Mathematical Programming

Cht. 2, 3, 4, 5, 9, 10.

Mathematical Programming (MP)

• Mathematical programming (MP) is an optimization method for “deterministic decision making” with very many decision alternatives.

• MP is also called “restricted decision making model”.

• It includes linear programming (cht. 2, 3, 4), integer programming (cht. 5), multicriteria programming (cht. 9), and non-linear programming (cht. 10).

Two Steps of Using MP to Solve a Decision Making Problem

• Step 1. Represent the problem with a “program with variables”, which has an objective function(s) and many constraints.

• Step 2. Solve the program by computer. The variables’ values in the solution are the decision.

Chapter 2

Linear Programming:

Fundamentals

Linear Programming (LP)

• Linear programming is the mathematical programming in which the objective function and all constraints are linear.

• A term is linear if it is a constant or its variable’s exponent is 1.

An Application Example

Resource Requirements

ProductsLabor

(hr/unit)Clay

(lb/unit)Profit

($/unit)Bowl 1 4 40Mug 2 3 50

Resource Available

40 hours 120 lb

Find how many bowls and mugs should be produced to maximize the profit.

LP Components

• Decision variables - their values are to be found in the solution.

• One linear objective function.

• Linear constraints - reflect limitations.

A Linear Program

Max 7X1 + 4X2

S.T. 3X1 + X2 <= 580

2X1 + 5X2 <= 720

X1 >= 20

X2 <= 100

X1, X2 >= 0

Format of a Linear Program

• No variable is in denominator.

• At most one term for each variable.

• Variable terms are at left, constant terms are at right (called right-hand-side, RHS).

• Align columns of inequality signs, variable terms, and constants.

• Put non-negative constraints in at last.

Solution

• A solution is a set of values each for a variable.

• A feasible solution satisfies all constraints.

• An infeasible solution violates at least one constraint.

• The optimal solution is a feasible solution that meets the objective.

LP Solution Methods

• Trial-and-Error

(brute force)• Graphic Method

(Won’t work if more than 2 variables)• Simplex Method

(Elegant, but time-taking if by hand)• Computerized simplex method

(We’ll use it programmed in QM)

Process of Solving a Problem By Using LP

Step 1. Formulate the problem into a linear program (by us)

Step 2. Solving the linear program (by computer)

LP Formulation

• Before using QM to solve a problem, we must first formulate the problem into a linear program, which is a description of the problem in terms of LP.

• Therefore, the process of formulating a problem in LP is a process of describing the problem by using an objective function and a couple of constraints.

LP Example 1, p.32-33

Resource Requirements

ProductsLabor

(hr/unit)Clay

(lb/unit)Profit

($/unit)Bowl 1 4 40Mug 2 3 50

Resource Available

40 hours 120 lb

Find how many bowls and mugs should be produced to maximize the profit.

Steps for LP Formulating

• Define variables unambiguously.

• Describe the objective function by using the variables.

• Describing restrictions one at a time by using the variables, which form constraints.

LP Example 2 p.47-49

BrandNitrogen (lb/bag)

Phosphate (lb/bag)

Cost ($/bag)

Super-gro 2 4 6Crop-quick 4 3 3Minimum

requirements16 24

Chemical Contribution

How many bags of each brand should be purchased in order to minimize the total cost?

Irregular LP Problems

• A regular LP has one optimal solution.

• Irregular cases:

– Multiple optimal solutions

– Infeasible problem

– Unbounded problem

Chapter 3

Linear Programming:

Sensitivity Analysis

Sensitivity Analysis (SA)

• SA is the analysis of the effect of parameter changes on the optimal solution.

• SA is conducted after the optimal solution is obtained.

Shadow Price (Dual Value)

• A shadow price is associated with a constraint in the solution.

In a product-mix problem

• as in example of ‘bowls and mugs’, a shadow price means:– the marginal value of a resource, i.e., – the contribution of an additional unit of a

resource to the objective function value, i.e.,

– The highest “price” the company would be willing to pay for one additional unit of a resource.

What Is “Dual”?

• Each linear program has another LP associated with it. They are called a pair of primal and dual.

• The dual LP is the “transposition” of the primal LP.

• Primal and dual have equal optimal objective function values.

• The solution of the dual is the shadow prices of the primal, and vice versa.

More General on Shadow Price:

• The shadow price of a constraint shows how much the objective function value would be better off if there were one unit increase on the RHS of the constraint.

• A shadow price can be negative, which shows a negative contribution (i.e., worse off) to the objective function value by an additional unit of RHS of the constraint.

S.A. on RHS

• Sensitivity range for a RHS value is the range over which the RHS value can change without changing the current shadow price.

S.A. on Objective Coefficients

• Sensitivity range for an objective coefficient is the range over which the objective coefficient can change without changing the current optimal solution.

S.A. on other changes

• To see sensitivities on following changes, one must solve the changed LP again: – Changing constraint coefficients

– Adding a new constraint

– Adding a new variable

Why doing S.A.?

• LP is used for decision making on something in the future.

• Rarely does a manager know all of the parameters exactly. Many parameters are inaccurate “estimates” when a model is formed and solved.

• We want to see to what extent the optimal solution is stable to the inaccurate parameters.

Sensitive or In-sensitive?

• Do we want a model more sensitive or less sensitive to the inaccuracies (changes) of parameters in it ?

• Answer: Less sensitive.

• Why? – An optimal solution that is insensitive to

inaccuracies of parameters is more likely valid in the real world situation.

Chapter 4

Linear Programming:

Modeling Examples

LP Modeling

• To model a decision making problem with LP:– Understand the problem thoroughly;– Identify the variables and objective;– Describe the problem in terms of the

variables, objective function, and constraints.

Examples Covered:

• Product mix

• Investment

• Marketing

• Blend (?)

• Transportation (?)

A product Mix Example. p.112

Printing on front side

Printing on both sides

Printing on front side

Printing on both sides

Capacity available

Work hour 0.1/dozen 0.25/dozen 0.08/dozen 0.21/dozen 72 hours

Space taken3 std.

boxes/dozen3 std.

boxes/dozen1 std.

box/dozen1std.

box/dozen1200 std.

boxesCost($)/dozen 36 48 25 35 $25,000Profit($)/dozen 90 125 45 65no more than

How many of each of four products should be produced so that the total profit is maximized ?

500 dozen 500 dozen

Sweatshirts T-shirts

An Investment Example. p.120-122

municipal bonds

CDstreasury

billsgrowth stock

fundAnnual Return 8.50% 5.00% 6.50% 13.00%

* Municipal bonds <= 20%.* CDs <= sum of other three.* (Treasury bills + CDs) >= 30%.* (Treasury bills + CDs) / (municipal bonds + stock fund) >= 1.2/1.* Total amount of investment is up to $70,000.

How much should be put in each of the four investment alternatives so that the total return is maximized ?

Investment Alternatives

A Marketing Example. p.126-127

television commercials

radio commercials

newspaper ads

Exposures (people / ad)

20,000 12,000 9,000

Cost $ 15,000 6,000 4,000

limits on number of ads

<=4 <=10 <=7

limits on total number of adsTotal budget

limit

How many ads of each type should be used so that the total

<=15

Types of Advertising

$100,000

exposure is maximized ?

A Blend Example, p.133-135

Component 1 Component 2 Component 3 Selling Price Minimum

Super Grade >=50% <=30% ---- $23 3,000 barrels

Premium Grade

>=40% ---- <=25% $20 3,000 barrels

Extra Grade >=60% >=10% ---- $18 3,000 barrels

Barrels Available

4,500 2,700 3,500

Cost $/barrel $12 $10 $14

How mnay barrels of each grade of motor oil (made of three componets) should be produced to maximize total profit?