Linear Programming

Jessica Faith Worrell

What is Linear Programming?

• A specialized mathematical decision-making aid used in industry and government

• Helps interpret data and examine the way things work or should work

• Goal is to find optimal solutions to problems

History

• Early 1800s

– Fourier, a French mathematician, formulated the linear programming problem

• 1900s

– Kantorovich, a Russian mathematician, also developed the problem

– Goal was to improve economic planning in the USSR

History

• World War II provided both the urgency and the funds for such research.

• Efficient resource allocation was required for large scale military planning such as fleets of cargo ships and convoys

History

• 1947- George Dantzig along with associates at the US Department of the Air Force developed the simplex method

• 1951- T.C. Koopmans developed a special linear programming solution used to plan the optimal movement of ships back and forth across the Atlantic during the war

• 1975 Nobel Prize in Economic Science

Applications

• Petroleum refineries• Maximize the value of oil inputs subject to

constraints on refinery equipment and gasoline blend requirements

• Find best locations for pipelines

• Find best routes and schedules for tankers

• 5-10% of total computing time

Applications

• Armour Company• Processed cheese spread specifications

• HJ Heinz Company• Shipment schedules between factories and

warehouses

• Agriculture• Minimize cost for cattle feed

Most Common Applications

• Minimize cost while meeting product specifications

• Maximize profit with optimal production processes or products

• Minimize cost in transportation routes

• Determine best schedules for production and sales

Problem

• A company manufactures two types of hand calculators, Model 1 and Model 2. Model 1 takes one hour to manufacture while Model 2 takes four hours. The cost of manufacturing is $30 and $20 for each Model 1 and 2, respectively. The company has 1,600 hours of labor time available and $18,000 in running costs. The profit on each Model 1 is $10 and each Model 2 is $8.

Solution

• Involves analyzing systems of linear equations

• Two methods• Geometric

• Computational

Geometric Solution

• Four inequalities, or constraints, to consider:

• Time constraint

x + 4y < 1,600

• Monetary constraint

30x + 20y < 18,000

• Nonnegativity constraint

x > 0 and y > 0

Feasible Set

• The set of all possible solutions to the family of inequalities

• Size depends on the amount of constraints

Geometric Solution

• Goal is to maximize the profit where

P = 10x + 8y

• Must be a point that lies in the feasible set

Geometric Solution

• Maximum will occur at a vertex of the feasible set.

P = 0 at A (0, 0)

P = 3200 at B (0, 400)

P = 6400 at C (400, 300)

P = 6000 at D (600, 0)

• Maximum profit is $6,400 when 400 Model 1 calculators are produced and 300 Model 2.

Computational Method

• Simplex method developed by Dantzig

• Involves using matrices to systematically check each corner of the feasible set

• Dantzig chose to go along an edge guaranteed to maximize profit

• Must have linear equations

Simplex Method

• Change inequalities to equalities

x + 4y + u = 1600

30x + 20y + v = 18000

• Rewrite profit function

-10x - 8y + f = 0

Simplex Method

• Initial Simplex Tableau:

1 4 1 0 0 1600 u

30 20 0 1 0 18000 v

-10 -8 0 0 1 0 f

x y u v f

• Corresponds to vertex A

• Basic variables

• Nonbasic variables

Simplex Method

• After using elementary row operations to produce a one in place of the pivot, and zeroes in the rest of the column, we get a second matrix

0 10/3 1 -1/30 0 1000 u

1 2/3 0 1/30 0 600 x

0 -4/3 0 1/3 1 6000 f

• Corresponds to vertex D

Simplex Method

• Now, with 10/3 as the pivot, the resulting matrix is as follows:

0 1 3/10 -1/100 0 300 y

1 0 -1/5 6/150 0 400 x

0 0 2/5 24/75 1 6400 f

• Corresponds to the point C

• Final Simplex Tableau

Simplex Method

• The final matrix gives the following equations:

y + 3/10 u - 1/100 v = 300

x - 1/5 u + 6/150v = 400

2/5 u + 24/75 v + f = 6400

where u = 0, v = 0, y = 300, x = 400 and f = 6400.

• Thus, maximum profit is $6400.

Advantages of Simplex Method

• Yields the same answer

• More practical for problems involving more than two variables

• Readily programmable for a computer

Minimization

• Must find the maximum of -f

• Then the minimum is the negative value of that maximum

Possible Outcomes

• Every linear programming problem falls into one of three categories:

• The feasible set is empty.

If constraints are contradictory, such as

x + 2y > 4 and x + 2y < -2

Possible Outcomes

• The cost function is unbounded on the set.

If two vertices of the feasible set satisfy the maximum or minimum, then every point of the line also does.

Leads to flexibility in production schedule.

Possible Outcomes

• The cost has a maximum or a minimum on the feasible set

There is one point that is the optimal solution

• The first two possibilities are uncommon for real problems in economics

Requirements

• Requires linearly proportional relationships

Resources to be consumed by an activity must be linearly proportional to the activity

• All activities must obey a materials balance

Sum of the resource inputs = Sum of product outputs

Conclusion

• Linear Programming offers industry a way to inform the decision makers of all the important information and the most favorable decision.

• It is an asset to companies in today’s growing economy.