CHAPTER 2
LINEAR PROGRAMMING MODELS: GRAPHICAL AND
COMPUTER METHODS
Note:
Permission to use the computer program GLP for all LP graphical solution screenshots in this
chapter granted by its author, Jeffrey H. Moore, Graduate School of Business, Stanford University.
Software copyrighted by Board of Trustees of the Leland Stanford Junior University. All rights reserved.
SOLUTIONS TO DISCUSSION QUESTIONS
21.
The requirements for an LP problem are listed in Section 2.2. It is also assumed that conditions of
certainty
exist; that is, coefficients in the objective function and constraints are known with certainty and
do not change during the period being studied. Another basic assumption that mathematically
sophisticated students should be made aware of is
proportionality
in the objective function and
constraints. For example, if one product uses 5 hours of a machine resource, then making 10 of that
product uses 50 hours of machine time.
LP also assumes
additivity.
This means that the total of all activities equals the sum of each
individual activity. For example, if the objective function is to maximize Profit = 6X
1
+ 4X
2
, and if X
1
=
X
2
= 1, the profit contributions of 6 and 4 must add up to produce a sum of 10.
22.
If we consider the feasible region of an LP problem to be continuous (i.e., we accept noninteger
solutions as valid), there will be an infinite number of feasible combinations of decision variable values
(unless of course, only a single solution satisfies all the constraints). In most cases, only one of these
feasible solutions yields the
optimal
solution.
23.
A problem can have alternative optimal solutions if the level profit or level cost line runs parallel to
one of the problem’s binding constraints (refer to Section 2.6 in the chapter).
24.
A problem can be unbounded if one or more constraints are missing, such that the objective value
can be made infinitely larger or smaller without violating any constraints (refer to Section 2.6 in the
chapter).
25.
This question involves the student using a little originality to develop his or her own LP constraints
that fit the three conditions of (1) unbounded solution, (2) infeasibility, and (3) redundant constraints.
These conditions are discussed in Section 2.6, but each student’s graphical displays should be different.
26.
The manager’s statement indeed has merit if he/she understood the deterministic nature of LP input
data. LP assumes that data pertaining to demand, supply, materials, costs, and resources are known with
certainty and are constant during the time period being analyzed. If the firm operates in a very unstable
environment (for example, prices and availability of raw materials change daily, or even hourly), the LP
model’s results may be too sensitive and volatile to be trusted. The application of sensitivity analysis
might, however, be useful to determine whether LP would still be a good approximating tool in decision
making in this environment.
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 Spring '10
 Bryan

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