Problems:
1. When McCoy wakes up Saturday morning, she remembers that she promised the PTA she would make some cakes and/ or homemade bread for its bake sale that afternoon. However, she does not have time to go to the store to get ingredients, and she has only a short time to bake things in her oven. Because cakes and breads require different baking temperatures, she cannot bake them simultaneously, and she has only 3 hours available to bake. A cake requires 45 minutes to bake, and loaf of bread requires 30 minutes. The PTA will sell a cake for $10 and loaf of bread for $6. Marie wants to decide how many cakes and loaves of bread she should make.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
2. The Kalo Fertilizer Company makes a fertilizer using two chemical that provide nitrogen and 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 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
3. Solve the following linear programming model graphically:
Maximize Z= 1.5x1 + x 2
Subject to
x1 ≤ 4
x2 ≤ 6
x1+ x2 ≤ 5
x1, x2 ≤ 0
4. A manufacturing firm produces two products. Each product must undergo an assembly process and finishing process. It is then transferred to the warehouse , which has space for only a limited number of items. The firm has 80 hours available for assembly and 112 hours for finishing, and it can store a maximum of 10 units in the warehouse. Each unit of product 1 has a profit or $30 and requires 4 hours to assemble and 14 hours to finish. Each unit of product 2 has a profit of $70 and requires 10 hour to assemble and 8 hours to finish. The firm wants to determine the quantity of each product to produce in order to maximize profit.
A. Formulate a linear programming model for this problem.
B. Solve this model by using graphical analysis.