Management Science Approaches: 'Linear
Programming Takes Centre Stage in Optimisation'
Paul Sagala
Phantom Solutions, Ltd
Introduction
Decision-making has grown by leaps over the years, with concepts of
'management science' dating back to the 18th and
19th centuries. However, World War II was to mark a turning point,
propelled by the desire to use scarce resources more effectively. Since
then, applications of a 'mathematical / scientific
approach' have assumed wide-ranging applications in business,
government and industry operations.
It has been variously said that the scientific approach endeavours to
model complex real life situations into forms that reasonably
'describe' them for purposes of analysis, and getting
solutions for the convenience of man. In that spirit, management science
has been the centre of attention in getting 'as much as possible
at as low a cost as possible'.
Today's cut-throat competition in business means 'survival of the
fittest'. If your business is unable to incur a lesser cost than an
identical competitor for failure to use minimum cost solutions, the
outcome is not difficult to imagine. Either you offer your product on the
market with a lower profit margin, or face difficulties of selling for
asking for a higher price, or worse still ponder leaving the business to
the more efficient competitors.
For the situation of Uganda, it is even more important to cut costs /
seek greater efficiency, given the several disadvantaged positions we
face, compared to producers in several other places. We needed management
science in all business endeavours yesterday!
We will now introduce one of such approaches, that probably commands
the widest application, 'linear programming', through a simple example.
This approach has been at the centre of research, and, can be applied to
very large problems, using computing power and speed to advantage.
Process
The procedure comprises of the following steps:
- Formulate the problem;
- Build a model;
- Perform analysis;
- Determine variability in solution; and,
- Implement findings.
An Example
Kampala Machinists Ltd have received orders to make two types of bolts,
x1 and x2 in quantities that are equal to or greater than
O1 and O2 but
not exceeding M1 and M2 respectively for every 'shift'. For each of these
types, there are two processes that are used. In one process, they 'forge'
the bolt head, and in the other, they 'cut' the threads using engine lathes.
Their interest or 'objective' is to minimise the cost of utilities,
where for each bolt they incur oil costs of shs.50.00 and shs.70.00 for
types x1 and x2 respectively. Likewise, for each bolt they incur
electricity costs of shs.35.00 and shs.20.00 for types x1 and x2
respectively.
The process time requirements for each bolt are of 1.5 and 2.2 minutes
for types x1 and x2 on the forge respectively. As for thread cutting, time
requirements for each bolt are 27 and 18 minutes for types x1 and x2
respectively. The company has one forge and 8 lathes, each available for
7.5 hours per shift, and the company works three shifts per day.
For delivery purposes, total number of bolts made per shift should not
be less than Q.
Determine the 'optimal' production plan.
General model is:
Minimise Utility Costs - (0)
Subject to (s.t.):
Meeting minimum production quantities required - (1)
Not exceeding maximum production quantities - (2)
Time required for production less than or equal to time
available - (3)
Production quantities being 'non-negative' - (5)
Note: The work that follows is given as an indication
of what is possible. The reader need not be put off if it
is not easy to follow, since such services would be
undertaken for your enterprise / business. |
Observations
The following may be said about the above problem, its nature,
and resulting 'easier' solution approach:
- By virtue of having two variables, x1 and x2, this problem can
be solved using the 'graphical' method.
-
All lines in the graph represent either a 'constraint' or an
'objective function', labelled in two ways, one with a number
in parentheses, indicating equivalences with the model,
both in words and formulae above, with an arrow showing
'areas' for which the constraint is 'feasible'. The feasible
side includes the 'boundary'line.
-
For a problem to be feasible, there must be an 'area' which
'satisfies' all constraints. In this case, it is the 'polygon ABCDEF'.
Should there be no such 'space' satisfying all constraints,
then the problem is not feasible.
-
The 'objective function' is represented by the dotted thicker
line, usually distinguished by drawing a few short lines parallel
to it beside it, with the 'arrow', both dotted, the latter
indicating the 'direction' of 'optimisation', 'maximisation' or
'minimisation' as in this case.
-
The 'optimal solution' is the production represented by point
A, namely values of x1 and x2 at that point.
Larger Scale Problems
In many real life problems, problems are often much bigger,
sometimes having hundreds of variables and several constraints.
As the graphical solution is only applicable to those
problems with only two variables, another solution approach,
'linear programming' is in use, either manually or with computer
programmes.
The linear programming approach is also applicable to the twovariable
problem as well.
In the April 2004 issue, we demonstrated how big savings can
be achieved with the 'transportation problem'. Similar benefits
are equally likely in linear programming problems. For instance,
point D represents the 'worst case' or highest cost feasible
solution, with x1 assuming its highest allowable production
value of M1 and x2 bound by one of the production capacity
'tight' or 'binding' constraint in (3).
