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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.
Kyle, a runner, is interested in mixing two cereals to minimize caloric intake while maintaining at least 96 units of calcium and 24 units of iron in each serving. Cheerios has 6 units of calcium, 3 units of iron and 14 calories per ounce. Fruitloops has 12 units of calcium, 2 units of iron and 11 calories per ounce. How much of each cereal should Kyle consume to minimize calorie intake while maintaining the minimum nutrient levels
A trading company buys and sells 10,000 bottles of pain-balm every year. The
company’s cost of placing an order of pain-balm is Rs. 100. The holding cost per
bottle on inventory is Rs. 0.3. (5)
(i) Determine the optimum order quantity and inventory cycle time for the pain-balm,
bottles.
(ii) How many orders should be placed each year?
Which of the following statements are true and which are false? Give a short proof or a
counter example in support of your answer. (10)
(a) The optimal solution for the following LPP is 30 :
*
Z =
Max 1 2 3 3 Z = x − x + x
Subject to x1 + x2 + x3 ≤10
x1
, x2
, x3 ≥ .0
(b) The optimal solution of an ILLP can be obtained by rounding off the optimal solution
of its LP relaxation.
(c) If the availabilities and requirements of a balanced transportation problem are
integers, the optimal solution to the problem will have integer values.
(d) The following max /3/4 F / F problem can be reduced to a machine problem.
Job
Processing time on
M1 M2 M3
1 8 6 10
2 5 2 13
3 4 11 11
4 6 7 10
(e) For the mixed generator rn+1 = 5( rn + )7 (mod ),8 if r0 = ,4 then 3
r is zero.
1. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients,
oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats
and rice it should include in each box of cereal to meet the minimum requirements of 48 mil-
ligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats
contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice
contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce
of rice costs $0.03.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
2. What would be the effect on the optimal solution in (Question 1) if the cost of rice increased from
$0.03 per ounce to $0.06 per ounce?
when Tracy 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 3 cups of flour, and a loaf of bread requires 8 cups; Tracy has 20 cups of flour.
A cake requires 45 minutes to bake, and a loaf of bread requires 30 minutes. The PTA will sell a
cake for $10 and a loaf of bread for $6. Tracy 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.
QUESTION 1
Food For U Sdn Bhd., a catering company is to make breakfast for a business meeting. It will serve chicken sandwiches, egg sandwiches, and vegetarian sandwiches. A chicken sandwich has 1 serving of vegetables, 4 slices of chicken, 2 slices of cheese, and 2 slices of bread. An egg sandwich has 2 serving of vegetables, 2 servings of egg, 2 slices of cheese and 2 slices of bread. A vegetarian sandwich has 3 servings of vegetables, 2 slices of cheese, and 2 slices of bread. A total of 5 bags of chicken are available, each of which has 40 slices; 19 loaves of bread are available, each with 14 slices; 200 servings of vegetables and eggs are available, and 15 bags of cheese, each with 60 slices, are available.
a) Formulate linear programming model based on the above problem by using Microsoft Excel. Show the answer and sensitivity report. (9 marks)
