality condition is satisfied, then the solution is
optimal feasible (though degenerate) solution to the given
problem. The constraints are consistent though
redundancy may exist in them. By redundancy is meant
that the problem has more than the required number of
constraints.
3. If at least one artificial variable appears in the basis at a
non-zero level (with positive value in b-column) and the
optimality condition is satisfied, then the original problem
has no feasible solution
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Remarks:
1. Slack variables are added to the left-hand
sides of the
constraints of (≤) type and surplus
variables are
subtracted from the left hand side of the constraints
of ≥ type.
2. Artificial variables are added to the constraints of (≥) and
(=) type. Equality constraints require neither slack nor
surplus variables.
3. Variables, other than the artificial variables, once driven
out in iteration, may re-enter in a subsequent iteration.
But, an artificial variable, once driven, can never re-enter,
because of the large penalty coefficient M associated with
it in the objective function. Advantage can be taken of this
fact by not computing its column in iterations subsequent
to the one from which it was driven out.
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Example
• Food X contains 6 units of vitamin A per gram and 7 units of vitamin B
per gram and costs 12 birr per gram. Food Y contains 8 units of
vitamin A per gram and 12 units of vitamin B per gram and costs 20
birr per gram. The daily minimum requirement of vitamin A is 100
units and that of vitamin B is 120 units. Find the minimum cost of
product mix by the simplex method.
Solution
Step 1: Model the LPP
• Let x1 and x2 be the grams of food X and Y to be purchased. Then the
problem can be formulated as follows:
Foods
Vitamin A Vitamin B Costs
DV
• The objective is to minimize cost:
Minimize Z =12x1 + 20x2
X1
x
requireme
nt/gram of
x and y
6
X2
y
8
12
Daily
min
require
ment
100
120
• Constraints are
Requirement of vitamin A: 6x1 + 8x2≥100
Requirement of vitamin B: 7x1 + 12x2 ≥120
where, x1,x2 ≥ 0
requireme per
nt/gram of gram
x and y
7
12
20
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Step 2: Express the problem in standard form
Use surplus variables s1 and s2
Min Z = 12x1 + 20x2 + 0s1 + 0s2
Subject to
6x1 + 8x2 -s1=100
7x1 + 12x2 -s2 =120
x1,x2, s1 , s2 ≥ 0
Variables s1 represents units of vitamin A in
product mix in excess of the minimum
requirement of 100, s2 represents units of vitamin
B in product mix in excess of the minimum
requirement of 120.
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Step 3: Find Initial Basic Feasible Solution
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Note that we start with a very heavy cost (compare
it with zero profit in maximization problem) which
we shall minimize during the solution procedure.
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Step 4: Optimality Test
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Step 5: Iterate Towards an Optimal Solution
Performing iteration results in the following
table
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Therefore, the optimal solution is:
x1= 15, x2 = 5/4
Zmin = 205 Birr
Hence 15 grams of food X and 5/4 grams of food Y
should be the required product mix with minimum
cost of Birr 205
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