October 6 University
Faculty of Information Systems
& Computer Science
3rd Year IS & CS Operations Research I Spring 2021/2022
Sheet -2 Linear Programing: Model Formulation and Graphical
Solution
Problems:
1.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.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
2.A clothier makes coats and slacks. The two resources required are
wool cloth and labor. The clothier has 150 square yards of wool and 200
hours of labor available. Each coat requires 3 square yards of wool and
10 hours of labor, whereas each pair of slacks requires 5 square yards of
wool and 4 hours of labor. The profit for a coat is $50, and the profit for
slacks is $40. The clothier wants to determine the number of coats and
pairs of slacks to make so that profit will be maximized.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
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3.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, while 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 the minimum requirements of 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.
4.The Pinewood Furniture Company produces chairs and tables from
two resources—labor and wood. The company has 80 hours of labor and
36 board-ft. of wood available each day. Demand for chairs is limited to
6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood,
whereas a table requires 10 hours of labor and 6 board-ft. 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 in order to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
5.The Crumb and Custard Bakery makes coffee cakes and Danish
pastries 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 5. Five pounds of flour and 2 pounds of
sugar are required to make a pan of coffee cakes, and 5 pounds of flour
and 4 pounds of sugar are required to make a pan of Danish. A pan of
coffee cakes has a profit of $1, and a pan of Danish has a profit of $5.
Determine the number of pans of cakes and Danish to produce each day
so that profit will be maximized.
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a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
6.A jewelry store makes necklaces and bracelets from gold and
platinum. The store has 18 ounces of gold and 20 ounces of platinum.
Each necklace requires 3 ounces of gold and 2 ounces of platinum,
whereas each bracelet requires 2 ounces of gold and 4 ounces of
platinum. The demand for bracelets is no more than four. A necklace
earns $300 in profit and a bracelet, $400. The store wants to determine
the number of necklaces and bracelets to make in order to maximize
profit.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis
7.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 milligrams 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.
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