Chapter 6. Toward Proper Modeling. Mathematical Modeling. Part art, part science Considerable room for judgment Assumptions concerning variables, constraints, coefficients Assumptions affect results How do you know if the model is “right”?. Structural Component Identification.
Toward Proper Modeling
There are cases when multiple variables must be defined for what may, at first, seem to be a single activity.
Criteria may also be developed where two variables may be treated as one. The simplest case occurs when the coefficients of one variable are simple multiples of another
(aij = Kaim and cj = Kcm).
The second case occurs when one variable uniquely determines another; i.e., when n units of the first variable always implies exactly m units of the second.
Consider the following example:
Max 3X + 2Y
s.t. X - Y ? 0
X LE 10
Y LE 15
Where ? is the constraint type, X depicts sales and Y production. Suppose we have made a mistake and have specified the cost of production as a revenue item (i.e., the +2Y should be -2Y in the objective function).
If the relation is an equality, then the optimal solution is X = Y = 10 and we do not discover the error. On the other hand, if the relation is LE then we would produce Y = 15 units while selling only X = 10 units and we would see the error.
Development of Model Structure
Example: A profit maximizing firm produces 4 crops s.t. land and labor constraints. Crops are grown at different times of the year.
Crop1 is planted in the spring and harvested in the summer. Crop2-3 are planted in the spring and harvested in the fall. Crop4 is planted following crop1 and harvested in the fall.
1. Make a table laying out potential variables
across the top and constraints /objective function down the side.
2. Enter profits for crops and resource coefficient and endowment.
Last constraints is 1/3 Y because chickens weigh 3 pounds. Equal to
Y Le 4500 (total pounds), which would probably be the way I would write it.
Here, we are trying to disassemble across a row. Look at what happens
if we set Y=1. If we have Y equal to 1, then X1 (breast quarters in lbs) can
equal up to 2 pounds, with all else zero. Alternatively, we could get about 3
pounds of leg quarters from this one pound of chicken, or10 pounds of necks.
All would be feasible in this formulation.
Now let us be able to sell mixed quarter packs, which
is an arbitrary combination of legs and breasts.
And let us be able to debone the chicken and sell
the meat for $1.20 a pound.
Here, instead of having one use for some items,
there are various uses – selling them as is, deboning, or
putting them in packs. These alternative uses
require new variables.
This specification allows you to use BQ’s and LQ’s twice,
once as direct sales or meat, the other as part of
a quarter pack.
It is easy to see why this is wrong. Even if only
one chair is produced, the company has to hire
overtime. Also, the overtime limit becomes binding
at 9 chairs, when in fact more chairs could be
produced using “regular” labor.
In this specification, the cheaper labor would
be used first.
It can be useful to set the model (either in
your mind or with a special restriction) to
one unit of a decision variable and
check that all the inputs and other
relationships make sense before running
the model “for real.”