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Using The Excel Solver To Solve Mathematical Programs
summarize “Chapter 8: Using The Excel Solver To Solve Mathematical Programs.” I want pictures. I want animations in each slide for written language, pictures and slid.
Chapter Overview
8.1
Introduction
8.2
Formulating Mathematical Programs
8.2.1
Parts of the Mathematical Program
8.2.2
Linear, Integer, and Nonlinear Programming
8.3
The Excel Solver
8.3.1
The Solver Steps
8.3.1.1
Standard Solver
8.3.1.2
Premium Solver
8.3.2
A Solver Example
8.3.2.1
Product Mix
8.3.2.2
Infeasibility
8.3.2.3
Unboundedness
8.3.3
Understanding Solver Reports
8.4
Applications of the Solver
8.4.1
Transportation Problem
8.4.2
Workforce Scheduling
8.4.3
Capital Budgeting
8.4.4
Warehouse Location
8.5
Limitations and Manipulations of the Solver
8.6
Summary
8.7
Exercises
Chapter 8
Using The Excel Solver
To Solve Mathematical Programs
Chapter 8: Using The Excel Solver
To Solve Mathematical Programs
2
8.1
Introduction
This chapter illustrates how to use the Excel Solver as a tool to solve mathematical
programs. We review the basic parts of formulating a mathematical program and present
several examples of how the Solver interprets these parts of the program from the
spreadsheet. We give examples of linear, integer, and non-linear programming problems
to show how the Solver can be used to solve a variety of mathematical programs. We
also give an overview of the Premium Solver and its benefits. This chapter is important
for the reader to understand as many DSS applications involve solving optimization
problems, which are mathematical programs. The reader should be comfortable with
preparing the spreadsheet for use with the Solver. In Chapter 19, we revisit the Solver
using VBA commands. We have several examples of DSS applications which use the
Solver to solve optimization problems, su
ch as Portfolio Management and Optimization.
8.2
Formulating Mathematical Programs
The Excel spreadsheet is unique because it is capable of working with complex
mathematical models. Mathematical models
transform a word problem into a set of
equations that clearly define the values that we are seeking, given the limitations of the
problem. Mathematical models are employed in
many fields, including all disciplines of
engineering. In order to solve a mathemat
ical model, we develop a mathematical
program which can numerically be solved and re
translated into a qualitative solution to
the mathematical model.
8.2.1
Parts of the Mathematical Program
A mathematical program consists of three main parts. The first is the
decision
variables
.
Decision variables
are assigned to a quantity or response that we must
determine in a problem. For example, if a toy manufacturer wants to determine how
many toy boats and toy cars to produce, we assign a variable to represent the quantity
of toy boats produced,
x
1
, and the quantity of toy cars produced,
x
2
. Decision
variables
are defined as
negative
,
non-negative,
or
unrestricted
. An
unrestricted
variable can be
either
negative
or
non-negative.
These variables represent all other relationships in a
mathematical program, including the objective, the limitations, and the requirements.
The second part of the math program, called the
objective function
, is an equation that
states the goal, or objective, of the model. In the same example of the toy manufacturer,
we want to know the quantities of toy boats and toy cars to produce. However, the goal
of the manufacturing plant’s production may be to increase profit. If we know that we can
profit $5 for every toy boat and $4 for every toy car, then our objective function is:
Maximize 5x
1
+ 4x
2
In other words, we want profit to drive us in determining the quantity of boats and cars to
produce. Objective functions are either
maximized
or
minimized
; most applications
involve maximizing profit or minimizing cost.
The third part of the math progam, the
constraints
, are the limitations of the problem.
That is, if we want to maximize our profit, as in the toy manufacturer example, we could
produce as many toys as possible if we di
d not have any limits. However, in most
Chapter 8: Using The Excel Solver
To Solve Mathematical Programs
5
8.3
The Excel Solver
We will now discuss how to operate these two versions of the Solver. In general, the
Solver must understand the problem’s mathematical program parts, which we take care
of by preparing our spreadsheet to contain distinct cells for the decision variables,
constraints, and objective function. We must then tell the Solver if we want to minimize
or maximize the problem, or if we want to solve it for a particular value of the objective
function. There are also several options that we can apply to give more specific
instructions to the Solver for solving the problem.
(Note: To find the Solver, go to
Tools > Solver
from the menu options. If you do not see
Solver
in the
Tools
menu, you must first choose the
Solver Add-In
. To do so, select the
Add-In
option from the
Tools
menu. A small dialog box will appear; from there, select
Solver
Add-In
from the list. If you do not see
Solver Add-In
in the
Add-In
list, click
Browse
and look for the
Solver.xla
file from the following directory:
C Drive
>
Program
Files
>
Microsoft Office
>
Office (or Office10
) >
Library
>
Solver
. Double-click this file.
Now you should find
Solver
Add-In
in the list; check the box next to it. Restart Excel. If
you do not find the
Solver.xla
file, go to the
Add-Ins
window as explained above; select
Solver Add-in
and press
OK
. Insert the MS Office CD in CD-ROM drive when asked.)
8.3.1
The Solver Steps
To operate the
Solver
, we must follow a short sequence of steps: 1) read and interpret
the problem; 2) prepare the spreadsheet; and 3) solve the model and review the results.
We will now describe these steps in detail for both the Standard Solver and the Premium
Solver.
The Standard Solver
STEP 1: Read and Interpret the Problem
We must first determine the type of problem that we are dealing with (linear
programming, Integer Programming, or nonlinear programming) and outline the model
parts. Whether the problem is an LP, IP, or NLP model does not affect the model parts
but does affect the Options that we specify for the Solver. They may also require some
additional constraint specifications. In each case, we still need to determine the decision
variables, the objective function, and the constraints. We need to write these
mathematically, with the objective function and constraints in terms of the decision
variables.
STEP 2: Prepare the spreadsheet
Next, we transfer these parts of the model into our Excel spreadsheet, clearly defining
each part of our model in the spreadsheet. The
Solver
interprets our model according to
the location of these model parts on the spreadsheet.
Chapter 8: Using The Excel Solver
To Solve Mathematical Programs
6
STEP 2.1
: Place the Input Table
Usually the input for the problem is provided for us. We just need to place it on the
spreadsheet in the form of a table. We reference this input when forming our constraint
and objective function formulas.
STEP 2.2
: Set the Decision Variables Cells
Next, we list the decision variables in individual cells with an empty cell next to each one.
The
Solver
places values in these cells for each
decision variable as it solves the model.
We recommend naming the range of decision variables for easier reference in constraint
and objective function formulas.
STEP 2.3
: Enter the Constraint Formulas
Now we place the constraint equations in the spreadsheet; we enter them separately
using formulas, with an optional description
next to each constraint. As each constraint
is in terms of the decision variables, all of these formulas must be in terms of the
decision variable cells already defined.
Another important consideration when laying out the constraints in preparation for the
Solver
is that there must be individual cells for the right-hand side (RHS) values as well.
We should also place all inequality signs in their own cells. This organization will become
clear once we explain how the
Solver
interprets our model.
Another advantageous way to keep our constraints organized as we use the
Solver
is to
name cells. We can also group constraints that have the same inequality signs. The
benefit of this habit will become apparent once we input the model parts for the
Solver
.
STEP 2.4
: Enter the Objective Function Formula
We can now place our objective function in a cell by transforming this equation into a
formula in terms of the decision variables. The spreadsheet is now prepared for the
Solver
with all three parts of the model clearly displayed.
STEP 3: Solve the Model with the
Solver
The
Solver
can now interpret this information and use algorithms to solve the model. The
Solver
receives the decision variables, constraint equations, and objective function
equation as input into a hidden programming code that applies the algorithm to the data.
We will explain in more detail how this programming works when we discuss VBA. To
use
Solver
, we choose
Tools > Solver
from the menu; the window in Figure 8.3 then
