Managerial decision modeling with spreadsheets by stair render chapter 06

Chapter 6
Integer, Goal, and Nonlinear
Programming Models

© 2007 Pearson Education


Variations of Basic
Linear Programming
• Integer Programming
• Goal Programming
• Nonlinear Programming


Integer Programming (IP)
Where some or all decision variables are
required to be whole numbers.
• General Integer Variables (0,1,2,3,etc.)
Values that count how many
• Binary Integer Variables (0 or 1)
Usually represent a Yes/No decision


General Integer Example:
Harrison Electric Co.
Produce 2 products (lamps and ceiling fans)
using 2 limited resources
Decision: How many of each product to
make? (must be integers)
Objective: Maximize profit

Decision Variables
L = number of lamps to make
F = number of ceiling fans to make
Lamps

Fans

(per lamp)

(per fan)

Profit
Contribution

$600

$700

Hours
Available

Wiring Hours

2 hrs

3 hrs

12

Assembly Hours

6 hrs

5 hr

30


LP Model Summary
Max 600 L + 700 F

($ of profit)

Subject to the constraints:

2L + 3F < 12

(wiring hours)

6L + 5F < 30

(assembly hours)

L, F > 0


Graphical Solution

Properties of Integer Solutions
• Rounding off the LP solution might not
yield the optimal IP solution
• The IP objective function value is usually
worse than the LP value
• IP solutions are usually not at corner
points


Using Solver for IP
• IP models are formulated in Excel in the
same way as LP models
• The additional integer restriction is entered
like an additional constraint
int - Means general integer variables
bin - Means binary variables
Go to file 6-1.xls


Binary Integer Example:
Portfolio Selection
Choosing stocks to include in portfolio
Decision: Which of 7 stocks to include?
Objective: Maximize expected annual
return (in $1000’s)


Stock Data

Decision Variables
Use the first letter of each stock’s name
Example for Trans-Texas Oil:
T = 1 if Trans-Texas Oil is included
T = 0 if not included


Restrictions
•
•
•
•
•

Invest up to $3 million
Include at least 2 Texas companies
Include no more than 1 foreign company
Include exactly 1 California company
If British Petro is included, then
Trans-Texas Oil must also be included


Objective Function (in $1000’s return)
Max 50T + 80B + 90D + 120H + 110L +
40S + 75C
Subject to the constraints:

Invest up to $3 Million
480T + 540B + 680D + 1000H
+ 700L + 510S + 900C < 3000

Include At Least 2 Texas Companies
T+H+L > 2
Include No More Than 1 Foreign Company
B+D < 1
Include Exactly 1 California Company
S+C = 1


If British Petro is included (B=1), then
Trans-Texas Oil must also be included (T=1)

Combinations
of B and T

B=0

T=0

T=1

ok

ok

B=1 not ok

ok

Ballows the 3 acceptable combinations and
prevents the unacceptable one
Go to file 6-3.xls


Mixed Integer Models:
Fixed Charge Problem
• Involves both fixed and variable costs
• Use a binary variable to determine if a
fixed cost is incurred or not
• Either linear or general integer variables
deal with variable cost


Fixed Charge Example:
Hardgrave Machine Co.
Has 3 plants and 4 warehouses and is
considering 2 locations for a 4th plant
Decisions:
• Which location to choose for 4th plant?
• How much to ship from each plant to each
warehouse?
Objective: Minimize total production and
shipping cost


Supply and Demand Data

Warehouse

Detroit

Monthly
Demand

Plant

Production
Monthly
Cost
Supply
(per unit)

10,000

Cincinnati

15,000

$48

Houston

12,000

Kansas
City

6,000

$50

New York

15,000

Pittsburgh

14,000

$52

Los Angeles

9,000

Total

46,000

35,000

Note: New plant must supply 11,000 units per month


Possible Locations for New Plant
Production Cost
(per unit)

Fixed Cost

(per month)

Seattle

$53

$400,000

Birmingham

$49

$325,000


Shipping Cost Data


Decision Variables
Binary Variables
Ys = 1 if Seattle is chosen
= 0 if not
YB = 1 if Birmingham is chosen
= 0 if not
Regular Variables
Xij = number of units shipped from plant i
to warehouse j


Objective Function (in $ of cost)

Min 73XCD + 103XCH + 88XCN + 108XCL +
85XKD + 80XKH + 100XKN + 90XKL +
88XPD + 97XPH + 78XPN + 118XPL +
113XSD + 91XSH + 118XSN + 80XSL +
84XBD + 79XBH + 90XBN + 99XBL +
400,000YS + 325,000YB
Subject to the constraints:
(see next slide)


Supply Constraints
-(XCD + XCH + XCN + XCL) = -15,000 (Cincinnati)
-(XKD + XKH + XKN + XKL) = - 6,000 (Kansas City)
-(XPD + XPH + XPN + XPL) = -15,000 (Pittsburgh)
Possible Locations for New Plant

-(XSD + XSH + XSN + XSL) = -11,000YS (Seattle)
-(XBD + XBH + XBN + XBL) = -11,000YB (B’ham)


Demand Constraints
XCD + XKD + XPD +XSD + XBD = 10,000
XCH + XKH + XPH +XSH + XBH = 12,000
XCN + XKN + XPN +XSN + XBN = 15,000
XCL + XKL + XPL +XSL + XBL = 9,000

(Detroit)
(Houston)
(New York)
(L. A.)

Choose 1 New Plant Location
YS + YB =1
Go to File 6-5.xls