Operations Research Answers

Solve the following linear programming model using the simplex method:

maximize Z = 100x1 + 20x2 + 60x3

subject to

x3 smaller than or equal to 40

2x1 + 2x2 + 2x3 smaller than or equal to 100

3x1 + 5x2 smaller than or equal to 60

x1, x2, x3 bigger than or equal to 0

A clothing shop makes suits and blazers. Three main resources are used: material, rack space, and labor. The shop has developed this linear programming model for determining the number of suits

and blazers to make ( and ) to maximize profits

maximize Z = 100x1 + 150x2 (profit, $)

Subject to

10x1 + 20x2 smaller than or equal to 300 (material, yd.2

)

x1 + x2 smaller than or equal to 20 (rack space)

10x1 + 4x2 smaller than or equal to160 (labor, hr.)

x1, x2 bigger than or equal to 0

A jewelry store makes both necklaces and bracelets from gold and platinum.

The store has developed the following linear programming model for determining the number of necklaces and bracelets

(x1 and x2) that it needs to make to maximize profit: maximize Z = 300x1+ 400x2(profit, S) subject to 3x1 + 2x2 less than equal to 18 (gold, oz)

2x1 + 4x2 less than equal to 20 (platinum,oz.)

X2 less than equal to 4 (demand, bracelets) X1,x2 greater than equal to 0

Solve this model using the simplex method.

A marketing firm has contracted to do a survey on a political issue for a Spokane television station. The firm conducts interviews during the day and at night, by telephone and in person. Each hour an interviewer works at each type of interview results in an average number of interviews. In order to have a representative survey, the firm has determined that there must be at least 400 day interviews, 100 personal interviews, and 1,200 interviews overall. The company has developed the following linear programming model to determine the number of hours of telephone interviews during the day (x1), telephone interviews at night personal interviews at night (x4) (x2), personal interviews during the day (x3), and that should be conducted to minimize cost:

minimize Z = 2x1 + 3x2 + 5x3 + 7x4 (cost, $)

subject to :

10x1 + 4x3 >= 400 (day interviews)

4x3 + 5x4 >= 100 (personal interviews)

x1 + x2 + x3 + x4 >= 1,200 (total interviews)

x1, x2, x3, x4 >= 0

Solve this model using the simplex method.

The Copperfield Mining Company owns two mines, both of which produce three grades of ore— high, medium, and low. The company has a contract to supply a smelting company with at least 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore each hour it is in operation. Mine 1 produces 6 tons of high-grade, 2 tons of medium-grade, and 4 tons of low-grade ore per hour. Mine 2 produces 2 tons of high-grade, 2 tons of medium-grade, and 12 tons of low-grade ore per hour. It costs $200 per hour to mine each ton of ore from mine 1, and it costs $160 per hour to mine a ton of ore from mine 2. The company wants to determine the number of hours it needs to operate each mine so that contractual obligations can be met at the lowest cost. Formulate a linear programming model for this problem and solve using the simplex method.

Solve the following model using the simplex method:

minimize

Z = 0.06x1 + 0.10x2

subject to:

 4x1 + 3x2>=12
 3x1 + 6x2 >= 12
 5x1 + 2x2 >= 10
 x1,  x2 >= 0

The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, 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 phosphate, 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.

The Crumb and Custard Bakery makes both coffee cakes and Danish in large pans. The main ingredients 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.

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.

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.