Spreadsheet-Based Decision Support Systems
Chapter 8: Solving Mathematical Programs
Prof. Name
Position
University Name
name@email.com
(123) 456-7890
Overview





Introduction
Formulating Mathematical Programs
The Risk Solver Platform
Applications of the Solver
Summary
2
Introduction

Formulating a mathematical program by determining its decision
variables, constraints, and objective function

The difference between linear, integer, and nonlinear programming
problems

Using the Risk Solver Platform to solve a mathematical program

Preparing the spreadsheet with the model parts.

Updating the Risk Solver Task Pane

Reading the Solver reports

Example linear, integer, and nonlinear programming problems
3
Formulating Mathematical Programs

Parts of the Mathematical Program

Linear, Integer, and Nonlinear Programming
4
Parts of the Mathematical Program

Decision Variables = variables assigned to a quantity or response that
must be determined in the problem

Objective Function = equation which states the goal of the model
– Maximize
– Minimize

Constraints = equations or inequalities which state limitations or
requirements of the problem

To solve the model, each constraint must be considered simultaneously
in conjunction with the objective function
5
Linear Integer and Nonlinear Programming

Linear Programming problem = there is a linear relationship among all
constraints and the objective function

Integer Programming problem = decision variables can only take
integer values in a given range (these integer values can also be
boolean = 0 or 1 only)

Nonlinear Programming problem = do not have a linear objective
function and/or constraints. NLP problems must use more challenging
methods to solve these complex equations.
6
The Risk Solver Platform

The Risk Solver Platform Steps

Understanding Solver Reports
7
The Solver Steps

Step 1: Read and Interpret the Problem
– Step 1.1: Determine the Decision Variables
– Step 1.2: State the Objective Function
– Step 1.3: Define any Constraints

Step 2: Prepare the Spreadsheet
–
–
–
–

Step 2.1: Place the Input Table
Step 2.2: Set the Decision Variables Cells
Step 2.3: Enter the Constraint Formulas
Step 2.4: Enter the Objective Function Formula
Step 3: Solve the model with Risk Solver Platform
–
–
–
–
–
Step 3.1: Set Objective.
Step 3.2: Select Variables
Step 3.3: Add Constraints
Step 3.4: Set Solver Options
Step 3.5: Solve the model and review Results
8
Product Mix Problem

A company produces six different types of products. They want to
schedule their production to determine how much of each product type
should be produced in order to maximize the profits. This is known as the
Product Mix problem.

Production of each product type requires labor and raw materials; but the
company is limited by the amount of resources available.

There is also a limited demand for each product, and no more than this
demand per product type can be produced. Input tables for the
necessary resources and the demand are given.
9
Step 1

Decision Variables: The amount produced of each product type
– x1, x2, x3, x4, x5, x6

Objective Function: Maximize Profit
– z = p1*x1 + p2*x2 + p3*x3 + p4*x4 + p5*x5 + p6*x6

Constraints:
– Labor Constraint:
 l1*x1 + l2*x2 + l3*x3 + l4*x4 + l5*x5 + l6*x6 <= available labor = 4500
– Raw Material Constraint
 r1*x1 + r2*x2 + r3*x3 + r4*x4 + r5*x5 + r6*x6 <= available raw material = 1600
– Demand Constraint:
 xi <= Di
for i = 1 to 6
10
Step 2

The spreadsheet should have each part of the model clearly defined.
11
Figure 8.2

The Labor and Raw Material constraint formulas use the SUMPRODUCT
function.
12
Figures 8.3 and 8.4

For the demand constraint, ensure that the values in our decision
variable range are less than each of the corresponding values in the
“Demand” range.

The objective function formula is also written with the SUMPRODUCT
function.
13
Step 3

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.

To use the Solver:
– We activate the Solver Task Pane:
 Click on: Risk Solver Platform > Model > Model
command.
– Use the Optimization tools listed in the Task
Pane to update problem’s Objective, Variables,
and Constraints
– Alternatively, use the commands listed in Risk
Solver Platform tab on the Ribbon.
14
Figure 8.6
Set the objective function:
 Select cell where the objective function formula is typed.
 Click on: Risk Solver Platform > Optimization Model > Objective
command.
 From the drop-down list that appears, select Max, and click on the
Normal option from the fly-out menu.
15
Figure 8.7

The selected Objective cell will appear in the
task pane.

Use the Objective window located in the
bottom of the task pane to change the
address, sense or value of the objective cell.
16
Figure 8.8
Select decision variables:
 Highlight the decision variable cells
 Click on: Risk Solver Platform > Optimization Model > Decisions
command.
 From the drop-down list that appears, select Normal option.
17
Figure 8.9

Select the objective cell; and click on: Risk Solver Platform >
Optimization Problem > Decisions command.

Select Plot from
the Decisions dropdown menu.

The graph confirms
that the objective
function of the
Product Mix
problem is linear.
18
Figure 8.10
Add constraints:
 Click on: Risk Solver Platform > Optimization Model > Constraints
command.
 From the Constraints drop-down menu, select Normal and then click
on the <= sign on the fly-out menu
19
Figures 8.11 and 8.12

Each added constraint should include:
– The cell with the constraint formula.
– The cell with the RHS value, or a directly entered numerical value.

We can define more than one constraint at a
time.

The task pane lists all of the constraints of a
problem.

Use the Normal Constraint window of the
task pane to change the left hand side, the
RHS and sense of a constraint.
20
Figure 8.13

The final Risk Solver Task Pane lists
– Decision variables
– Constraints
– Objective function of the model.

Click on the Analyze without Solving button
located in the upper-right corner of the task pane.

The Model Diagnosis window in the bottom half of
the task pane presents a summary of model
characteristics.
21
Figure 8.14

We modify some of the Risk Solver’s
Platform and Engine related options
before we solve the model.
– Change the lower bound of the decision
variables (Decision Vars Lower) to 0.
– Set the Solve Mode to Solve Complete
Problem.
– Set the Intended Model Type as Linear.

The rest of the options is kept at their
default values.
22
Figure 8.15

Max Time, is the maximum time that the
Solver should take to find a solution to the
model.

The number of Iterations affects the
number of iterations for which the Solver’s
algorithm will run.

Primal (Dual) Tolerance is an upper bound
on the amount by which the primal (dual)
constraints can be violated and be
considered feasible.

Set the value of Show Iterations to True if
you want Solver to pause at every iteration.
23
Figure 8.15 (cont’d)

If you set the value of Use Automatic Scaling to True, then Solver rescales the values of the objective function and constraints internally.

Set the value of Bypass Solver Reports to False if you do not need the
reports related to the current solution run.

Set the value of Presolve to True to allow Solver to perform a presolve
step prior to applying the Primal or Dual Simplex method.

Select either option from Derivates drop-down list to determine how
Solver computes derivates when solving Quadratic Programming (QP)
problems.
24
Figure 8.16

To run the solver:
– Click on: Risk Solver Platform > Solve
Action > Optimize command.
– From the Optimize drop-down menu
select Solve Complete Problem.
– Alternatively, click on the green arrow on
the top-right corner of the task pane.

During the time that Solver seeks for a
solution, the Output tab of the task pane
becomes active and presents a
description of the different events that
occur while the problem is being solved.
25
Figure 8.17


If Solver finds an optimal solution, then, we observe this solution in the
background of our spreadsheet.
This solution indicates that all constraints are met.
26
Infeasibility

In an infeasible problem at least one of
the constraints cannot be met.

Example: Let consider this problem is
infeasible due to “Demand” not being met.

Select the “Demand” (Bound) Constraint
from the Model tab of task pane, and
change the Relation property to >=
Constraint.
27
Figure 8.19

The Output window of the task pane
conveys that the Solver could not find
a feasible solution with this modified
constraint.

This solution is infeasible because
there are not enough resources
available to meet the demand,.
28
Figure 8.20

The Feasibility report identifies the exact constraints that are violated
by the current solution.

The feasibility report for this example indicates that constraint
H13 >= H15 is violated.
29
Figure 8.21

The results of this infeasible solution are shown.
30
Unboundedness

In an unbounded problem the
objective function can reach an
unreasonably large number (if we are
maximizing) or small number (if we are
minimizing).

Such a situation implies that the
constraints are not inclusive enough.
31
Figure 8.23

To observe unboundedness:
– Select Objective from Solver task pane
and change its sense to Minimize.
– Set the value of Assume NonNegative to False from the General
properties window of the Engine tab of
the task pane.
– Solve the problem.

The Solver indicates that objective
values did not converge.
32
Solver Reports

The three main reports available when using the Risk Solver Platform
are the Answer report, the Sensitivity report, and the Limits report.

To generate these reports, click on: Risk Solver Platform > Analysis
> Reports command.

Select Optimization from the Reports drop-down menu. Next, select
Answer, Sensitivity or Limits reports from the fly-out menu.
33
Answer Report

The Answer Report
provides the original and
final values of the
Objective Cell, Decision
Variable Cells, and the
Constraints.
34
Sensitivity Report

The Sensitivity Report provides information about the Decision
Variable Cells and the Constraints as well as their final values.
35
Limits Report


The Limits Report provides information about the Objective Cell and
the Decision Variable Cells.
It also includes the value of each cell.
36
Applications of the Solver

Transportation Problem

Workforce Scheduling

Capital Budgeting

Warehouse Location
37
Transportation Problem

A company ships their products from three different plants (one in LA,
one in Atlanta, and one in New York City) to four regions of the United
States (East, Midwest, South, West).

Each plant has a capacity on how many products can be sent out, and
each region has a demand of products they must receive.

There is a different transportation cost between each plant, or each city,
and each region.

The company wants to determine how many products each plant should
ship to each region in order to minimize the total transportation cost.
38
Transportation Problem (cont’d)

Decision variables:
– The amount to ship from each plant to each region

Constraints :
– Demand: the total number of products received by a region (from each plant)
is greater than or equal to its demand
– Capacity: the total number of products shipped from a plant (to each region)
is less than or equal to its capacity

Objective function:
– Minimize the total transportation costs
39
Figure 8.27

Prepare the spreadsheet
40
Figure 8.28





Set the Objective Cell and choose Min for our objective function.
Set the Variables
Add both the capacity and demand constraints to the constraint list.
Select Standard LP/Quadratic Engine as the solution method.
Set the value of Assume Non-Negative property to True.
41
Figure 8.29

The solution
42
Workforce Scheduling

A company wants to schedule its employees for every day of the week.

Employees work 5 days consecutively, so the company wants to
schedule on which day each employee starts working; that is how many
employees start working each day.

There is a certain number of employees needed each day of the week.

The objective function is to find the schedule which minimizes the total
number of employees working for the week.
43
Workforce Scheduling (cont’d)

Decision variables:
– The number of employees that will begin working (for 5 consecutive days) on
each day of the week

Constraints:
– The total number of employees working on a given day (regardless of which
day they started working) is greater than or equal to the number of
employees needed on that particular day

Objective function:
– Minimize the total number of employees needed
44
Figure 8.30

Prepare the spreadsheet
45
Figure 8.31





Specify the Objective and choose Min for the objective function.
Set the Variables
Add the Constraint to the constraint list..
Specify Standard LP/Quadratic Engine for solving the problem.
Set the Assume Non-Negative property to True.
46
Figure 8.32

The solution
47
Figures 8.33 and 8.34

Integer constraint must be added
48
Figure 8.35

Updated solution
49
Figure 8.36

What happens to the total
number of employees required
when the number of employs
needed on a Saturday increases
from 9 to 16?

Modify the model:
-Cell I19: =PsiOptParam(M10:M17)
-Cell N10: = PsiOptValue($C22$,L10)
-Copy N10 to N11:N17
-Set the value of Optimizations to
Run property to 8 in the Platform tab
of Solver task pane.

Solve the problem.
50
Figure 8.37

The PsiOptParam() function
of Risk Solver Platform
supports multiple
parameterized optimizations.

The results present the total
number of employees needed
as Saturday requirements
increase.

To observe the solution of a
run, select the corresponding
Opt # from Risk Solver
Platform > Tools group.
51
Capital Budgeting

There are 20 projects that a company, or individual, can invest in.

Each project has a net present value (NPV) and cost per year given.

The company, or investor, wants to determine how much to invest in
each project, given a limited amount of yearly funds available, in order to
maximize the total NPV of the investment.
52
Capital Budgeting (cont’d)

Decision variables:
– Which projects we do and do not invest in.

Constraints:
– No more than the yearly available funds can be spent each year.

Objective Function:
– Maximize the total NPV.
53
Figure 8.38

Input table for the capital budgeting problem
54
Figure 8.39

Prepare the spreadsheet
55
Figures 8.40 and 8.41

Add binary constraint.
56
Figure 8.42

Select the Standard Evolutionary Engine
as the solution method for this problem.

Set the value of Assume Non-Negative
property to True.

Set the values of Convergence, and
Maximum Time without Improvement as
shown. Keep the rest of the parameters at
their default values.
57
Figure 8.43

The Solver runs several iterations of
the Genetic Algorithm to solve the
problem.

The objective function value for
each iteration is plotted in the task
pane.

The Integer Gap for the solution
found is zero. This solution is
optimal.
58
Figure 8.44

The solution
59
Warehouse Location

A company stores all of its products in one warehouse.

The company has customers in cities around the United States. The
company wants to determine the best location of their warehouse in
order to minimize the total transportations costs.

Each city’s location is given by its latitude and longitude. The number of
shipments made to each city is also given. We are to determine the
warehouse location based on its latitude and longitude values.
60
Warehouse Location (cont’d)

Decision variables:
– The latitude and longitude values of the location of the warehouse.

Constraints:
– The latitude and longitude for the warehouse location must be between the
values of 0 and 120.

Objective function:
– Minimize the total distance traveled from the warehouse to each city.
61
Figure 8.45

Prepare the spreadsheet
62
Figure 8.46

Calculating the distance between each city and the possible warehouse
location
63
Figure 8.47

The objective function formula
64
Figures 8.48 and 8.49


The completed optimization model is shown.
The Standard GRG Nonlinear Engine is used to solve this nonlinear
programming problem.
65
Figure 8.50

The Output tab of Solver’s task pane
indicates that the Solver has converged
to the current solution.
66
Figure 8.51

The solution
67
Summary

The three parts of a mathematical model are decision variables, objective
function, and constraints.

The three primary types of mathematical models are linear, integer, and
nonlinear programming problems.

Using Risk Solver Platform involves three main steps: reading and interpreting
the problem to determine the three parts of the model; preparing the spreadsheet
so that Solver can read the data; and running the Solver.

LP examples are Transportation and Workforce Scheduling. An IP example is
Capital Budgeting, and an NLP example is the Warehouse Location problem.

We use the multiple parameterized optimization capabilities of Risk Solver
Platform to solve a multi-scenario Workforce Scheduling problem.
68
Additional Links

(place links here)
69