Linear Programming Problems

A. Two variable - maximization problems

1. Maximize z = 16x + 8y subject to:
2x + y ≤ 30
x + 2y ≤ 24
x ≥ 0
y ≥ 0

a.Graph the feasibility region.
b.Identify all applicable corner points of the feasibility region.
c.Find the point(s) (x,y) that maximizes the objective function z = 16x + 8y.

2. A company is planning to purchase and store two items, gadgets and widgets. Each gadget costs $2.00 and occupies 2 square meters of floor space; each widget costs $3.00 and occupies 1 square meter of floor space. $1,200 is available for purchasing these items and 800 square meters of floor space is available to store them. Each gadget contributes $3.00 to profit and each widget contributes $2.00 to profit.

a.Identify all constraints.
b.Identify all applicable corner points of the feasibility region.
c.What combination of gadgets and widgets produces maximum profit?

3. A manufacturer produces two items, bookcases and library tables. Each item requires processing in each of two departments. Department I has 40 hours available and department II has 36 hours available each week for production. To manufacture a bookcase requires 2 hours in department I and 4 hours in department II, while a library table requires 3 hours in department I and 2 hours in department II. Profits on the items are $6.00 for a bookcase and $7.00 for a library table.

a.Identify all constraints.
b.Identify all applicable corner points of the feasibility region.
c.If all units produced can be sold, how many of each should be made in order to maximize profits?

4. A manufacturer has a maximum of 240, 360, and 180 kilograms of wood, plastic and steel available. The company produces two products, A and B. Each unit of A requires 1, 3 and 2 kilograms of wood, plastic and steel respectively; each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively, and each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively. The profit per unit of A and B is $4.00 and $6.00 respectively.

a.Identify all constraints.
b.Identify all applicable corner points of the feasibility region.
c.How many units of A and B should be manufactured in order to maximize profits? What would the maximum profit be?

B. Two variable - minimization problems

5. Minimize z = 3x + 6y subject to:
4x + y ≥ 20
x + y ≤ 20
x + y ≥ 10
x ≥ 0
y ≥ 0

a.Graph the feasibility region.
b.Identify all applicable corner points of the feasibility region.
c.Find the point(s) (x,y) that minimizes the objective function z = 3x + 6y.

6. A feed company is developing a feed supplement from two grains A and B. Each kilogram of A contains 0.3 grams of protein and 0.2 grams of carbohydrates; each kilogram of B contains 0.9 grams of protein and 0.1 grams of carbohydrates. There must be at least 27 grams of protein and at least 8 grams of carbohydrates.

a.Identify all of the constraints.
b.Identify all applicable corner points of the feasibility region.
c.If each kilogram of A costs 30 cents and each kilogram of B costs 50 cents what combination of quantities of grain A and grain B will minimize the cost of a package of the new feed supplement?