Computational Methods for Management and Economics

Carla Gomes

Module 2 (addendum)Revisiting the Divisibility Assumption

(Textbook – Hillier and Lieberman)

LP AssumptionsDivisibility

 Decision variables in an LP model are allowed to have any values, including noninteger values, that satisfy the functional and nonnegativity constraints. i.e., activities can be run at fractional levels.

What to do when divisibility assumption violated:realm of integer programming!!!

TBA Airlines Problem

An Example where Integrality Matters

The TBA Airlines Problem

• TBA Airlines is a small regional company that specializes in short flights in small airplanes.

• The company has been doing well and has decided to expand its operations.

• The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both.

Question: How many airplanes of each type should be purchased to maximize their total net annual profit?

Data for the TBA Airlines Problem

SmallAirplane

LargeAirplane

CapitalAvailable

Net annual profit per airplane

$1 million $5 million

Purchase cost per airplane

5 million 50 million $100 million

Maximum purchase quantity

2 —

Linear Programming FormulationLet S = Number of small airplanes to purchase

L = Number of large airplanes to purchase

Maximize Profit = S + 5L ($millions)

subject to

Capital Available: 5S + 50L ≤ 100 ($millions)

Max Small Planes: S ≤ 2

and

S ≥ 0, L ≥ 0.

Graphical Method for Linear Programming

3

2

1

0 1 2 3 S

L

Feasible region

Number of large airplanes purchased

Number of small airplanes purchased

(2, 1) = Rounded solution (Profit = 7)

(2, 1.8) = Optimal solution

Profit = 11 = S + 5 L

Violates Divisibility Assumption of LP

• Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values.

• Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.

Integer Programming Formulation

Let S = Number of small airplanes to purchase

L = Number of large airplanes to purchase

Maximize Profit = S + 5L ($millions)

subject to

Capital Available: 5S + 50L ≤ 100 ($millions)

Max Small Planes: S ≤ 2

and

S ≥ 0, L ≥ 0

S, L are integers.

Graphical Method for Integer Programming

3

2

1

0 1 2 3 S

LNumber of large airplanes purchased

Number of small airplanes purchased

(2, 1) = Rounded solution (Profit = 7)

(2, 1.8) = Optimal solution for the LP relaxation (Profit = 11)

Profit = 10 = S + 5 L

(0, 2) = Optimal solution for the integer programming problem (Profit = 10)

Graphical Method for Integer Programming• When an integer programming problem has just two decision

variables, its optimal solution can be found by applying the graphical method for linear programming with just one change at the end.

• We begin as usual by graphing the feasible region for the LP relaxation, determining the slope of the objective function lines, and moving a straight edge with this slope through this feasible region in the direction of improving values of the objective function.

• However, rather than stopping at the last instant the straight edge passes through this feasible region, we now stop at the last instant the straight edge passes through an integer point that lies within this feasible region.

• This integer point is the optimal solution.

Why integer programs?

• Advantages of restricting variables to take on integer values– More realistic– More flexibility

• Disadvantages– More difficult to model – Can be much more difficult to solve

Integer Programming• When are “non-integer” solutions okay?

– Solution is naturally divisible• e.g., $, pounds, hours

– Solution represents a rate• e.g., units per week

– Solution only for planning purposes

• When is rounding okay?– When numbers are large

• e.g., rounding 114.286 to 114 is probably okay.

• When is rounding not okay?– When numbers are small

• e.g., rounding 2.6 to 2 or 3 may be a problem.

– Binary variables• yes-or-no decisions

LP AssumptionsCertaintyThe parameters of the model, (coefficients of the objective function and of the functional constraints, and the righ-hand sides of the functional constraints) are assumed to be known constants.

T Rarely the case – sometimes we use approximations important to perform sensitivity analysis to identify sensitive parameters (the parameters that cannot be changed without changing the value of the objective function).

What to do when certainty assumption violated:treat parameters as random variables

LP Assumptions - Comments 

Quite often the assumptions are not 100% applicable!

Except for the divisibility assumption, minor disparities are to be expected. This is especially true for the certainty assumption, so sensitivity analysis normally is a must to compensate for the violation of

this assumption! 

LP Assumptions - CommentsMathematical model --- abstractions of the real world problem. Approximations and simplifying assumptions generally are required in order for the model to be tractable. Adding too much detail and precision can make the model too complex for useful analysis of the problem.

Main issues a correlation between the prediction of the model and what would actually happen in the real problem.

 

Additional Examples

• Giapetto’s Woodcarving (from Winston and Venkataramanan)

• Dorian Auto (from Winston and Venkataramanan)

• Radiation Therapy (from Hillier and Lieberman)

3.2 – Graphical Solution to a 2-Variable LPBinding and Nonbinding constraints

A constraint is binding if the left-hand and right-hand side of the constraint are equal when the optimal values of the decision variables are substituted into the constraint. In the Giapetto LP, the finishing and carpentry constraints are binding.

Once the optimal solution to an LP is found, it is useful to classify each constraint as being a binding or nonbinding constraint.

3.2 – Graphical Solution to a 2-Variable LP

A constraint is nonbinding if the left-hand side and the right-hand side of the constraint are unequal when the optimal values of the decision variables are substituted into the constraint. In the Giapetto LP, the demand constraint for wooden soldiers is nonbinding since at the optimal solution (x1 = 20), x1 < 40.

3.2 – Graphical Solution to a 2-Variable LPConvex sets, Extreme Points, and LPA set of points S is a convex set if the line segment jointing any two pairs of points in S is wholly contained in S.

For any convex set S, a point p in S is an extreme point if each line segment that lines completely in S and contains the point P has P as an endpoint of the line segment.

Consider the figures (a) – (d) below:

A E B

DC

A B

A B

(a) (b) (c) (d)

3.2 – Graphical Solution to a 2-Variable LP

For example, in figures (a) and (b) below, each line segment joining points in S contains only points in S. Thus is convex for (a) and (b). In both figures (c) and (d), there are points in the line segment AB that are not in S. S in not convex for (c) and (d).

A E B

DC

A B

A B

(a) (b) (c) (d)

3.2 – Graphical Solution to a 2-Variable LP

A E B

DC

A B

A B

(a) (b) (c) (d)

In figure (a), each point on the circumference of the circle is an extreme point of the circle. In figure (b), A, B, C, and D are extreme points of S. Point E is not an extreme point since E is not an end point of the line segment AB.

3.2 – Graphical Solution to a 2-Variable LP

Extreme points are sometimes called corner points, because if the set S is a polygon, the extreme points will be the vertices, or corners, of the polygon.

The feasible region for the Giapetto LP will be a convex set.

3.2 – Graphical Solution to a 2-Variable LP

It can be shown that:

• The feasible region for any LP will be a convex set.

• The feasible region for any LP has only a finite number of extreme points.

1.Any LP that has an optimal solution has an extreme point that is optimal.

Dorian Auto

Dorian Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income men and women. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games. Each comedy commercial is seen by 7 million high-income women and 2 million high-income men. Each football commercial is seen by 2 million high-income women and 12 million high-income men. A 1-minute comedy ad costs $50,000, and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen by at least 28 million high-income women and 24 million high-income men. Use LP to determine how Dorian Auto can meet its advertising requirements at minimum cost.

(from Winston)