The Solver and Mathematical Programing

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The Solver and MathematicalProgramming


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Overview

Introduction

Formulating Mathematical Programs

The Excel SolverApplications of the Solver

Summary


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Introduction

Formulating a mathematical program by determining its decision

variables, constraints, and objective function

The difference between linear, integer, and nonlinear programmingproblems

Using the Excel Solver to solve a mathematical program

Preparing the spreadsheet with the model parts and then enter thecorresponding cells into the Solver window

Reading the Solver reports

Example linear, integer, and nonlinear programming problems


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Formulating Mathematical Programs

Parts of the Mathematical Program

Linear, Integer, and Nonlinear Programming


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Parts of the Mathematical Program

Decisio n Variables= variables assigned to a quantity or response thatmust be determined in the problem

Object ive Funct io n= equation which states the goal of the model Maximize

Minimize

Constraints= equations which state limitations of the problem

To solve the model, each constraint must be considered simultaneously

in conjunction with the objectivefunction


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Linear Integer and Nonlinear Programming

Linear Programm ingproblem = there is a linear relationship among all

constraints and the objectivefunction

Integer Programm ingproblem = decisionvariables can only take

integer values in a given range (these integer values can also be

boolean= 0 or 1 only)

Nonl inear Programm ing problem = do not have a linear objective

function and/or constraints. NLP problems must use more challenging

methods to solve these complex equations.


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The Excel Solver

The Solver Steps

Understanding Solver Reports


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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: state any constraints

Step 2: Prepare the Spreadsheet

Step 2.1: Enter the decision variables Step 2.2: Enter the constraints

Step 2. 3: Enter the objective function

Step 3: Solve the model with the Solver

Step 3.1: Set the Target Celland choose Minor Max Step 3.2: Select Changin g Cel ls

Step 3.3: Add Constraints

Step 3.4: Set SolverOpt ions

Step 3.5: Solve and review Results


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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 inorder to maximize their profits. This is known as the Product Mix problem.

Production of each product type requires labor and raw materials; but thecompany is limited by the amount of resources available. (Labor hours available4500 and Raw Material available 1600 boards)

There is also a limited demand for each product, and no more than this demandper product type can be produced. Input tables for the necessary resources andthe demand are given.

Product

Type 1

Product

Type 2

Product

Type 3

Product

Type 4

Product

Type 5

Product

Type 6

Labor 6 5 4 3 2.5 1.5

Raw Material 3.2 2.6 1.5 0.8 0.7 0.3

Unit price $12.50 $11.00 $9.00 $7.00 $6.00 $3.00

Variable cost $6.50 $5.70 $3.60 $2.80 $2.20 $1.20

Demand 960 928 1041 977 1084 1055


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Step 1

Decision Variables: The amount produced of each product type

x1, x2, x3, x4, x5, x6

Objective Function: Maximize Profitz = 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


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Step 2

The spreadsheet should have each part of the model clearly entered


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Step 3

The Solverparameters can now be set according to the cell references

with the appropriate model parts


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Step 3 (contd)

SolverOptionsshould also now be set


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Figure 8.11

The results of the Solver are shown

All constraints are met


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Applications of the Solver

Transportation Problem (Linear Programming)

Workforce Scheduling (Integer Programming)

Capital Budgeting (Integer Binary Programming)

Warehouse Location (Non-Linear Programming)


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Transportation Problem

A company ships their products from three different plants (one in LA, one inAtlanta, 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 eachregion has a demand of products they must receive.

There is a different transportation cost between each plant, or each city, andeach region.

The company wants to determine how many products each plant should ship toeach region in order to minimize the total transportation cost.

EAST MIDWEST SOUTH WEST CAPACITYLA $5.00 $3.50 $4.20 $2.20 10000

ATLANTA $3.20 $2.60 $1.80 $4.80 12000

NEW YORK CITY $2.50 $3.10 $3.30 $5.40 14000

DEMAND 9000 6000 6000 13000


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Transportation Problem (contd)

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


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Prepare the spreadsheet


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The Solver window


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The solution


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Workforce Scheduling

A company wants to schedule its employees for every day of the week.

Employees work 5 days consecutively, so the company wants toschedule on which day each employee starts working; that is how manyemployees 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 totalnumber of employees working for the week.

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Number needed 17 13 15 17 9 9 12


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Workforce Scheduling (contd)

Decision variables:

The number of employees that will begin working (for 5 consecutive days) oneach 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 ofemployees needed on that particular day

objective function

Minimize the total number of employees needed


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Figure 8.23



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Figure 8.24

The Solver window


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Figure 8.25

The solution


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Integer constraint must be added


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Updated solution


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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 ineach project, given a limited amount of yearly funds available, in order tomaximize the total NPV of the investment.


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Capital Budgeting (contd)

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


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Inputs

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NPV Cost Year 1 Cost Year 2 Cost Year 3 Cost Year 4 Cost Year 5 Cost Year 6

Project 1 $928 $398 $180 $368 $111 $108 $123

Project 2 $908 $151 $269 $248 $139 $86 $83

Project 3 $801 $129 $189 $308 $56 $61 $23

Project 4 $543 $275 $218 $220 $54 $70 $59

Project 5 $944 $291 $252 $228 $123 $141 $70

Project 6 $848 $80 $283 $285 $119 $84 $37

Project 7 $545 $203 $220 $77 $54 $44 $42Project 8 $808 $150 $113 $143 $67 $101 $43

Project 9 $638 $282 $141 $160 $37 $55 $64

Project 10 $841 $214 $254 $355 $130 $72 $62

Project 11 $664 $224 $271 $130 $51 $79 $58

Project 12 $546 $225 $150 $33 $35 $107 $63

Project 13 $699 $101 $218 $272 $43 $90 $71Project 14 $599 $255 $202 $70 $3 $75 $83

Project 15 $903 $228 $351 $240 $60 $93 $80

Project 16 $859 $303 $173 $431 $60 $90 $41

Project 17 $748 $133 $427 $220 $59 $40 $39

Project 18 $668 $197 $98 $214 $95 $96 $74

Project 19 $888 $313 $278 $291 $66 $75 $74

Project 20 $655 $152 $211 $134 $85 $59 $70Available $2,500 $2,800 $2,900 $900 $900 $900


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A binary constraint

The Solver window


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The solution


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Warehouse Location

A company stores all of its products in one warehouse.

It has customers in cities around the United States and is trying todetermine the best location of their warehouse in order to minimize thetotal transportations costs.

Each citys location is given by its latitude and longitude. The number of

shipments made to each city is also given. We are to determine thewarehouse location based on its latitude and longitude values.


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Warehouse Location (contd)

Decision variable:

The latitude and longitude values of the location of the warehouse

Constraints:

The latitude and longitude for the warehouse location must be between thevalues of 0 and 120

Objective function:

Minimize the total distance traveled from the warehouse to each city

Distance=69*SQRT((WHLat-state_lat)^2+(WHLong-state_long)^2)


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The Solver window


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The solution

S


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Summary

The three parts of a mathematical model, decision variables, objective

function, and constraints.

The three primary types of mathematical models are linear, integer, andnonlinear programming problems.

Using Solver involves three main steps: reading and interpreting theproblem to determine the three parts of the model; preparing thespreadsheet so that Solver can read the data; and running the Solver.

LP examples are Transportation and Workforce Scheduling. An IPexample is Capital Budgeting, and an NLP example is the WarehouseLocation problem.

The Solver and Mathematical Programing

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