Reference no: EM13495077

1. J. Sells the dietician at Kalous Hospital must prepare a breakfast menu for patients with Lazzari's Disease. The table below summarizes the relevant data per ounce for each of the three food groups. The meal must consist of at least 10 ounces. Lazzari's patients require high Vitamin A low Vitamin B diets. Thus the diet must contain at least 20% Vitamin A and no more than 10% Vitamin B. At least 20 percent of the calories should come from cereals. Formulate the LP model to minimize cost.

Cost               per ounce    Vitamin A    Vitamin B       Calories

Eggs                   $0.50        10%            10%        60

Bacon Strips        $0.90        30%             20%        120 

Cereal                  $0.20        15%            5%         90  

2. Lakeside Boatworks is planning the production of their fiberglass recreational boats for the next four months. The forecasted demand is 450 for April, 420 for May, 450 for June and 500 for July. The production capacity is 470, 450, 380 and 470 for April, May, June and July respectively. The projected production costs are $7,000 for April, $8,000 for May, and $9,000 for June and July. Boats that are not sold in the months in which they are produced can be put into storage for sale during the following months. Each month, the inventory costs are $50 per boat. A maximum of 70 boats can be put into storage at the end of each month. It is possible to use overtime labor in April to produce an additional 50 boats at a cost of $7,500 per boat. Formulate the LP model to minimize production and inventory costs and meet the demand for the next four months.

3. At the end of each week, General Appliance ships (by truck) refrigerators that they produce in plants located in Atlanta and Boston to distribution centers located in Chicago, Denver, and Eugene. Relevant shipping costs are presented in the table below. The route from Boston to Eugene is serviced by only one truck and is therefore limited to 50 units shipped per week on this route. In addition, refrigerators can be sent from the distribution center in Chicago to Denver at a cost of $30 per unit. Finally, it is possible to ship refrigerators by freight from Boston to Eugene for a price of $80 per unit, but only if the weekly shipment amount is exactly 200 units per week. Formulate the LP model to minimize weekly shipping cost.

Distribution Centers

Plants            Chicago           Denver           Eugene           Supply

Atlanta           $40              $65        $120       500

Boston            $50              $60        $130       400

Demand          300              300        300

4. Union City Al has decided to upgrade their tornado warning system. The city leaders have identified six potential locations for erecting a warning siren tower. They want to ensure that each of the seven schools in Union City is within three miles of a siren, yet they want to build as few warning siren towers as possible. Relevant information on the six potential locations of the warning siren towers and distance in miles to each of the seven schools is presented in the table below. Formulate the LP model for Union City to minimize towers built.

Potential Locations                                         

Schools           A    B    C    D    E    F   

1. Washington      2    1    4    8    11   7   

2. Adams          1    2    5    12   9    2

3. Jefferson        4    1    2    5    6    13  

4. Madison        1    5    4    2    6    1

5. Jackson         6    8    2    4    1    7

6. Lincoln         7    4    1    2    6    1

7. Kennedy        9    12   4    1    2    8

5. Kenton owns two fruit stands, one in Montego Bay and one in Kingston. One of Kenton's most popular products is the guava. Kenton relies on five orchards to provide him with fresh guava each day. The table below summarizes the daily demand at each fruit stand, the daily supply at each orchard, and the cost (per guava delivered to the city of the fruit stand in Jamaican dollars). The supply values in the last row of the table are provided by the orchard. However Kenton knows that the orchards are very unreliable and may not be able to provide as many as claimed on any given day. Therefore, Kenton views the supply figures as the maximum amount possible to obtain from each orchard. He would like to distribute the shipments from the various orchards to the fruit stands to the extent possible. That is, he does not want to rely on one orchard to provide a large portion of the demand to either of the two fruit stands he owns (or put another way, he wants each fruit stand to rely as little as possible on each of the various orchards). The budget for purchasing guavas is limited to 820 Jamaican dollars. Formulate the LP model to ensure daily demand is met within the budget and minimize the number of guava supplied to either fruit stand by any one orchard.

Orchards       

 Fruit Stands   Adric   Bevaun    Carnell   Dallan    Ervan    Demand

Montego Bay   0.50     0.70      0.60     0.40     0.90       500

Kingston      0.70     0.60      0.50     0.55     0.75       700.

Supply        150      200      300     100       600

6. Barnwest Airlines must staff the daily flights between St. Louis and Atlanta as shown in the table below. Barnwest has crews that live in both cities. Each day, a crew must fly one St. Louis to Atlanta flight and one Atlanta to St. Louis flight (the order of the flights for the crew is not important - i.e. which flight is first just depends on where the crew is located and there are plenty of crews in both cities). There must be at least one hour of down time between the two flights. For example, a St. Louis based crew can fly the 9 - 11AM St. Louis to Atlanta flight and return on the noon to 2PM Atlanta to St. Louis flight. This incurs a downtime of one hour between the two fights. They can also return to St Louis on any of the later flights, but the downtime would be longer. Barnwest wants to schedule crews to cover all flights and minimize the total downtime. Formulate LP model.

Flight Leave St. Louis Arrive Atlanta  Flight   Leave Atlanta Arrive St. Louis

  1         6A.M.      8A.M.      1          7A.M.       9A.M.

  2         9A.M.      11A.M.     2          8A.M.      10A.M. 

  3         NOON           2P.M.      3          10A.M.     NOON

  4         3P.M.      5P.M.      4          NOON           2P.M.

  5         5P.M.      7P.M.      5          2P.M.      4P.M.

  6         7P.M.      9P.M.      6          4P.M.      6P.M.

  7         8P.M.      10P.M.           7          7P.M.      9P.M..

Objective function: ______________________________________________

ST.

7. ABS Advertising uses linear programming to develop an advertising campaign to promote the 2012 annual fair that will be held in July. The model is formulated to maximize audience contact (in thousands) and determine the number of radio, newspaper and television ads to purchase. The effectiveness of TV ads (audience contact) decreases with repeated use. For large volume customers, the newspaper offers discounts ads on every other ad purchased.

7a. How many TV ads can be purchased before the effective audience decreases? ________

7b. How many newspaper ads must be purchased before discount ads can be purchased?  _____________ __

7c. According to the solution, what is the total audience contact through TV ads? _______

7d. How much will be spent on newspaper ads? ______________

7e According to the model formulation, what percentage of the radio audience is in the target market age group? ______________

  1) 500R+400N1+200N2+1000TV1+1000TV2<30000 (Budget)

  2) 3R+2N1+2N2+32TV1+24TV2>500      (Target Market age group)

  3) 1N1-1N2>2

  4) 1TV1<10

  5) .7R-.3N1-.3N2-.3TV1-.3TV2<0

  6) 1TV1+1TV2<22

Objective Function Value =    976.000

Variable      Value      Reduced Costs

  --------------  ---------------  ------------------

  R          12.000         0.000

  N1          4.000         0.000

  N2          2.000         0.000

TV1         10.000         0.000

TV2         12.000         0.000

  Constraint   Slack/Surplus     Dual Prices

  --------------  ---------------  ------------------

  1          0.000         0.024

  2         156.000         0.000

  3          0.000        -2.417

  4          0.000        10.000

  5          0.000         4.167

  6          0.000         7.083

OBJECTIVE COEFFICIENT RANGES

  Variable   Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

R        10.000      15.000     27.143

N1       -18.857      6.000     12.000

N2        1.953      6.000     12.000

  TV1        30.000      40.000 No Upper Limit

  TV2        22.917      30.000     40.000

RIGHT HAND SIDE RANGES

Constraint  Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

  1      27942.857    30000.000 No Upper Limit

  2    No Lower Limit     500.000    656.000

  3        -7.931      2.000     5.349

  4        0.000      10.000     22.000

  5       -14.400      0.000     2.880

  6        13.100      22.000     23.694

8. Oceanic Airlines transports Doohickeys, Gizmos, and Thingamagigs from Mobile to Dog Island. The airplanes have two storage compartments, one in the Front and one in the Back. The LP model is formulated to determine how to load the products in the plane in order to maximize revenue.

8a The binding constraints will not change as long as the capacity of the back storage does not exceed what value?  _____________

8b According to the solution, how much total profit will be made on the transport of Thingamagigs?  _______________

8c. What would be the total profit if 15 gizmos needed to be transported on each trip? _____

8d. What does the fourth constraint require in terms of the amount stored in the front and back?

8e. The solution value for DB is zero and the reduced cost is equal to zero. What does this indicate about the nature of the solution to the problem?

MAX 20DF+20DB+25GF+25GB+80TF+80TB

  S.T.

  1) 1DF+1DB+1.25GF+1.25GB+2TF+2TB<100

  2) 5DF+8GF+20TF<200          (capacity of the Front)

  3) 5DB+8GB+20TB<400          (capacity of the back)

  4) 10DF-5DB+16GF-8GB+40TF-20TB<0 (weight restriction)

  5) 1GF+1GB>10

  6) 1DF+1DB-2GF-2GB>0

Objective Function Value =   2330.000

Variable      Value      Reduced Costs

  --------------  ---------------  ------------------

  DF         20.000         0.000

  DB          0.000         0.000

  GF          0.000         0.000

  GB         10.000         0.000

  TF          5.000         0.000

  TB         16.000         0.000

  Constraint   Slack/Surplus     Dual Prices

  --------------  ---------------  ------------------

  1          25.500         0.000

  2          0.000         4.000

  3          0.000         4.000

  4          0.000         0.000

  5          0.000        -7.000

  6          0.000         0.000

OBJECTIVE COEFFICIENT RANGES

  Variable   Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

DF        20.000      20.000     20.000

DB    No Lower Limit      20.000     20.000

GF    No Lower Limit      25.000     25.000

GB        25.000      25.000     32.000

TF        80.000      80.000     80.000

TB        80.000      80.000     80.000

RIGHT HAND SIDE RANGES

Constraint  Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

  1        74.500     100.000 No Upper Limit

  2       100.000     200.000    200.000

  3       400.000     400.000    655.000

  4        0.000      0.000 No Upper Limit

  5        0.000      10.000     20.000

  6       -20.000      0.000     20.000

  9. Allied Grocers can choose between three types of trucks for lease by Camel Trucking. For example, Type A trucks can haul 20 m³ of refrigerated products and 40 m³ of non-refrigerated products on each trip. Allied Grocers needs to haul 300 m³ of refrigerated products each day and 720 m³ of non-refrigerated products. Allied Grocers wants to determine how many of each type of truck to lease in order to minimize the average daily leasing costs.

9a. How much would the daily leasing costs change if 320 m3 refrigerated products needed to be transported?

_________

9b. How much refrigerated products will be hauled each day by Type B trucks? _________

9c. If Camel Trucking decides they must increase the daily leasing cost of Type A and Type B trucks by 10 percent and Type C Trucks by 20 percent, what would be the impact on the solution. Provide support for your answer.

9d. According to the solution, what will be the total daily amount spent on leasing Type C trucks?

  ________________

9e. How would the solution change if Camel Trucking made four more Type B trucks available each day.

MIN 60A+50B+40C

  S.T.

  1) 20A+30B>300      (Refrigerated Products)

  2) 40A+30B+100C>720    (Non-refrigerated Products)

  3) 1A<8

  4) 1B<6

Objective Function Value =    780.000

Variable      Value      Reduced Costs

  --------------  ---------------  ------------------

  A          6.000         0.000

  B          6.000         0.000

  C          3.000         0.000

  Constraint   Slack/Surplus     Dual Prices

  --------------  ---------------  ------------------

  1          0.000        -2.200

  2          0.000        -0.400

  3          2.000         0.000

  4          0.000        28.000

OBJECTIVE COEFFICIENT RANGES

  Variable   Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

A        41.333      60.000 No Upper Limit

B    No Lower Limit      50.000     78.000

C         0.000      40.000    133.333

RIGHT HAND SIDE RANGES

Constraint  Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

  1       180.000     300.000    340.000

  2       420.000     720.000 No Upper Limit

  3        6.000      8.000 No Upper Limit

  4        4.667      6.000     10.000

10.As part of a settlement for a class action lawsuit, Sunjoe must provide sufficient funds to make annual payments of $150,000, $160,000, $130,000, $100,000 and $80,000 on May 1^st over the next five years beginning with 2012. The plan involves the purchase of the following investment alternatives; Government Securities Type 1, which cost $1,000 each and have an annual return of 5 percent; Government Securities Type II which cost $1,500 each and return of 6 percent annually; and one year CD's which are expected to have 4 percent return each year. The LP model is formulated on the next page.

10a. What is the total amount that will be put into CD's over the four years? ______

10b. How many years does it take for Government Security Type II to mature? _________

10c. How much money does Sunjoe need to provide on May 1^st of 2012 in order to cover the class action lawsuit? _________

10d. Assume the plaintiffs would like a larger payout on May 1^st of 2012 and are willing to decrease the payout in year five to $60,000. What is the maximum additional funds Sunjoe should be willing to make available on May 1^st 2012 in order to reduce the payout on year five to $60,000?  _________

10e. Demonstrate whether it would be beneficial to Sunjoe to add $9,000 to the initial payment on May 1^st of 2012 (i.e, $159,000) in order to reduce the payout during year 3 to $120,000.

MIN 1F

  S.T.

  1) 1F-1GS1-1.5GS2-1S1=150       (Year 2012)

  2) 0.05GS1+0.09GS2+1.04S1-1S2=160

  3) 1.05GS1+0.09GS2+1.04S2-1S3=130

  4) 1.09GS2+1.04S3-1S4=100

  5) 1.04S4>80

Objective Function Value =    575.931

Variable      Value      Reduced Costs

  --------------  ---------------  ------------------

  F         575.931         0.000

GS1         285.827         0.000

GS2          0.000         0.382

  S1         140.104         0.000

  S2          0.000         0.019

  S3         170.118         0.000

  S4         76.923         0.000

  Constraint   Slack/Surplus     Dual Prices

  --------------  ---------------  ------------------

  1          0.000        -1.000

  2          0.000        -0.962

  3          0.000        -0.907

  4          0.000        -0.872

  5          0.000        -0.838

OBJECTIVE COEFFICIENT RANGES

  Variable   Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

F         0.000      1.000 No Upper Limit

  GS1        -0.952      0.000     0.019

  GS2        -0.382      0.000 No Upper Limit

S1        -0.019      0.000     11.086

S2        -0.019      0.000  No Upper Limit

S3        -0.907      0.000     0.364

S4        -0.872      0.000 No Upper Limit

RIGHT HAND SIDE RANGES

Constraint  Lower Limit   Current Value  Upper Limit

 ------------ --------------- --------------- ---------------

  1       -425.931     150.000 No Upper Limit

  2        14.291     160.000 No Upper Limit

  3       -170.118     130.000    3189.882

  4       -76.923     100.000    3282.277

  5        0.000      80.000    3389.568

41. The GPSS program on the next three pages simulates customer arrivals at a catalog store to pick up an order and then pay for the product ordered. Customers first attempt to park their car on a lot if a space is available. Assume customers that are unable to park their car balk. Customers that are able to park their car enter the store. Customers go through two sequential lines. Customers first enter a line with two servers to Receive their package. Customers then enter one of two available lines to Pay for their products from one of two cashiers. Each question is worth two points.

41a). The expected number of cars on the parking lot is equal to what value? _____________

41b). What was the average time required to pay for packages at the second cashier?

  _____________

41c. How much simulation time was required to process 500 customers? ____________

41d). For customers who actually had to wait in line to receive their package, what was the average waiting time?  _____________

41e). The cars spent an average of how many seconds in a parking space? _____________

 41f). According to the program, what is the longest possible time between customer arrivals?  ___________

41g). How much time on average did a customer spend in the system (from the point of entry to the point of exiting the system)?

41h) What is the expected number of customers in line to pay at cashier #1?_____________

41). How many customers balked? _________

41j). How many customers had a transit time less than 1,250 seconds? ____________