ms lecture011 lp formulation

7/29/2019 MS Lecture011 LP Formulation

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More on LP Formulation

OPIM 101 Management Science


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Example: Product-Mix Problem of BM

Semester 2 2011/20122

BM produces three rubber-based products using threepolymers and a base.

The amount (in oz) of each ingredient used per lb of eachproduct is given in following Table.

(Note: 1 lb = 16 oz )


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Problem

Semester 2 2011/20123

For coming week, BM has commitment to produce at least 1000 lb of

Airtex, 500 lb of Extendex, and 400 lb of Resistex, but BM knows it can sellmore of each product.

Current inventories of ingredients are 500 lb of polymer A, 425 lb ofpolymer B, 650 lb of polymer C, and 1100 lb of base.

Each lb of Airtex nets a profit of $7, each lb of Extendex a profit of $7, andeach lb of Resistex a profit of $6.

As production manager, you need to determine optimal

production plan for coming week.


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Decision Variables

Semester 2 2011/20124

What can you control ? What is a plan ?-variables which requires your planning/decision

Let

A = no. of lb of Airtex to produce

E = no. of lb of Extendex to produce

R = no. of lb of Resistex to produce


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Objective Function

Semester 2 2011/20125

What do we mean by optimalplan ?

Profit = 7A + 7E + 6R (in $)

Want to choose decision variables such that profit is

maximized.


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Constraints

Semester 2 2011/20126

In order to maximize profit we would like to choosevalues of our decision variablesA, E, and R to be as large

as possible.

However, the problem has imposed some restrictions onthe values which these variables can take.

Such restrictions are called constraints.


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Demand Constraints:

Semester 2 2011/20127

A 1,000E 500

R 400

Why?


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Resource Constraints:

Semester 2 2011/20128

Amount of polymer A used = 4A + 3E + 6R (oz)Amount available = 500 lb = 8,000 oz

Therefore, 4A + 3E + 6R 8,000 (polymer A)

Similarly,

2A + 2E + 3R 6,800 (polymer B)

4A + 2E + 5R 10,400 (polymer C)

6A + 9E + 2R 17,600 (base)


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Nonnegativity Constraints:

Semester 2 2011/20129

A, E, R, 0

May be very obvious!

In this course, we will still state these constraints.


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Formulation of BMs Product-Mix Problem

Semester 2 2011/201210


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Semester 2 2011/201211

Although the nonnegativity conditions are clearly redundanthere, they are often included unless unwanted.

Note that the objective function and constraints are all linear

in the decision variables.

Hence, we call the above a linear program (LP) or linear

programming problem (LP problem).


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Other modeling examples:

Scheduling/Allocation Problem

Semester 2 2011/201212

A bus company is in charge of allocating buses on a 6-hourday. Each bus shift is 4 hours and cost $100 in terms ofoperating cost. The demand for buses for the 6 hours are12,20,35,20,10 and 15 respectively.

Formulate a model to find the best strategy to allocate thebuses?


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RECALL:

Semester 2 2011/201213

Note:By convention, constraints are written such that all the

decision variables are on the LHS, while the constant is

on the RHS.

So far, we have only learn how to model the problem. We

will learn how to solve in the next lecture.

ms lecture011 lp formulation

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