Managerial decision modeling with spreadsheets by stair render chapter 06

Chapter 6 Integer, Goal, and Nonlinear Programming Models... Variations of Basic Linear Programming• Integer Programming • Goal Programming • Nonlinear Programming... Integer Programming

Chapter 6 Integer, Goal, and Nonlinear

Programming Models

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

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

Profit

Contribution $600 $700

Wiring Hours 2 hrs 3 hrs 12

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

Binary Integer Example:

Portfolio SelectionChoosing stocks to include in portfolio

Decision: Which of 7 stocks to include? Objective: Maximize expected annual

return (in $1000’s)

Stock Data

Decision VariablesUse 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

• 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)

Include At Least 2 Texas Companies

T + H + L > 2

Include No More Than 1 Foreign Company

B + D < 1Include Exactly 1 California Company

S + C = 1

If British Petro is included (B=1), then

Trans-Texas Oil must also be included (T=1)

T=0 T=1 B=0 ok ok

B=1 not ok ok

B < T allows the 3 acceptable combinations and prevents the unacceptable one

Go to file 6-3.xls

Combinations

of B and T

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

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

Monthly

Monthly Supply

Production Cost

(per unit)

Los Angeles 9,000

Shipping Cost Data

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 +

Supply Constraints-(XCD + XCH + XCN + XCL) = -15,000 (Cincinnati)

Goal Programming Models

• Permit multiple objectives

• Try to “satisfy” goals rather than optimize

• Objective is to minimize

underachievement of goals

Goal Programming Example:

Wilson Doors Co.

Makes 3 types of doors from 3 limited

resources

Decision: How many of each of 3 types of

doors to make?

Data

1 Total sales at least $180,000

2 Exterior door sales at least $70,000

3 Interior door sales at lest $60,000

4 Commercial door sales at least $35,000

Regular Decision Variables

E = number of exterior doors made

I = number of interior doors made

C = number of commercial doors made

Deviation Variables

di+ = amount by which goal i is overachieved

di- = amount by which goal i is underachieved

Goal ConstraintsGoal 1: Total sales at least $180,000

70E + 110I + 110C + dT- - dT+ = 180,000Goal 2: Exterior door sales at least $70,000

70E + dE- - dE+ = 70,000

Goal 3: Interior door sales at least $60,000

110 I + dI- - dI+ = 60,000

Goal 4: Commercial door sales at least

$35,000110C + dC- - dC+ = 35,000

Objective Function

Minimize total goal underachievementMin dT- + dE- + dI- + dC-

Subject to the constraints:

• The 4 goal constraints

• The “regular” constraints (3 limited

C-Properties of Weighted Goals

• Solution may differ depending on the

weights used

• Appropriate only if goals are measured in the same units

Ranked Goals

• Lower ranked goals are considered only if all higher ranked goals are achieved

• Suppose they added a 5th goal

Goal 5: Steel usage as close to 9000 lb

as possible4E + 3I + 7C + dS- = 9000 (lbs steel)(no dS+ is needed because we cannot

exceed 9000 pounds)

• Rank R1: Goal 1

• Rank R2: Goal 5

• Rank R3: Goals 2, 3, and 4

A series of LP models must be solved

1) Solve for the R1 goal while ignoring the

other goals

Objective Function: Min d

2) If the R1 goal can be achieved (dT- = 0), then this is added as a constraint and we attempt to satisfy the R2 goal (Goal 5)

S-3) If the R2 goal can be achieved (dS- = 0), then this is added as a constraint and we solve for the R3 goals (Goals 2, 3, and 4)

Nonlinear Programming Models

• Linear models (LP, IP, and GP) have linear objective function and constraints

• If a model has one or more nonlinear

equations (objective or constraint) then the model is nonlinear

Characteristics of Nonlinear Programming (NLP) Models

• Solution may depend on starting point

• Starting point is usually arbitrary

Nonlinear Programming Example:

Pickens Memorial HospitalPatient demand exceeds hospital’s capacity

Decision: How many of each of 3 types of

patients to admit per week?

Objective: Maximize profit

Decision Variables

M = number of Medical patients to admit

S = number of Surgical patients to admit

P = number of Pediatric patients to admit

Profit FunctionProfit per patient increases as the number of patients increases (i.e nonlinear profit

function)

• Hospital capacity: 200 total patients

• X-ray capacity: 560 x-rays per week

• Marketing budget: $1000 per week

• Lab capacity: 140 hours per week

Objective Function (in $ of profit)

Using Solver for NLP Models

• Solver uses the Generalized Reduced

Gradient (GRG) method

• GRG uses the path of steepest ascent (or descent)

• Moves from one feasible solution to

another until the objective function value stops improving (converges)