# A-46

# Module A

# The Simplex Solution Method

6 milligrams of vitamin A and 2 milligrams of vitamin B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. Formulate a linear programming model for this problem and solve using the simplex method.

8. A company makes product 1 and product 2 from two resources. The linear programming model for determining the amounts of product 1 and 2 to produce (x_{1 }and x_{2}) is

maximize Z = 8x_{1 }+ 2x_{2 }(profit, $)

subject to

( r e s o u r c e 1 , l b . ) ( r e s o u r c e 2 , l b . ) x 1 , x 2 Ú 0 2 x 1 + 6 x 2 … 1 8 4 x 1 + 5 x 2 … 2 0

Solve this model using the simplex method.

9.

A company produces two products that are processed on two assembly lines. Assembly line 1 has 100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours of processing time on line 1, while on line 2 product 1 requires 7 hours and product 2 requires 3 hours. The profit for product 1 is $6 per unit, and the profit for product 2 is $4 per unit. Formulate a linear programming model for this problem and solve using the simplex method.

10.

The Pinewood Furniture Company produces chairs and tables from two resources—labor and wood. The company has 80 hours of labor and 36 board feet of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board feet of wood to produce, while a table requires 10 hours of labor and 6 board feet of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day to maximize profit. Formulate a linear programming model for this problem and solve using the simplex method.

11.

The Crumb and Custard Bakery makes both coffee cakes and Danish in large pans. The main ingre- dients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available and the demand for coffee cakes is 8. Five pounds of flour and 2 pounds of sugar are required to make one pan of coffee cake, and 5 pounds of flour and 4 pounds of sugar are required to make one pan of Danish. One pan of coffee cake has a profit of $1, and one pan of Danish has a profit of $5. Determine the number of pans of cake and Danish that the bakery must produce each day so that profit will be maximized. Formulate a linear programming model for this problem and solve using the simplex method.

12.

The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phos- phate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, whereas a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phos- phate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of fertilizer to meet minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost. Formulate a linear programming model for this problem and solve using the simplex method.

13. Solve the following model using the simplex method: minimize Z = 0.06x_{1 }+ 0.10x_{2 }

subject to