Forecasting: Time series and trend analysis

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Quantitative Analysis for Management

Thirteenth Edition

Chapter 10

Integer Programming, Goal Programming, and Nonlinear Programming

Learning Objectives

After completing this chapter, students will be able to:

10.1 Understand the difference between LP and integer programming.

10.2 Understand and solve the three types of integer programming problems.

10.3 Formulate and solve goal programming problems using Excel and QM for Windows.

10.4 Formulate and solve nonlinear programming problems using Excel.

Chapter Outline

10.1 Integer Programming

10.2 Modeling with 0-1 (Binary) Variables

10.3 Goal Programming

10.4 Nonlinear Programming

Introduction

There are other mathematical programming models that can be used when the assumptions of LP are not met

A large number of business problems require variables have integer values

Many business problems have multiple objectives

Goal programming permits multiple objectives

Nonlinear programming allows objectives and constraints to be nonlinear

Max profit = 25X1 − 0.4X12 + 30X2 − 0.5X22

Integer Programming (1 of 2)

An integer programming model is one where one or more of the decision variables has to take on an integer value in the final solution

Three types of integer programming problems

Pure integer programming – all variables have integer values

Mixed-integer programming – some but not all of the variables will have integer values

Zero-one integer programming – special cases in which all the decision variables must have integer solution values of 0 or 1

Integer Programming (2 of 2)

Solving an integer programming problem is much more difficult than solving an LP problem

Solution time required may be excessive

Harrison Electric Company Example of Integer Programming (1 of 6)

Company produces two products, old-fashioned chandeliers and ceiling fans

Both require a two-step production process involving wiring and assembly

It takes about 2 hours to wire each chandelier and 3 hours to wire a ceiling fan

Final assembly of the chandeliers and fans requires 6 and 5 hours, respectively

Only 12 hours of wiring time and 30 hours of assembly time are available

Each chandelier produced nets the firm $7 and each fan $6

Harrison Electric Company Example of Integer Programming (2 of 6)

Production mix LP formulation

Maximize profit = $7X1 + $6X2

subject to 2X1 + 3X2 ≤ 12 (wiring hours)

6X1 + 5X2 ≤ 30 (assembly hours)

X1, X2 ≥ 0

where

X1 = number of chandeliers produced

X2 = number of ceiling fans produced

Harrison Electric Company Example of Integer Programming (3 of 6)

FIGURE 10.1 Harrison Electric Problem

Harrison Electric Company Example of Integer Programming (4 of 6)

Production planner recognizes this is an integer problem

First attempt at solving it is to round the values to

X1 = 4 and X2 = 2

However, this is not feasible

Rounding X2 down to 1 gives a feasible solution, but it may not be optimal

This could be solved using the enumeration method

Generally not possible for large problems

Harrison Electric Company Example of Integer Programming (5 of 6)

TABLE 10.1 Integer solutions to the Harrison Electric Company Problem

CHANDELIERS (X1)	CEILING FANS (X2)	PROFIT ($7X1 + $6X2)
0	0	$0
1	0	7
2	0	14
3	0	21
4	0	28
5	0	35
0	1	6
1	1	13
2	1	20
3	1	27
4	1	34
0	2	12
1	2	19
2	2	26
3	2	33
0	3	18
1	3	25
0	4	24

Optimal solution to integer programming problem

Solution if rounding is used

Harrison Electric Company Example of Integer Programming (6 of 6)

The optimal integer solution is less than the optimal LP solution of $35.25

An integer solution can never be better than the LP solution and is usually a lesser value

Using Software (1 of 4)

PROGRAM 10.1A QM for Windows Input Screen for Harrison Electric Problem

Using Software (2 of 4)

PROGRAM 10.1B QM for Windows Solution Screen for Harrison Electric Problem

Using Software (3 of 4)

PROGRAM 10.2 Excel 2016 Solver Solution for Harrison Electric Problem

Using Software (4 of 4)

PROGRAM 10.2 Excel 2016 Solver Solution for Harrison Electric Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: D5 By Changing cells: B4:C4 To: Max Subject to the Constraints: D8:D9 >= F8:F9 B4:C4 = integer Solving Method: Simplex LP  Make Variables Non-Negative	Blank	Copy D5 to D8:D9

Mixed-Integer Programming Problem Example (1 of 3)

Many situations in which only some of the variables are restricted to integers

Bagwell Chemical Company produces two industrial chemicals

Xyline must be produced in 50-pound bags

Hexall is sold by the pound and can be produced in any quantity

Both xyline and hexall are composed of three ingredients – A, B, and C

Bagwell sells xyline for $85 a bag and hexall for $1.50 per pound

Mixed-Integer Programming Problem Example (2 of 3)

AMOUNT PER 50-POUND BAG OF XYLINE (LB)	AMOUNT PER POUND OF HEXALL (LB)	AMOUNT OF INGREDIENTS AVAILABLE
30	0.5	2,000 lb–ingredient A
18	0.4	800 lb–ingredient B
2	0.1	200 lb–ingredient C

Objective is to maximize profit

Mixed-Integer Programming Problem Example (3 of 3)

Let X = number of 50-pound bags of xyline

Let Y = number of pounds of hexall

A mixed-integer programming problem as Y is not required to be an integer

Maximize profit =	$85X +	$1.50Y	Blank	Blank
subject to	30X +	0.5Y	≤	2,000
Blank	18X +	0.4Y	≤	800
Blank	2X +	0.1Y	≤	200
Blank	Blank	X, Y	≥	0 and X integer

Using Software (1 of 3)

PROGRAM 10.3 QM for Windows Solution for Bagwell Chemical Problem

Using Software (2 of 3)

PROGRAM 10.4 Excel 2016 Solver Solution for Bagwell Chemical Problem

Using Software (3 of 3)

PROGRAM 10.4 Excel 2016 Solver Solution for Bagwell Chemical Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: D5 By Changing cells: B4:C4 To: Max Subject to the Constraints: D8:D10 <= F8:F10 B4 = integer Solving Method: Simplex LP  Make Variables Non-Negative	Blank	Copy D5 to D8:D10

Modeling With 0-1 (Binary) Variables

Demonstrate how 0-1 variables can be used to model several diverse situations

Typically a 0-1 variable is assigned a value of 0 if a certain condition is not met and a 1 if the condition is met

This is also called a binary variable

Capital Budgeting Example (1 of 3)

Common capital budgeting problem – select from a set of possible projects when budget limitations make it impossible to select them all

A 0-1 variable is defined for each project

Quemo Chemical Company is considering three possible improvement projects for its plant

A new catalytic converter

A new software program for controlling operations

Expanding the storage warehouse

It cannot do them all

Capital Budgeting Example (2 of 3)

Objective is to maximize net present value of projects undertaken

subject to Total funds used in year 1 ≤ $20,000

Total funds used in year 2 ≤ $16,000

TABLE 10.2 Quemo Chemical Company Information

PROJECT	NET PRESENT VALUE	YEAR 1	YEAR 2
Catalytic Converter	$25,000	$8,000	$7,000
Software	$18,000	$6,000	$4,000
Warehouse expansion	$32,000	$12,000	$8,000
Available funds	Blank	$20,000	$16,000

Capital Budgeting Example (3 of 3)

Decision variables

Formulation	Blank	Blank	Blank	Blank	Blank
Maximize NPV =	25,000X1 +	18,000X2 +	32,000X3	Blank	Blank
subject to	8,000X1 +	6,000X2 +	12,000X3	≤	20,000
Blank	7,000X1 +	4,000X2 +	8,000X3	≤	16,000
Blank	Blank	Blank	X1, X2, X3	=	0 or 1

Using Software (1 of 3)

PROGRAM 10.5 Excel 2016 Solver Solution for Quemo Chemical Problem

Using Software (2 of 3)

Optimal Solution

X1 = 1, X2 = 0, X3 = 1

Fund the catalytic converter and warehouse projects but not the software project

NPV = $57,000

Using Software (3 of 3)

PROGRAM 10.5 Excel 2016 Solver Solution for Quemo Chemical Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: E5 By Changing cells: B4:D4 To: Max Subject to the Constraints: E8:E9 <= G8:G9 B4:D4 = binary Solving Method: Simplex LP  Make Variables Non-Negative	Blank	Copy E5 to E8:E9
Blank	Blank	Blank

Limiting the Number of Alternatives Selected

One common use of 0-1 variables involves limiting the number of projects or items that are selected from a group

Suppose Quemo Chemical is required to select no more than two of the three projects regardless of the funds available

This would require adding a constraint

X1 + X2 + X3 ≤ 2

If they had to fund exactly two projects the constraint would be

X1 + X2 + X3 = 2

Dependent Selections

At times the selection of one project depends on the selection of another project

Suppose Quemo’s catalytic converter could only be purchased if the software was purchased

The following constraint would force this to occur

X1 ≤ X2 or X1 − X2 ≤ 0

If we wished for the catalytic converter and software projects to either both be selected or both not be selected, the constraint would be

X1 = X2 or X1 − X2 = 0

Fixed-Charge Problem Example (1 of 5)

Often businesses are faced with decisions involving a fixed charge that will affect the cost of future operations

Sitka Manufacturing is planning to build at least one new plant and three cities are being considered

Baytown, Texas

Lake Charles, Louisiana

Mobile, Alabama

Fixed-Charge Problem Example (2 of 5)

Constraints

Total production capacity at least 38,000 units each year

Number of units produced at the Baytown plant is 0 if the plant is not built and no more than 21,000 if the plant is built

Number of units produced at the Lake Charles plant is 0 if the plant is not built and no more than 20,000 if the plant is built

Number of units produced at the Mobile plant is 0 if the plant is not built and no more than 19,000 if the plant is built

Fixed-Charge Problem Example (3 of 5)

TABLE 10.3 Fixed and Variable Costs for Sitka Manufacturing

SITE	ANNUAL FIXED COST	VARIABLE COST PER UNIT	ANNUAL CAPACITY
Baytown, TX	$340,000	$32	21,000
Lake Charles, LA	$270,000	$33	20,000
Mobile, AL	$290,000	$30	19,000

Fixed-Charge Problem Example (4 of 5)

Decision variables

X4 = number of units produced at Baytown plant

X5 = number of units produced at Lake Charles plant

X6 = number of units produced at Mobile plant

Fixed-Charge Problem Example (5 of 5)

Formulation

Minimize cost = 340,000X1 + 270,000X2 + 290,000X3

+ 32X4 + 33X5 + 30X6

subject to X4 + X5 + X6 ≥ 38,000

X4 ≤ 21,000X1

X5 ≤ 20,000X2

X6 ≤ 19,000X3

X1, X2, X3 = 0 or 1

X4, X5, X6 ≥ 0 and integer

Using Software (1 of 3)

PROGRAM 10.6 Excel 2016 Solver Solution for Sitka Manufacturing Problem

Using Software (2 of 3)

Optimal solution

X1 = 0, X2 = 1, X3 = 1, X4 = 0, X5 = 19,000, X6 = 19,000

Objective function value = $1,757,000

Using Software (3 of 3)

PROGRAM 10.6 Excel 2016 Solver Solution for Sitka Manufacturing Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: H5 By Changing cells: B4:G4 To: Min Subject to the Constraints: H8 >= J8 H9:H11 <= J9:J11 B4:D4 = binary E4:G4 = integer Solving Method: Simplex LP  Make Variables Non-Negative	Blank	Copy H5 to H8:H11
Blank	Blank	Blank

Financial Investment Example (1 of 3)

Simkin, Simkin, and Steinberg specialize in recommending oil stock portfolios

One client has the following specifications

At least two Texas firms must be in the portfolio

No more than one investment can be made in a foreign oil company

One of the two California oil stocks must be purchased

The client has $3 million to invest and wants to buy large blocks of shares

Financial Investment Example (2 of 3)

TABLE 10.4 Oil Investment Opportunities

STOCK	COMPANY NAME	EXPECTED ANNUAL RETURN ($1,000s)	COST FOR BLOCK OF SHARES ($1,000s)
1	Trans-Texas Oil	50	480
2	British Petroleum	80	540
3	Dutch Shell	90	680
4	Houston Drilling	120	1,000
5	Texas Petroleum	110	700
6	San Diego Oil	40	510
7	California Petro	75	900

Financial Investment Example (3 of 3)

Formulation

Maximize return = 50X1 + 80X2 + 90X3 + 120X4 + 110X5 + 40X6 + 75X7

subject to

X1 + X4 + X5 ≥ 2 (Texas constraint)

X2 + X3 ≤ 1 (foreign oil constraint)

X6 + X7 = 1 (California constraint)

480X1 + 540X2 + 680X3 + 1,000X4 + 700X5 + 510X6 + 900X7 ≤ 3,000

($3 million limit)

Xi = 0 or 1 for all i

Using Software (1 of 2)

PROGRAM 10.7 Excel 2016 Solver Solution for Financial Investment Problem

Using Software (2 of 2)

PROGRAM 10.7 Excel 2016 Solver Solution for Financial Investment Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: I5 By Changing cells: B4:H4 To: Max Subject to the Constraints: I7 >= K7 I8 <= K8 I9 = K9 I10 <= K10 B4:H4 = binary Solving Method: Simplex LP  Make Variables Non-Negative	Blank	Copy I5 to I7:I10
Blank	Blank	Blank

Goal Programming (1 of 3)

Firms often have more than one goal

In linear and integer programming methods the objective function is measured in one dimension only

It is not possible for LP to have multiple goals unless they are all measured in the same units

Highly unusual situation

Goal programming developed to supplement LP

Goal Programming (2 of 3)

Typically goals set by management can be achieved only at the expense of other goals

Establish a hierarchy of importance so that higher-priority goals are satisfied before lower-priority goals

Not always possible to satisfy every goal

Goal programming attempts to reach a satisfactory level of multiple objectives

May not optimize but have to satisfice

Goal Programming (3 of 3)

Main difference is in the objective function

Goal programming tries to minimize the deviations between goals and what can be achieved given the constraints

Objective is to minimize deviational variables

Harrison Electric Company Revisited (1 of 4)

Production mix LP formulation

Maximize profit = $7X1 + $6X2

subject to 2X1 + 3X2 ≤ 12 (wiring hours)

6X1 + 5X2 ≤ 30 (assembly hours)

X1, X2 ≥ 0

where

X1 = number of chandeliers produced

X2 = number of ceiling fans produced

Harrison Electric Company Revisited (2 of 4)

Moving to a new location and maximizing profit is not a realistic objective

A profit level of $30 would be satisfactory during this period

The goal programming problem is to find the production mix that achieves this goal as closely as possible given the production time constraints

Define two deviational variables

d1− = underachievement of the profit target

d1+ = overachievement of the profit target

Harrison Electric Company Revisited (3 of 4)

Single-goal programming formulation

Minimize under or overachievement of profit target = d1− + d1+

subject to

$7X1 + $6X2 + d1− − d1+ = $30 (profit goal constraint)

2X1 + 3X2 ≤ 12 (wiring hours)

6X1 + 5X2 ≤ 30 (assembly hours)

X1, X2, d1−, d1+ ≥ 0

Harrison Electric Company Revisited (4 of 4)

Analyze each goal to see if underachievement or overachievement of that goal is acceptable

If overachievement is acceptable, eliminate the appropriate d + variable from the objective function

If underachievement is okay, the d − variable should be dropped

If a goal must be attained exactly, both d − and d + must appear in the objective function

Extension to Equally Important Multiple Goals (1 of 3)

Achieve several goals that are equal in priority

Goal 1: to produce a profit of $30 if possible during the production period

Goal 2: to fully utilize the available wiring department hours

Goal 3: to avoid overtime in the assembly department

Goal 4: to meet a contract requirement to produce at least seven ceiling fans

Extension to Equally Important Multiple Goals (2 of 3)

The deviational variables can be defined as

d1− = underachievement of the profit target

d1+ = overachievement of the profit target

d2− = idle time in the wiring department (underutilization)

d2+ = overtime in the wiring department (overutilization)

d3− = idle time in the assembly department (underutilization)

d3+ = overtime in the assembly department (overutilization)

d4− = underachievement of the ceiling fan goal

d4+ = overachievement of the ceiling fan goal

Extension to Equally Important Multiple Goals (3 of 3)

Management is unconcerned about d1+, d2+, d3−, and

d4+ so these may be omitted from the objective function

New objective function and constraints

Minimize total deviation = d1− + d2− + d3+ + d4−

subject to

$7X1 + $6X2 + d1− − d1+ = $30 (profit constraint)

2X1 + 3X2 + d2− − d2+ = 12 (wiring hours constraint)

6X1 + 5X2 + d3− − d3+ = 30 (assembly hours constraint)

X2 + d4− − d4+ = 7 (ceiling fan constraint)

All Xi, di variables ≥ 0

Ranking Goals with Priority Levels (1 of 3)

In most goal programming problems, one goal will be more important than another

Lower-order goals considered only after higher-order goals are met

Priorities (Pis) are assigned to each deviational variable

P1 is the most important goal

P2 the next most important

P3 the third, and so on

Ranking Goals with Priority Levels (2 of 3)

Harrison Electric has set the following priorities for their four goals

GOAL	PRIORITY
Reach a profit as much above $30 as possible	P1
Fully use wiring department hours available	P2
Avoid assembly department overtime	P3
Produce at least seven ceiling fans	P4

Priority 1 is infinitely more important than Priority 2, which is infinitely more important than the next goal, and so on

Ranking Goals with Priority Levels (3 of 3)

Harrison Electric has set the following priorities for their four goals

GOAL	PRIORITY
Reach a profit as much above $30 as possible	P1
Fully use wiring department hours available	P2
Avoid assembly department overtime	P3
Produce at least seven ceiling fans	P4

With ranking of goals considered, the new objective function is

Minimize total deviation = P1d1− + P2d2− + P3d3+ + P4d4−

Goal Programming with Weighted Goals (1 of 2)

Priority levels assume that each level is infinitely more important than the level below it

However a goal may be only two or three times more important than another

Instead of placing these goals on different levels, they are placed on the same level but with different weights

The coefficients of the deviation variables in the objective function include both the priority level and the weight

Goal Programming with Weighted Goals (2 of 2)

Suppose Harrison decides to add another goal of producing at least two chandeliers

The goal of producing seven ceiling fans is considered twice as important as this goal

The goal of two chandeliers is assigned a weight of 1 and the goal of seven ceiling fans is assigned a weight of 2 and both of these will be priority level 4

The new constraint and objective function are

X1 + d5− − d5+ = 2 (chandeliers)

Minimize

total = P1d1− + P2d2− + P3d3+ + P4(2d4−) + P4d5−

deviation

Using Software (1 of 2)

PROGRAM 10.8A Harrison Electric’s Goal Programming Analysis Using QM for Windows: Inputs

Using Software (2 of 2)

PROGRAM 10.8B Summary Solution Screen for Harrison Electric’s Goal Programming Problem Using QM for Windows

Nonlinear Programming

The methods seen so far have assumed that the objective function and constraints are linear

However, there are many nonlinear relationships in the real world that would require the objective function and/or constraint equations to be nonlinear

Computational procedures for nonlinear programming (NLP) may only provide a local optimum solution rather than a global optimum

Nonlinear Objective Function and Linear Constraints (1 of 3)

The Great Western Appliance Company sells two models of toaster ovens, the Microtoaster (X1) and the Self-Clean Toaster Oven (X2)

They earn a profit of $28 for each Microtoaster no matter the number of units sold

For the Self-Clean oven, profits increase as more units are sold due to a fixed overhead

The profit function for the Self-Clean oven

21X2 + 0.25X22

Nonlinear Objective Function and Linear Constraints (2 of 3)

The objective function is nonlinear and there are two linear constraints on production capacity and sales time available

Maximize profit = 28X1 + 21X2 + 0.25X22

subject to

X1 + X2 ≤ 1,000 (units of production capacity)

0.5X1 + 0.4X2 ≤ 500 (hours of sales time available)

X1, X2 ≥ 0

Nonlinear Objective Function and Linear Constraints (3 of 3)

When an objective function contains a squared term and the problem constraints are linear, it is called a quadratic programming problem

Using Software (1 of 2)

PROGRAM 10.9 Excel 2016 Solver Solution for Great Western Appliance NLP Problem

Using Software (2 of 2)

PROGRAM 10.9 Excel 2016 Solver Solution for Great Western Appliance NLP Problem

Solver Parameter Inputs and Selections	Blank	Key Formulas
Set Objective: E8 By Changing cells: B4:C4 To: Max Subject to the Constraints: E11:E12 <= G11:G12 Solving Method: GRG Nonlinear  Make Variables Non-Negative	Blank
Blank	Blank	Blank

Both Nonlinear Objective Function and Nonlinear Constraints (1 of 2)

The annual profit at a medium-sized (200-400 beds) Hospicare Corporation hospital depends on

The number of medical patients admitted (X1)

The number of surgical patients admitted (X2)

The objective function for the hospital is nonlinear

There are three constraints, two of which are nonlinear

Nursing capacity - nonlinear

X-ray capacity - nonlinear

Marketing budget required

Both Nonlinear Objective Function and Nonlinear Constraints (2 of 2)

Objective function and constraint equations

Maximize profit = $13X1 + $6X1X2 + $5X2 + $1÷X2

subject to

2X12 + 4X2 ≤ 90 (nursing capacity in thousands of labor-days)

X1 + X23 ≤ 75 (x-ray capacity in thousands)

8X1 − 2X2 ≤ 61 (marketing budget required in thousands of $)

Using Software (1 of 2)

PROGRAM 10.10 Excel 2016 Solution for Hospicare NLP Problem

Using Software (2 of 2)

PROGRAM 10.10 Excel 2016 Solution for Hospicare NLP Problem

Solver Parameter Inputs and Selections		Key Formulas
Set Objective: H8 By Changing cells: B4:C4 To: Max Subject to the Constraints: H11:H13 <= J11:J13 Solving Method: GRG Nonlinear  Make Variables Non-Negative		Copy H8 to H11:H13

Linear Objective Function and Nonlinear Constraints (1 of 2)

Thermlock Corp. produces massive rubber washers and gaskets like the type used to seal joints on the NASA Space Shuttles

It combines two ingredients, rubber (X1) and oil (X2)

The cost of the industrial quality rubber is $5 per pound and the cost of high viscosity oil is $7 per pound

Two of the three constraints are nonlinear

Linear Objective Function and Nonlinear Constraints (2 of 2)

Objective function and constraints

Minimize costs = $5X1 + $7X2

subject to

3X1 + 0.25X12 + 4X2 + 0.3X22 ≥ 125 (hardness constraint)

13X1 + X13 ≥ 80 (tensile strength)

0.7X1 + X2 ≥ 17 (elasticity)

Using Software (1 of 2)

PROGRAM 10.11 Excel 2016 Solution to the Thermlock NLP Problem

Using Software (2 of 2)

PROGRAM 10.11 Excel 2016 Solution to the Thermlock NLP Problem

Solver Parameter Inputs and Selections		Key Formulas
Set Objective: D5 By Changing cells: B4:C4 To: Min Subject to the Constraints: G10:G12 >= I10:I12 Solving Method: GRG Nonlinear  Make Variables Non-Negative		Copy G10 to G11:G12

Forecasting: Time series and trend analysis

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