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Imagine that you have a 5-week business commitment between Fayetteville (FYV) and Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesdays.

**OPERATIONS RESEARCH MODELS**

Imagine
that you have a 5-week business commitment between Fayetteville (FYV) and
Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesdays.
A regular round-trip ticket costs $400, but a 20% discount is granted if the
dates of the ticket span a weekend. A one-way ticket in either direction costs
75% of the reg-ular price. How should you buy the tickets for the 5-week
period?

We can
look at the situation as a decision-making problem whose solution re-quires
answering three questions:

1. What are the decision alternatives?

2. Under what restrictions is the decision made?

3. What is an appropriate objective criterion for
evaluating the alternati-ves?

Three
alternatives are considered:

Buy five regular FYV-DEN-FYV for departure on
Monday and return on Wednes-day of the same week.

Buy one FYV-DEN, four DEN-FYV-DEN that span
weekends, and one DEN-FYV.

Buy one FYV-DEN-FYV to cover Monday of the first
week and Wednesday of the last week and
four DEN-FYV-DEN to cover the remaining legs. All tickets in this alternative
span at least one weekend.

The
restriction on these options is that you should be able to leave FYV on Monday and return on
Wednesday of the same week.

An
obvious objective criterion for evaluating the proposed alternative is the
price of the tickets. The alternative that yields the smallest cost is the
best. Specifically, we have

Alternative 1 cost = 5 X
400 = $2000

Alternative
2 cost = .75 X 400 + 4 X (.8 X
400) + .75 X 400
= $1880

Alternative
3 cost =
5 X (.8 X 400) =
$1600

Thus, you
should choose alternative 3.

Though
the preceding example illustrates the three main components of an OR model-alternatives, objective
criterion, and constraints-situations differ in the de-tails of how each
component is developed and constructed. To illustrate this point, con-sider
forming a maximum-area rectangle out of a piece of wire of length *L* inches.
What should be the width and height of the rectangle?

In
contrast with the tickets example, the number of alternatives in the present
ex-ample is not finite; namely, the width and height of the rectangle can
assume an infinite number of values. To formalize this observation, the
alternatives of the problem are identified by defining the width and height as
continuous (algebraic) variables.

Let

*w *=* *width of the rectangle in inches

*h *= height of the rectangle in
inches

Based on
these definitions, the restrictions of the situation can be expressed verbally
as

Width of rectangle + Height
of rectangle = Half the length of the wire

Width and height cannot be negative

These
restrictions are translated algebraically as

*2(w *+* h) *=* L*

*w *>=* *0,* h *>=* *0

The only
remaining component now is the objective of the problem; namely, maximization
of the area of the rectangle. Let *z*
be the area of the rectangle, then the complete model becomes

Maximize *z* = *wh*

subject
to

*2(w
*+* h) *=* L*

*w, h *2 >=* *0

The
optimal solution of this model is *w* = *h* = L/4, which
calls for constructing a square shape.

Based on
the preceding two examples, the general OR model can be organized in the
following general format:

A
solution of the mode is feasible if it satisfies all the constraints. It is optimal if, in addition to
being feasible, it yields the best (maximum or minimum) value of the objective
function. In the tickets example, the problem presents three feasible alternatives,
with the third alternative yielding the optimal solution. In the rectangle
problem, a feasible alternative must satisfy the condition *w* + *h* = L/2 with wand *h* assuming nonnegative values. This leads to an infinite number of
feasible solutions and, unlike the tickets problem, the optimum solution is
determined by an appropriate mathematical tool (in this case, differential
calculus).

Though OR
models are designed to "optimize" a specific objective criterion subject
to a set of constraints, the quality of the resulting solution depends on the
completeness of the model in representing the real system. Take, for example,
the tickets model. If one is
not able to identify all the dominant alternatives for purchasing the tick-ets,
then the resulting solution is optimum only relative to the choices represented
in the model. To be specific, if alternative 3 is left out of the model, then
the resulting "optimum" solution would call for purchasing the
tickets for $1880, which is a suboptimal solution. The conclusion is that
"the" optimum solution of a model is best only for *that* model. If the model happens to represent
the real system reasonably well, then its solution is optimum also for the real
situation.

**PROBLEM
SET 1.1A**

1. In the tickets example, identify a fourth feasible
alternative.

2. In the rectangle problem, identify two feasible
solutions and determine which one is better.

3. Determine the optimal solution of the rectangle
problem. *(Hint:* Use the constraint to
express the
objective function in terms of one variable, then use differential calculus.)

4. Amy, Jim, John, and Kelly are standing on the east
bank of a river and wish to croSs to the west side using a canoe. The canoe can
hold at most two people at a time. Amy, being the most athletic, can row across
the river in 1 minute. Jim, John, and Kelly would take 2, 5, and 10 minutes,
respectively. If two
people are in the canoe, the slower person dictates the crossing time. The
objective is for all four people to be on the other side of the river in the
shortest time possible.

a. Identify at least two feasible plans for
crossing the river (remember, the canoe is the only mode of transportation and
it cannot be shuttled empty).

b. Define the criterion for evaluating the
alternatives.

c. What is the smallest time for moving all four
people to the other side of the river?

In a
baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can
throw either a fast or a curve ball at random. If Joe correctly predicts a curve ball, he can main-tain a .500 batting
average, else if Jim throws a curve ball and Joe prepares for a fast ball, his
batting average is kept down to .200. On the other hand, if Joe correctly
predicts a fast ball, he gets a .300 batting average, else his batting average
is only .100.

a. Define the alternatives for this situation.

b. Define the objective function for the problem
and discuss how it differs from the familiar optimization (maximization or
minimization) of a criterion.

6. During the construction of a house, six joists of
24 feet each must be trimmed to the cor-rect length of 23 feet. The operations
for cutting a joist involve the following sequence:

Three
persons are involved: Two loaders must work simultaneously on operations 1,2,
and 5, and one cutter handles operations 3 and 4. There are two pairs of saw
horses on which untrimmed joists are placed in preparation for cutting, and
each pair can hold up to three side-by-side joists. Suggest a good schedule for
trimming the six joists.

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