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Linear Optimization Essay

4064 words - 17 pages


Linear Optimization

Linear Optimization
A constrained optimization model takes the form of a
performance measure to be optimized over a range of
feasible values of the decision variables.
The feasible values of the decision variables are
determined by a set of inequality constraints.
Values of the decision variables must be chosen such
that the inequality constraints are all satisfied while
either maximizing or minimizing the desired
performance variable.
These models can contain tens, hundreds, or thousands
of decision variables and constraints.

Linear Optimization
Very ...view middle of the document...

The use of a certain type of crude oil in producing
gasoline is restricted by the characteristics of the
gasoline (e.g., octane rating, etc.)

Formulating LP Models
A constrained optimization model represents the
problem of allocating scarce resources in such a way as
to optimize an objective of interest.
To illustrate this, we will return to the Oak Products
model. However, we will start with a simplified model.

Oak Products, Inc.
Based on economic forecasts for the next week, it has
been determined that it will be possible to sell as many
Captains or Mates chairs as the firm can produce.
So, the question becomes: How many Captains and
Mates chairs should be produced given our constraints
in order to maximize next week’s profit contribution?

Formulating LP Models
Consider the following major factors:
1. The unit contribution margin (price minus unit
variable cost) is $56 for each Captain sold and
$40 for each Mate.
2. Long dowels, short dowels, legs and one of two
types of seats are needed to assemble the chairs.
3. We have a limited inventory of 1280 long dowels
and 1600 short dowels for next week’s
Each Captain uses 8 long and 4 short dowels.
Each Mate uses 4 long and 12 short dowels.

Formulating LP Models
4. The inventory of legs is 760 units and each chair
produced of either type uses 4 legs.
5. The inventory of heavy and light seats is 140 and
120, respectively. Each Captain produced uses a
heavy seat and each Mate uses a light seat.
6. In order for management to honor an agreement
with the union, the total number of chairs
produced of both types cannot fall below 100.

Let’s summarize the information given so far in a table:

Given these considerations, now decide how many
Captains and Mates to produce next week.
This is called an optimal product mix problem or an
optimal production plan.
The first step in solving this is to identify the constraints
and objective function.

For notation purposes, let C = number of Captains
chairs and M = number of Mates chairs to be produced.

Start with the number of long dowels:
Both types of chairs require long dowels.
So, we can mathematically describe this as:
8(# Captains produced) + 4(# Mates produced) = 1280

8C + 4M = 1280

Once again, here is the equation for the long dowels:
8C + 4M = 1280
However, since only 1280 long dowels are available and
we can use at most, only that amount, we must make
this equation an inequality in order to satisfy this
restriction. In this case:
8C + 4M < 1280
We use the < (less than or equal to) inequality because,
we can use all 1280 long dowels, we can use less than
that amount, but we will never be able to use more than
This is called an inequality constraint.
Left hand side (LHS) is called a constraint function.
Right hand side (RHS) specifies the limitation.

Now let’s...

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