Q1. Challenge Problem: A student has a limited amount of time to complete an assignment. She assumes that if she completes a part of the assignment, she will get full credit. Parts that are only partially complete receive no credit. She has 204 minutes to complete the assignment. The table below gives the amount of time (in minutes) that each part of the assignment takes along with how many points each part is worth. What parts should she complete to maximize her score?
Note: Each part can only be completed once, or not be completed at all. Scientific notation can be considered zero.
Part Time Points
A 5 1
B 30 8
C 50 13
D 68 20
E 45 14
F 60 17
G 40 12
H 51 15
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
(c) There are 100 total points. What is the best letter grade she can earn? Use the standard scale.
Q2. A beer company makes a lager beer and a light beer and are sold in 12 ounces bottles. They mix imported beer and domestic beer to make a hybrid blend. (This is their niche!) The imported beer and domestic beer are made of hops and barley, as given in Table.
Demand is at least 10,000 bottles of the lager and at least 12,000 bottles of the light beer. Each bottle of lager must consist of at most 43% of hops and each bottle of light beer must consist of at least 54% of barley.
Assume that one bottle imported or domestic beer will yield one bottle of lager or light beer, and that the rest of the beer is made of insignificant ingredients.
How much imported beer and domestic beer should be put into the lager and the light beer to meet demand but minimizes cost?
Hops (%) Barley (%) cost/bottle ($)
Imported 38 60 $0.52
Domestic 45 50 $0.40
(d) Write the linear program, in standard form, that is needed to solve the problem.
(e) Put the linear program from (a) into Excel to find the optimal solution.
Q3. A company sells syrup, the kind that should be put on pancakes and waffles. They syrup is made of maple syrup, corn syrup solution, or both. The cost of water is negligible and an unlimited amount is available. The contract minimum and demand is for number of bottles per month. The contract minimum should always be met, and more can be made, but what is made should not exceed the demand. How many bottles of each product should be made to maximize profit?
(f) Write the linear program, in standard form, that is needed to solve the problem.
(g) Put the linear program from (a) into Excel to find the optimal solution.
Q4. A company has factories in Buffalo, Syracuse, and Allentown. There are stores in Queens, Philadelphia, Atlantic City, and Hoboken. The tables below give the production capacity of each factory and the demand for each store for next month, and the transportation cost per unit from the origin to destinations. Find the arcs (routes) that should be used to minimize transportation cost.
(a) Draw a network representation of the transportation problem.
(b) Write the linear program, in standard form, that is needed to solve the problem.
(c) Put the linear program from (b) into Excel to find the optimal solution.
Q5.
Below is a transshipment network for a shortest-route problem with origin node 1 and destination node 8. The numbers with the arcs represents the cost to use the arcs. Find the route from node 1 to node 8 that minimizes cost.
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
(c) What is the route of the optimal solution?
Q6. A company that makes telescopes has plants, warehouses, and retail outlets across the country. Table gives the cost per unit to ship from location to location. Table 6.2 gives the supply for the plants and demand for the retail outlets. How many units should be shipped on each route to minimize cost? The Jacksonville to Houston route needs a minimum of 100 units shipped. The Austin to Oakland route can only have a maximum of 250 units shipped.
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
Q7. A University has three theoretical physics professors and three topics to research. All the professors are capable of researching all the topics, but the dean knows that the number of grants each professor can secure is different for each topic, as given in the table. Write a linear program that assigns one professor to one topic that maximizes the number of grants that can be secured.
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
Q8. A wireless phone provider needs to staff a store for the evening hours, 5:00 PM to 11:00 PM, with full time and part time staff. Full time staff will work the entire time, and pat time staff will work three consecutive hours.
No part time staff can work the last hour (10:00 – 11:00).
40% of full time staff will go to a meeting (and not working with customers) at from 6:00 – 7:00, and
60% of full time staff will go from 7:00 – 8:00.
Full time staff must work 70% of all work hours.
Full time staff earns $120 for their 6 hour shift and the part time staff earns $42 for their 3 hour shift.
The table gives the minimum number of staff needed to work with customers for each hour.
Write a linear program that minimizes the cost of staffing the evening hours of the store.
Time Number of Staff
5:00 – 6:00 14
6:00 – 7:00 12
7:00 – 8:00 11
8:00 – 9:00 9
9:00- 10:00 7
10:00 – 11:00 4
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
Q9. An all-purpose store sells four costumes for Halloween in the month of October. The selling price of the costumes is given in the table. The demand (the exact number of costumes the store must have to sell) for each type of costume is in Table.
Mickey and cowboy are for boys, and Minnie and princess are for girls.
Mickey and Minnie are cartoon characters, and cowboy and princess are fantasy characters.
Write a linear program that maximizes the revenue for selling the Halloween costumes.
Table:
Price Demand
Mickey $20 Boys 35
Minnie $23 Girls 45
Cowboy $17 Cartoon 30
Princess $15 Fantasy 50
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
Q10. Challenge Problem:
A company makes outdoor cat houses and dog houses. Both houses use the same base, but each house has a different perimeter and roof.
The company can either manufacture or purchase the pieces, and the cost is given in the table below.
Also given is the manufacture time, in minutes.
The demand projected is 2800 for the cat houses and 2200 for the dog houses. The company has 220 hours (13200 minutes) of manufacture time available. (There is no overtime available).
Find the number of each piece that should be manufactured or purchased that meets demand but minimizes cost.
Cost per Unit
Component Manufacture Purchase
Base $5 $6
Cat Perimeter $38 $40
Dog Perimeter $34 $40
Cat Roof $7 $8
Dog Roof $8 $9
Component Manufacturing Time (minutes)
Base 1.1
Cat Perimeter 2.8
Dog Perimeter 2.6
Cat Roof 1.0
Dog Roof 1.7
(a) Write the linear program, in standard form, that is needed to solve the problem.
(b) Put the linear program from (a) into Excel to find the optimal solution.
(c) Which, if any, of the pieces were both purchased and manufactured?