Linear Programming Models for Fertilizer Production, Metal Stamping, and Coffee Blending
BA250 MANAGEMENT SCIENCE
EXAMPLE PROBLEMS
CHAPTER 4
#13,21,32
13) The Kalo Fertilizer Company produces two brands of lawn
fertilizerSuper Two and Green Growat plants in Fresno, California,
and Dearborn, Michigan. The plant at Fresno has resources
available to produce 5,000 pounds of fertilizer daily; the plant at
Dearborn has enough resources to produce 6,000 pounds daily. The
cost per pound of producing each brand at each plant is as follows:
The company has a daily budget of $45,000 for both plants
combined. Based on past sales, the company knows the maximum
demand (converted to a daily basis) is 6,000 pounds for Super Two
and 7,000 pounds for Green Grow. The selling price is $9 per pound
for Super Two and $7 per pound for Green Grow.
Formulate a linear programming model for this problem.
21) The Midland Tool Shop has four heavy presses it uses to stamp out prefabricated
metal covers and housings for electronic consumer products. All four presses
operate differently and are of different sizes. Currently the firm has a contract to
produce three products. The contract calls for 400 units of product 1; 570 units of
product 2; and 320 units of product 3. The time (in minutes) required for each
product to be produced on each machine is as follows:
Machine 1 is available for 150 hours, machine 2 for 240 hours, machine 3 for 200
hours, and machine 4 for 250 hours. The products also result in different profits,
according to the machine they are produced on, because of time, waste, and
operating cost. The profit per unit per machine for each product is summarized as
follows:
Formulate this problem as a linear programming
model.
32)
The Mill Mountain Coffee Shop blends coffee on the premises for its
customers. It sells three basic blends in 1-pound bags,
Special, Mountain
Dark, and Mill Regular
. It uses four different types of coffee to produce the
blends
Brazilian, mocha, Columbian, and mild
. The shop has the following
blend recipe requirements:
Blend
Mix Requirements
Selling Price/Pound
Special
At least 40% Columbian, at least 30% mocha
$6.50
Dark
At least 60% Brazilian, no more than 10% mild
5.25
Regular
No more than 60% mild, at least 30% Brazilian
3.75
The cost of Brazilian coffee is
$2.00
per pound, the cost of mocha is
$2.75
per
pound, the cost of Columbian is
$2.90
per pound, and the cost of mild is
$1.70
per pound.
The shop has
110
pounds of Brazilian coffee,
70
pounds of mocha,
80
pounds
of Columbian, and
150
pounds of mild coffee available per week.
•
The shop wants to know the amount of each blend it should prepare each
week to maximize profit. Formulate a linear programming model for this
problem.
The examples provide real-world scenarios requiring the formulation of linear programming models. The first involves the Kalo Fertilizer Company deciding on daily production quantities of two lawn fertilizer brands given resource constraints, costs, and demand. The second scenario explores the optimization of product manufacturing on different machines at Midland Tool Shop to maximize profits within time constraints. Lastly, the Mill Mountain Coffee Shop blends coffee varieties to maximize profit by meeting blend recipe requirements and ingredient availability. Linear programming is used to solve these business optimization problems.
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BA250 MANAGEMENT SCIENCE EXAMPLE PROBLEMS CHAPTER 4 #13,21,32
13) The Kalo Fertilizer Company produces two brands of lawn fertilizerSuper Two and Green Growat plants in Fresno, California, and Dearborn, Michigan. The plant at Fresno has resources available to produce 5,000 pounds of fertilizer daily; the plant at Dearborn has enough resources to produce 6,000 pounds daily. The cost per pound of producing each brand at each plant is as follows: The company has a daily budget of $45,000 for both plants combined. Based on past sales, the company knows the maximum demand (converted to a daily basis) is 6,000 pounds for Super Two and 7,000 pounds for Green Grow. The selling price is $9 per pound for Super Two and $7 per pound for Green Grow. Fresno Dearborn Product Super Two 2$ 4$ Green Grow 2$ 3$ Formulate a linear programming model for this problem.
21) The Midland Tool Shop has four heavy presses it uses to stamp out prefabricated metal covers and housings for electronic consumer products. All four presses operate differently and are of different sizes. Currently the firm has a contract to produce three products. The contract calls for 400 units of product 1; 570 units of product 2; and 320 units of product 3. The time (in minutes) required for each product to be produced on each machine is as follows: MACHINE PRODUCT 1 2 3 4 1 35 41 34 39 2 40 36 32 43 3 38 37 33 40 Machine 1 is available for 150 hours, machine 2 for 240 hours, machine 3 for 200 hours, and machine 4 for 250 hours. The products also result in different profits, according to the machine they are produced on, because of time, waste, and operating cost. The profit per unit per machine for each product is summarized as follows: PRODUCT 1 2 3 4 MACHINE 1 7.8$ 7.8 8.2 7.9 Formulate this problem as a linear programming model. 2 6.7 8.9 9.2 6.3 3 8.4 8.1 9.0 5.8
32) The Mill Mountain Coffee Shop blends coffee on the premises for its customers. It sells three basic blends in 1-pound bags, Special, Mountain Dark, and Mill Regular. It uses four different types of coffee to produce the blends Brazilian, mocha, Columbian, and mild. The shop has the following blend recipe requirements: Blend Mix Requirements At least 40% Columbian, at least 30% mocha At least 60% Brazilian, no more than 10% mild No more than 60% mild, at least 30% Brazilian Selling Price/Pound $6.50 5.25 3.75 Special Dark Regular The cost of Brazilian coffee is $2.00 per pound, the cost of mocha is $2.75 per pound, the cost of Columbian is $2.90 per pound, and the cost of mild is $1.70 per pound. The shop has 110 pounds of Brazilian coffee, 70 pounds of mocha, 80 pounds of Columbian, and 150 pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit. Formulate a linear programming model for this problem.
