LINEAR PROGRAMMING

PREPARED BY: MS. DONNA JANE M. DELIMA

What is Linear Programming?

HISTORYDeveloped by Leonid Kantorovich in 1937. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. Postwar, many industries found its use in their daily planning.

What is Linear Programming?Say you own a 500 square acre farm. On this

farm you can grow wheat, barley, corn or some combination of the 3. You have a limited supply of fertilizer and pesticide, both of which are needed (in different quantities) for each crop grown. Let’s say wheat sells at $7 a bushel, barley is $3, and corn is $3.50.

So, how many of each crop should you grow to maximize your profit?

LINEAR PROGRAMMING

Linear programming (also called linear

optimization) is a method to achieve the best outcome

in a mathematical model whose requirements are

represented by linear relationships.

The maximization or minimization of some

quantity is the objective in all linear

programming problems.

(1) Graphing the constraints

In linear programming we have two variables that must satisfy several linear inequalities. These are called the constraints because they restrict the variables to only certain values.Constraints – is a condition that a solution to a linear programming problem is required by the problem itself to satisfy.

(1) Graphing the constraints

Graph the solution set to the system of inequalities and identify each vertex of the region.

𝑥≥0 , 𝑦 ≥03 𝑥+2 𝑦 ≤12𝑥+2 𝑦 ≤8

GRAPH

(2) Writing the ConstraintsProblem 1.

One serving of food A contains 2 grams of protein and 6 grams of carbohydrates. One serving of food B contains 4 grams of protein and 3 grams of carbohydrates. A dietitian wants a meal that contains at least 12 grams of protein and at least 18 grams of carbohydrates. If the cost of food A is 9 cents per serving and the cost of food B is 20 cents per serving, then how many servings of each food to maximize the cost and satisfy the constraints?

(2) Writing the ConstraintsLet x – number of servings of food A y – number of servings of food B

Types of Food

No. of Servings of

FoodGrams of Protein

Grams of Carbohydra

tes

Cost of food per serving

( cents)

A x 2 6 9

B y 4 3 20

Minimum Nutrients 12 18

Objective Function – is a function in which the objective is to find the values on the decision variables that produce the optimal value.

Non-negative Constraints – is a condition that a solution to a linear programming problem is positive.

Regular constraints – are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant.

(2) Writing the Constraints

Objective

Function“Regular”Constraint

sNon-negativity Constraints

𝑃=9𝑥+20 𝑦2𝑥+4 𝑦 ≥126 𝑥+3 𝑦 ≥18𝑥≥0 , 𝑦≥0

Strategy for Linear Programming

1. Graph the region that satisfies all the constraints.

2. Determine the coordinate of each vertex of the region.

3. Evaluate the function at each vertex of the region.

4. Identify which vertex gives the maximum or minimum value of the function.

Problem 1.

One serving of food A contains 2 grams of protein and 6 grams of carbohydrates. One serving of food B contains 4 grams of protein and 3 grams of carbohydrates. A dietitian wants a meal that contains at least 12 grams of protein and at least 18 grams of carbohydrates. If the cost of food A is 9 cents per serving and the cost of food B is 20 cents per serving, then how many servings of each food to maximize the cost and satisfy the constraints?

Solution: (1) Write the constraints

𝑃=9𝑥+20 𝑦2𝑥+4 𝑦 ≥126 𝑥+3 𝑦 ≥18𝑥≥0 , 𝑦≥0

( 2)

GRAPH

A feasible solution satisfies all the problem's constraints.

An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).

(3) Determine the coordinatesof each vertex of the region

The vertices are (0,6), (6,0),(2,2).

(4) Evaluate the function at each vertex of the region.

𝑃 (0,6)=9(0)+20(6)=120𝑐𝑒𝑛𝑡𝑠𝑃 (6,0)=9 (6)+20 (0)=54𝑐𝑒𝑛𝑡𝑠𝑃 (2,2)=9(2)+20 (2)=58𝑐𝑒𝑛𝑡𝑠

(5) Identify which vertex gives the maximum

or minimum value of the function.

The minimum cost of 54 cents is attained by using six

servings of food A and no servings of food B.

Problem 2.A manufacturer of lightweight mountain tents makes a standard model and expedition model for national distribution. Each standard tents requires 1 labor hour from cutting department and 3 labor hours from the assembly department. Each expedition tent requires 2 labor hours from the cutting department and four labor hours from the assembly department.

The maximum labor hours available per day in the cutting department and the assembly department are 32 and 84 respectively. If the company makes a profit of S50 on each standard tent and S80 on each expedition tent, how many tents of each type should be manufactured each day to maximize the total daily profit( assuming that all tents can be sold)?