Hi k9queen!
I have downloaded a "lite" version of QM so I can get a feel of how
data is entered into this program. I only have access to the
Forecasting, Linear Progrmming and Waiting Lines modules. Although I
believe the right module for this program would be "Integer
Programming" or "Mixed Programming", I think the data entry is mostly
the same as in the Linear Programming Module.
Here's a way to write the problem. We can think of this as a problem
with 16 variables (let's call them X1, X2,... , X16) and 8
constraints. The 16 variables should be yes/no variables or integer
variables for reasons that will become apparent below.
The interpretation of the variables is the following. Let X1 be 1 if
Jones pitches against Des Moines, and 0 if he doesn't. For X2, the
same goes for Baker at Des Moines. For X3, again the same for Parker
at Des Moines; and X4 is the same for Wilson at Des Moines. Now, let
X5 be 1 if Jones pitches against Davenport, and 0 if he doesn't. X6,
X7 and X8 are defined similar to X2, X3 and X4, but for Davenport. We
apply an analogous interpretation for variables X9 up to X16, using
Omaha and Peoria instead of Des Moines and Davenport.
The problem then consists to choose which variables will be set to 1
and which ones will be set to 0. That is, the problem is to choose
which pitcher plays each game. Thus the function to maximize is:
Max 0.6X1 + 0.7X2 + 0.9X3 + 0.5X4 + 0.8X5 + 0.4X6 + 0.8X7 + 0.3X8 +
0.5X9 + 0.8X10 + 0.7X11 + 0.4X12 + 0.4X13 + 0.3X14 + 0.8X15 + 0.2X16
So for example, setting X1 to 1, which means to let Jones pitch
against Des Moines, adds 0.6 to the sum of probabilities of winning,
just as it is mentioned in the table in your question. The same goes
for every other variable.
Of course, there are constraints in this problem. If this weren't the
case, we would set all the variables to 1 in order to maximize this
function. Clearly, this wouldn't make sense: we would be having all
players pitching first against each opponent (when only one can be the
first!); and we would have each player pitching first against many
opponents (something that the manager doesn't want to do). Therefore,
the constraints must be such that these contingencies are taken into
account. We must apply constraints so that only 1 player pitches first
against an opponent, and such that each player pitches first only
once. Therefore, the 8 constraints are:
X1 + X2 + X3 + X4 = 1
X5 + X6 + X7 + X8 = 1
X9 + X10 + X11 + X12 = 1
X13 + X14 + X15 + X16 = 1
X1 + X5 + X9 + X13 = 1
X2 + X6 + X10 + X14 = 1
X3 + X7 + X11 + X15 = 1
X4 + X8 + X12 + X16 = 1
The first set of constraints imply that at each game, only 1 pitcher
pitches first. Recall, for example, that X1,X2,X3,X4 all refer to the
games against Des Moines. Thus in this first constraint we impose that
only 1 player pitches first against Des Moines. That is, we allow only
one of these variables to be set to 1; the other ones must be set to
0. The same goes for the other constraints in this set.
The second set of constraints imposes that each player pitches first
only once in all the games. For example, X1,X5,X9,X13 all refer to
pitcher Jones in each the games. With the constraint, we impose that
we set to 1 only one of these variables, and that all others must be
set to 0.
Now it's also clear why we must use Integer Programming or Mixed
Programming. These methods would allow only integer values for the
variables; and the constraints of the problem would allow that these
integers be either 0 or 1. In "normal" Linear Programming we can get
fractional values for the solutions, but it wouldn't make sense in
this problem. For example, setting X1=1/2 and X2=1/2 would still
fulfill the first constraint, but would mean that "half" Jones pitches
against Des Moines and "half" Baker also pitches against Des Moines.
Anyway, I went ahead and tried to use QM to solve this, so I had to
use Linear Programming. Fortunately, I got integer results, meaning
that it would have been the same to apply any of the other 2 methods.
But keep in mind that this is probably a feature of this problem
alone: it would have been perfectly possible that I got fractional
values using Linear Programming. The solution I got was:
X4=1, X5=1, X10=1, X15=1, and all other variables equal to 0. The
interpretation is:
Against Des Moines, let Wilson pitch first
Against Davenport, let Jones pitch first
Against Omaha, let Baker pitch first
Against Peoria, let Parker pitch first
The sum of probabilities of winning using this strategy is 2.9.
I hope this helps! If you have any doubt regarding my answer, please
request a clarification before rating it. Otherwise, I await your
rating and final comments.
Best wishes!
elmarto |
