Introduction to Solver
This document is intended to introduce the use of Excel Solver to solve linear
programming (LP) problems. We will use the example of Malcolm’s glass
molding operation. The sample Excel screens shown here are from Excel 2007;
other versions of Excel have similar features.
Maximize:
Z =
500X1 +
450X2
6 X1
1 X1
10 X1
1 X1
0 X1
5 X2
0 X2
20 X2
0 X2
1 X2
Subject to:
+
+
+
+
+





60
8
150
0
0
Figure 1 shows the LP Formulation worksheet.
Figure 1 LP Formulation worksheet
When you set up a problem for Solver, you follow four steps:
1. Create a spreadsheet with all the information from the LP problem. In the
example (Figure 1) you can see the LP information in the first 11 rows.
2. Specify the target cell (which contains the formula for the objective function)
that you want to minimize, maximize, or set to a certain value. In
MALCOLM.XLS, the target cell is E4.
3. Specify the changing cells (called decision variables in LP lexicon) that you
want to be adjusted until a solution is found. You can specify up to 200 changing
cells, but for this problem we only need 2.
The number of cases of 6-oz. glasses to be produced appears in cell B2 (in
hundreds of cases), and the number of cases of 10-oz. glasses to be produced
appears in cell C2. It doesn’t matter what exact values you put into these cells,
because Solver will change them anyway, but you do need to put some starting
values in the cells. In this example the starting values are set to 1.
4. Specify constraint cells that must fall within certain limits or satisfy target
values. You can specify up to 500 constraints — 2 for each changing cell — plus
100 additional constraints, representing a total of no more than 1000 cells in a
problem. In this example there are only five constraints.
In the example, the constraint cells are D7 through D11. Each of these cells
contains a formula which is a function of the decision variables. For example, cell
D7 contains a formula for how much molding capacity Malcolm is using, based
on the current levels of production for the two products.
After setting up the problem, select Add-Ins - Solver to open up the Solver
Parameters dialog box (Figure 2). Note that the drop-down menu on the right
(available with Premium Solver) says “Standard LP Simplex”. If you don’t have
this menu, don’t worry about it.
Figure 2 Blank Solver Parameters dialog box
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In this dialog box, you specify (1) the target cell containing the formula for the
objective function (E4 in the example); (2) the range of cells containing the
decision variables, which Solver calls changing cells (B2:C2 in the example); (3) a
set of constraints, clicking on the Add button to use the Add Constraint dialog
box (Figure 3). Don’t forget to choose the appropriate Equal to radio button for
the problem; in this case we want to maximize profit so we select Max.
Figure 3 Add Constraint dialog box
The completed Solver Parameters dialog box for the example problem appears as
Figure 4.
Figure 4 Completed Solver Parameters dialog box
If you don’t have Premium Solver, you’ll need to click on the “Options” button
and check “Assume Linear Model” (equivalent to the “Standard LP Simplex”
selection in the drop-down menu in Premium Solver).
Note that we have created non-negativity constraints in this model with the
constraint that says $D$10:$D$11>=$F$10:$F$11. An easier way to create nonnegativity constraints is to click on the “Options” button and check “Assume
Non-negative”. In English, that means that all decision variables must be at least
zero. Don’t worry about the other stuff in the Options box.
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Premium Solver
Non-Premium Solver
Figure 5 Completed Options dialog box
After setting up the problem, click on the Solve button and wait for Solver to
find a solution. You can create three types of reports that summarize the results
of a successful solution process. For the purposes of this course, we are interested
in the Answer Report (Figure 6) and the Sensitivity Report (Figure 7).
Figure 6 Answer Report
In the Answer Report (Figure 6) we see several important facts about the Solver
solution: (1) the solution will yield a profit of $5,142.86; (2) the solution calls for
Malcolm to produce 643 cases of 6-oz glasses and 429 cases of 10-oz glasses; (3)
the binding constraints (the constraints which prevent Malcolm from increasing
his profit) are the limits on molding capacity and storage space. This is important
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information because it quickly tells Malcolm that there is no point in trying to
increase sales of 6-oz glasses.
From the information in the Sensitivity Report (Figure 7), we can make a number
of inferences about the Solver solution, most of which are beyond the scope of
this course. However, you should pay attention to the column called Shadow
Price in the Constraints table. There is a shadow price associated with each of the
constraints, as you can see in the table.
The shadow price can be interpreted as the amount that the objective function
would improve for every unit of relaxation in the constraint. For example, the
shadow price associated with the Storage Space constraint is $2.86. This can be
interpreted to mean that Malcolm can increase his profit if he can acquire
additional storage space at less than $2.86 per unit. You may recall that storage
space was defined in hundredths in our formulation, so Malcolm should be
prepared to spend up to $.0286 per cubic foot per week for additional storage
space.
Figure 7 Sensitivity Report
Notes on using Solver:
(1) Notice that the Solver solution suggests that Malcolm produce only 643 cases
of 6-oz glasses. Remember that Malcolm only has one customer for this product
and that she will buy no more than 800 cases. This is an example of an LP
formulation that may be leaving out important information. It may be that this
customer needs more than 643 cases and will abandon Malcolm as a supplier
unless he can produce all the 6-oz glasses she needs. It would be possible to
reformulate the problem with a new constraint, setting some minimum quantity
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of 6-oz glasses, but many of the nuances of management cannot be captured in a
mathematical program.
Malcolm may want to call his customer before implementing the solution and
find out whether or not 643 cases will be satisfactory to her. He might decide to
implement a sub-optimal plan, forgoing some profit this week in order to keep a
loyal customer. The point here is that the results of any mathematical model
ought to be scrutinized with common sense before being implemented.
(2) Solver is a relatively new optimization tool, and does not yet completely
replace older and more powerful software (e. g. Lindo, Storm, AMPL). In this
class we are using Solver for its accessibility, as a vehicle to introduce basic
concepts of linear optimization with a familiar spreadsheet package. To solve
more complicated problems, you will probably want to take a course in
mathematical programming or solicit the help of someone who has.
(3) Some users have reported computational problems with Solver. There are
some problems, for example, for which Solver comes up with different answers
depending on the initial values of the changing cells. One way to help avoid
problems with this is to solve your problem several times using different sets of
initial changing cell values.
(4) When selecting reports for Solver to generate, hold down the Shift key to
select more than one type of report at a time.
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