Linear Programming

Introduction:

Linear programming(LP) is a mathematical optimization technique. By “Optimization” technique we mean a method which attempts to maximize or minimize some objective, e.g. maximize profits or minimize costs.In many problems in business and industry we are making decisions that will maximize or minimize some quantity. For example, a plant manager may want to determine the most economical way of shipping goods from the factory to the markets, a hospital may want to design a diet satisfying certain nutritional requirements at a minimum cost, an investor may want to select investments that will maximize profits, or a manufacturer may wish to blend ingredients, subject to given specifications, to maximize profit.

Linear Programming

Under this topic we give; i) Several examples of LP problems and show how mathematical

models can be formulated for them.

ii) Geometric solutions of above LP problems.

Linear Programming

Example 1 (A production problem)A small manufacturer of photographic products prepare two types (Fine and Extra Fine) of film developers each day, using solution A and B as the raw material. Suppose that each quart of Fine contains 2 ounces of solution A and 1 ounce of solution B, while each quart of Extra Fine contains 1 ounce of solution A and 2 ounces of solution B. Suppose also that the profit on each quart of Fine is 8 cents and that it is 10 cents on each quart of Extra Fine. If the firm has 50 ounces of solution A and 70 ounces of solution B available each day, how many quarts of Fine and Extra Fine should be made each day to maximize the profit.

Linear Programming

Mathematical Formulation;

Let be the number of quarts of Fine to be made , Let be the number of quarts of Extra Fine to be made. Since each quart of Fine contains 2 ounces of solution A and each quart of Extra Fine contains 1 ounce of solution A, the total amount of solution A required is

Similarly, since each quart of Fine contains 1 ounce of solution B and each quart of Extra Fine contains 2 ounces of solution B, the total amount of solution B required is

Since firm has 50 ounces of solution A and 70 ounces of solution B available, we must have

Of course, cannot be negative, so must have

x y

2 .x y

2 .x y

2 50x y 2 70x y

0, 0x y x and y

Linear Programming

Since the profit on each quart of Fine is 8 cents and 10 cents on each quart of Extra Fine, the total profit is

Our problem can be stated in mathematical form as: Find the values of that will maximize

Subject to constraints (restrictions) that must be satisfied by

,x y

8 10Z x y

8 10Z x y

2 50x y

2 70x y

0

0

x

y

x and y

Linear Programming

Example 2 (Pollution)Example 3 (The Diet problem)

Linear Programming

Linear programming A linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to linear constraints.The constraints may be equalities or inequalities.

Here is a simple exampleFind the numbers that maximize

Subject to constraints

and

3 2Z x y

2 18x y

,x y

0, 0x y

2 3 42x y 3 24x y

In this problem there are two unknowns and five constraints. All the constraints are inequalities and they are all linear in a sense that each involves an inequality in some linear function of the variables. The first two constraints are called nonnegative constraints and often found in linear programming problems. The other constraints are called main constraints. The function to be maximized or minimized is called objective function and here it is Since there are two variables, we can solve this problem by graphing the set of points in the plain that satisfies all the constraints (called constraint set) and then finding the which point of this set that maximizes the objective function.

Linear Programming

0, 0x y

3 2Z x y

Graphical Solution

To solve the problem by Graphical Method, we represent all constraints graphically. Take the first one and we draw the line that is obtained by considering the constraint as an equality. In the figure, this is represented with the A-B edge, and the region that defines this constraint is shown in YELLOW color. We repeat the process with the other restrictions, limiting BLUE and RED regions for the second and third constraint respectively.

Feasible region is determined for the intersection of every region defined by the constraints and the non-negativity condition of each variable, that is, both axis. This feasible region is represented by the O-F-H-G-C polygon, in VIOLET color.

Graphical Solution

Graphical Solution

Since there is a feasible region, we proceed to determine its extreme points, or vertices of the polygon that represents. These vertices are the candidate points for optimal solutions. In this example are the O-F-H-G-C points shown in the figure.

Finally, we evaluate the objective function Z= 3x+2y in each point (the results in the table below). As the G point provides the greatest value of the Z function, and the objective is to maximize, this point is the optimal solution: Z = 33 with x = 3 and y = 12.

Graphical Solution

Extreme point Coordinates (x,y) Objective value(Z)

O (0,0) 0

C (0,14) 28

G (3,12) 33

H (6,6) 30

F (8,0) 24