Problem Solving

 

PROBLEM SOLVING

1.

A sm small all pet petrol roleum eum com compan pany y ow owns ns ttwo wo rrefn efneri eries. es. Ref Refner nery y 1 ccost ostss $2 $25,00 5,000 0 pe perr da day y tto o operate, and it can produce 300 barrels o high-grade oil, 200 barrels o medium-grade oil, and 150 barrels o low-grade oil each day. Refnery 2 is newer and more modern. It costs cos ts $30,000 per day to operate, and it can produce 300 barrels o high-grade oil, 250 barrels o medium-grade oil, and 400 barrels o low-grade oil each day. The company has orders totaling 35,000 barrels o high-grade oil, 30,000 barrels o medium-grade oil, and 40,000 barrels o low-grade oil. How many days should the company run each refnery to minimize its costs and sll meet its orders? Variability: X=the number o days to run Refnery 1 Y= the number o days to run Refnery 2 Objecve Funcon: C= 25,000x + 30,000y Constraints: High oil requirement = 300x + 300y > 35,000 Medium oil requirement = 200x + 250y > 30,000 Low oil requirement = 150x + 400y > 40,000 Graphing: Intersecon Points

(0, 120) (47.059, 82.353) (266.667, 0)

C= 25,000x + 30,000y 25,000 (0) = 30,000 (120) = 3,600,000 25,000 (47.059) = 30,000 (82.353) = 3,647,065 25,000 (266.667) = 30,000 (0) = 6,666,675

Recommendaon: The company can achieve the minimum cost when the Refnery 1 will do its operaon in 0 days and refnery r efnery 2 will operate in 120 days.

 

2.

A st steel eel com compan pany y ha hass tw two o mi mills lls.. Mi Millll 1 cost costss $70 $70,00 ,000 0 pe perr da day y to ope operate rate,, an and d it can produce 400 tons o high-grade steel, 500 tons o medium-grade steel, and 450 tons o low-grade steel each day. Mill 2 costs $60,000 per day to operate, and it can produce 350 tons o high-grade steel, 600 tons o o  medium-grade steel, and 400 tons o low-grade steel each day. The company has orders totaling 100,000 tons o high-grade steel, 150,000 tons o medium-grade steel, and 124,500 tons o low-grade steel. How many days should the company run each mill to minimize its costs and sll fll the orders. Variability: X=the Y= the number number o o days daysto to operate operatemill mill12 Objecve Funcon: P= 70,000x + 60,000y Constraints: High-grade steel = 400x + 350y > 100,000 Medium-grade steel = 500x + 600y > 150,000 Low-grade steel = 450x + 450y > 124,000 y-Intercept

x-Intercept

400(0)+350y>100,000 350y/350>100,000/350

400x+350(0)>100,000 400x/400>100,000/400

 Y=285.71

X=166.67

500(0)+600y>150,000 350y/350>150,000/350

500x+600(0)>150,000 500x/500>150,000/500

 Y=250

X=300

450(0)+400y>124,500 400y/400>124,500/400

450x+400(0)>124,500 450x/450>124,500/450

 Y=311.25

X=276.67

Graphing: Intersecon Points

(0, 311.75) (300, 0) (210, 75)

P= 70,000x + 60,000y 70,000 (0) = 60,000 (311.75) = 18,705,000 70,000 (300) = 60,000 (0) = 21,000,000 70,000 (210) = 60,000 (75) = 19,200,000

 

Recommendaon: The Minimum cost will be achieved when the steel company will run the mill 2 in 311.25 days.

3.

Mic Michae haell is ttakin aking g an exa exam m in ord order er tto o be becom come e a vo volun luntee teerr frefgh frefghter ter.. The exa exam m ha hass 10 essay quesons and 50 short quesons. He has 90 minutes to take the exam. The essay quesons are worth 20 points each and the short s hort quesons are worth 5 points each. An essay queson takes 10 minutes to answer and a short queson takes 2 minutes. Michael must do at least 3 essay quesons and at least 10 short quesons. Michael knows the material well enough to get ull points on all quesons he aempts and wants to maximize the number o points he will get. Variability: X=the number o short quesons Y= the number o essay quesons Objecve Funcon: P = 5x + 20y Constraints: Time encoded to answer = 2x + 10y < 90 Short quesons = 10 < x < 50 Essay quesons = 3< x < 10

 

Graphing: Intersecon Points

(10, 3) (10, 7) (30, 3)

P = 5x + 20y 5 (10) = 10 (3) = 110 5 (10) = 10 (7) = 190 5 (30) = 10 (3) = 210

Recommendaon: To get the the maximum number o point points, s, Michael needs to answer 30 short short quesons and 3 essay quesons