Linear Programming. Jessica Faith Worrell

Transcription

1 Linear Programming Jessica Faith Worrell

2 Goal is to find optimal solutions to problems What is Linear Programming? A specialized mathematical decisionmaking aid used in industry and government Helps interpret data and examine the way things work or should work

3 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

4 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

5 History George Dantzig along with associates at the US Department of the Air Force developed the simplex method 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

6 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

7 Applications Armour Company Processed cheese spread specifications HJ Heinz Company Shipment schedules between factories and warehouses Agriculture Minimize cost for cattle feed

8 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

9 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.

10 Solution Involves analyzing systems of linear equations Two methods Geometric Computational

11 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

12 Feasible Set The set of all possible solutions to the family of inequalities Size depends on the amount of constraints

13 Geometric Solution Goal is to maximize the profit where P = 10x + 8y Must be a point that lies in the feasible set

14 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.

15 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

16 Simplex Method Change inequalities to equalities x + 4y + u = x + 20y + v = Rewrite profit function -10x - 8y + f = 0

17 Simplex Method Initial Simplex Tableau: u v f x y u v f Corresponds to vertex A Basic variables Nonbasic variables

18 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/ u 1 2/3 0 1/ x 0-4/3 0 1/ f Corresponds to vertex D

19 Simplex Method Now, with 10/3 as the pivot, the resulting matrix is as follows: 0 1 3/10-1/ y 1 0-1/5 6/ x 0 0 2/5 24/ f Corresponds to the point C Final Simplex Tableau

20 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 = Thus, maximum profit is $6400.

21 Advantages of Simplex Method Yields the same answer More practical for problems involving more than two variables Readily programmable for a computer

22 Minimization Must find the maximum of -f Then the minimum is the negative value of that maximum

23 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

24 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.

25 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

26 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

27 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.