Instructions: Answer each question. Use Lindo to solve all problems. Show all work by describing variables,
stating assumptions, illustrating model and showing output solution to the
problem.Download Lindo/Lingo software to
solve all problems
1. Shale Oil
Company (20 pts)
Shale Oil, located on the
island of Aruba, has a capacity of 600,000 barrels of crude oil per day. The final products from the refinery include
two types of unleaded gasoline; regular and premium. The refining process encompasses four stages:
(1)
The pure crude flows through a distillation
tower that produces a feedstock
(2)
The feedstock output breaks up into two paths;
the first path involves feedstock flowing into a cracker unit that refines the
mixture into a gasoline stock. The
second path has a portion of the feedstock flowing into the blender unit.
(3)
The gasoline stock (from the cracker unit) feeds
into the blender.
(4)
The blender unit produces the final product,
regular or premium gasoline. Both the
regular and premium gasoline can be produced from either the feedstock or the
gasoline stock during the blending process, although at different production
costs. The company estimates that the
net profit per barrel of regular gasoline is $5.20 from feedstock and $7.70
from gasoline stock. The corresponding
profit values for the premium are $10.40 from the feedstock and $12.30 from the
gasoline stock. According to design
specifications, it takes five barrels of crude oil to produce one barrel of
feedstock. The cracker units cannot use
more than 40,000 barrels of feedstock per day.
All remaining feedstock is used directly in the blender unit to produce
the end product gasoline. The demand
limits for regular and premium gasoline are 80,000 and 50,000 barrels per day.
a.
Using Linear Programming, determine the optimum
production schedule for the refinery.
Allow for the solution variables to be in fractions (non integer)
b.
Suppose the capacity of the distillation tower
can be increased to 650,000 barrels per day at cost of $40,000. Is it worth making this investment? (answer this question using sensitivity
analysis, stating shadow price, net profit or net loss and rationale. Extract all information from the lindo output
used in part a). What would the new profit be for this investment? (Note, if it
is not worth the investment, enter 0 for the new profit value)
Problems
Students
Answers
Problem 2a (Maximum Profit)
Problem 2b (Shadow price for
Additional Capacity)
Problem 2b (Is it worth the
investment?)
Problem 2b (New profit?, enter
value of 0 if there is no new profit )
2. Newcor Steel (25 pts) Newcors
steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 5% molybdenum
by weight. The steel is manufactured by combining
three types of metals; steel ingots, scrap steel and alloys. Steel ingots can only be purchased in lot
sizes of 1 at the specified weights given in the table below. The ingot can be cut to the desired amount
needed in the process. The weight, cost
per ton and chemical contents of each ingot is:
INGOT WEIGHT COST PER
TON CARBON% MOLYBDENUM%
1 5 tons $350 5 3
2 3 tons $330 4 3
3 4 tons $310 5 4
4 6 tons $280 3 4
Unlimited amounts of alloys and scrap
steel can be purchased. Scrap steel cost
$100 per ton and contains 3% carbon and 9% molybdenum. The cost per ton and chemical makeup of the
alloys is:
ALLOY COST PER TON CARBON% MOLYBDENUM%
1 $500 8 6
2 $450 7 7
3 $400 6 8
Determine the
appropriate mixture of the steel components to minimize the cost of
fulfilling the requirements.
Problems
Answers
Problem 4 (Minimal Cost Value)
3.
First West Chemical Company (25 Points)
First West Chemical Company produces two chemical ingredients
for pharmaceutical firms; formula X and formula Y. Production of each
ingredient requires two processes. A unit of Formula X requires 4 hours in
process 1 and 3 hours in process 2. A unit in formula Y requires 2 hours in
process 1 and 5 hours in process 2. The
normal operating production times for the two processes are as follows: process
1, 70 hours and process 2, 60 hours. The production process of formula X
results in one unit of a by-product called XZ.
Four (4) units of X produce 1 unit of XZ. The production process for
formula Y yields 5 units of a by-product, YK for each unit of formula Y. The
unit profits for formulas X and Y are $10,000 and $15,000 respectively. By
products XZ yields $6000 unit profit. By product YK yields a $3000 unit profit
for up to 15 units. Because of the limited market and the danger involved in
handling the material, any by product YK in excess of 15 units must be
destroyed at a unit cost of $4,000. Also due to storage regulations, no more
than 15 units of Formula X can be produced.
The management at this chemical company has established the
following goals in order of importance:
Avoid any
underutilization of normal operation hours of each of the two processes.
Each of equal importance.
Meet the outstanding
orders for 8 units of formula X and 7 units of Formula Y. Each of equal
importance.
Limit any overtime
operation of each of the two production process to 10 hours. Each of equal
importance.
Achieve a profit goal
of $220,000
Limit the production
of by-product YK to 15 units.
Minimize the overtime
operation of the production processes. Each of equal importance.
Formulate this problem into a mathematical model and determine
the solution that best satisfies these goals.
Problems
Answers
Problem 5 -First Goal Achieved
(yes, no)
Problem 5 -Second Goal
Achieved (yes, no)
Problem 5 -Third Goal
Achieved (yes, no)
Problem 5 -Fourth Goal Achieved
(yes, no)
Problem 5 -Fifth Goal Achieved
(yes, no)
Problem 5 -Sixth Goal Achieved
(yes, no)
Problem 5 – Production amount of
Formula X at conclusion
Problem 5 – Production amount of
Formula Y at conclusion
Problem 5 – Production amount of
By Product YK at conclusion
Problem 5 – Production amount of
By Product XZ at conclusion
Problem 5 – Profit at conclusion
Instructions: Answer each question. Use Lindo to solve all problems. Show all work by describing variables,
stating assumptions, illustrating model and showing output solution to the
problem.Download Lindo/Lingo software to
solve all problems1. Shale Oil
Company (20 pts)
Shale Oil, located on the
island of Aruba, has a capacity of 600,000 barrels of crude oil per day. The final products from the refinery include
two types of unleaded gasoline; regular and premium. The refining process encompasses four stages:
(1)
The pure crude flows through a distillation
tower that produces a feedstock (2)
The feedstock output breaks up into two paths;
the first path involves feedstock flowing into a cracker unit that refines the
mixture into a gasoline stock. The
second path has a portion of the feedstock flowing into the blender unit.(3)
The gasoline stock (from the cracker unit) feeds
into the blender. (4)
The blender unit produces the final product,
regular or premium gasoline. Both the
regular and premium gasoline can be produced from either the feedstock or the
gasoline stock during the blending process, although at different production
costs. The company estimates that the
net profit per barrel of regular gasoline is $5.20 from feedstock and $7.70
from gasoline stock. The corresponding
profit values for the premium are $10.40 from the feedstock and $12.30 from the
gasoline stock. According to design
specifications, it takes five barrels of crude oil to produce one barrel of
feedstock. The cracker units cannot use
more than 40,000 barrels of feedstock per day.
All remaining feedstock is used directly in the blender unit to produce
the end product gasoline. The demand
limits for regular and premium gasoline are 80,000 and 50,000 barrels per day.a.
Using Linear Programming, determine the optimum
production schedule for the refinery.
Allow for the solution variables to be in fractions (non integer)b.
Suppose the capacity of the distillation tower
can be increased to 650,000 barrels per day at cost of $40,000. Is it worth making this investment? (answer this question using sensitivity
analysis, stating shadow price, net profit or net loss and rationale. Extract all information from the lindo output
used in part a). What would the new profit be for this investment? (Note, if it
is not worth the investment, enter 0 for the new profit value)ProblemsStudents
AnswersProblem 2a (Maximum Profit) Problem 2b (Shadow price for
Additional Capacity) Problem 2b (Is it worth the
investment?) Problem 2b (New profit?, enter
value of 0 if there is no new profit ) 2. Newcor Steel (25 pts) Newcors
steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 5% molybdenum
by weight. The steel is manufactured by combining
three types of metals; steel ingots, scrap steel and alloys. Steel ingots can only be purchased in lot
sizes of 1 at the specified weights given in the table below. The ingot can be cut to the desired amount
needed in the process. The weight, cost
per ton and chemical contents of each ingot is: INGOT WEIGHT COST PER
TON CARBON% MOLYBDENUM% 1 5 tons $350 5 3 2 3 tons $330 4 3 3 4 tons $310 5 4 4 6 tons $280 3 4 Unlimited amounts of alloys and scrap
steel can be purchased. Scrap steel cost
$100 per ton and contains 3% carbon and 9% molybdenum. The cost per ton and chemical makeup of the
alloys is: ALLOY COST PER TON CARBON% MOLYBDENUM% 1 $500 8 6 2 $450 7 7 3 $400 6 8Determine the
appropriate mixture of the steel components to minimize the cost of
fulfilling the requirements.Problems AnswersProblem 4 (Minimal Cost Value) 3.
First West Chemical Company (25 Points)
First West Chemical Company produces two chemical ingredients
for pharmaceutical firms; formula X and formula Y. Production of each
ingredient requires two processes. A unit of Formula X requires 4 hours in
process 1 and 3 hours in process 2. A unit in formula Y requires 2 hours in
process 1 and 5 hours in process 2. The
normal operating production times for the two processes are as follows: process
1, 70 hours and process 2, 60 hours. The production process of formula X
results in one unit of a by-product called XZ.
Four (4) units of X produce 1 unit of XZ. The production process for
formula Y yields 5 units of a by-product, YK for each unit of formula Y. The
unit profits for formulas X and Y are $10,000 and $15,000 respectively. By
products XZ yields $6000 unit profit. By product YK yields a $3000 unit profit
for up to 15 units. Because of the limited market and the danger involved in
handling the material, any by product YK in excess of 15 units must be
destroyed at a unit cost of $4,000. Also due to storage regulations, no more
than 15 units of Formula X can be produced.The management at this chemical company has established the
following goals in order of importance:Formulate this problem into a mathematical model and determine
the solution that best satisfies these goals.Problems AnswersProblem 5 -First Goal Achieved
(yes, no) Problem 5 -Second Goal
Achieved (yes, no) Problem 5 -Third Goal
Achieved (yes, no) Problem 5 -Fourth Goal Achieved
(yes, no) Problem 5 -Fifth Goal Achieved
(yes, no) Problem 5 -Sixth Goal Achieved
(yes, no) Problem 5 – Production amount of
Formula X at conclusion Problem 5 – Production amount of
Formula Y at conclusion Problem 5 – Production amount of
By Product YK at conclusion Problem 5 – Production amount of
By Product XZ at conclusion Problem 5 – Profit at conclusion
