# Problem 10-21 (Algorithmic)

United Express Service (UES) uses large quantities of packaging materials at its four distribution hubs. After screening potential suppliers, UES identified six vendors that can provide packaging materials that will satisfy its quality standards. UES asked each of the six vendors to submit bids to satisfy annual demand at each of its four distribution hubs over the next year. The following table lists the bids received (in thousands of dollars). UES wants to ensure that each of the distribution hubs is serviced by a different vendor. Which bids should UES accept, and which vendors should UES select to supply each distribution hub?

The value of the objective function is fill in the blank  thousands of dollars.

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Problem 10-05

Premier Consulting’s two consultants, Avery and Baker, can be scheduled to work for clients up to a maximum of 160 hours each over the next four weeks. A third consultant, Campbell, has some administrative assignments already planned and is available for clients up to a maximum of 140 hours over the next four weeks. The company has four clients with projects in process. The estimated hourly requirements for each of the clients over the four-week period are as follows:

Hourly rates vary for the consultant–client combination and are based on several factors, including project type and the consultant’s experience. The rates (dollars per hour) for each consultant–client combination are as follows:

1. Choose the correct network representation of the problem.

Model______?_______

1. Formulate the problem as a linear program, with the optimal solution providing the hours each consultant should be scheduled for each client to maximize the consulting firm’s billings. What is the schedule and what is the total billing?
Let xij = number of hours from consultant i assigned to client j.
1. New information shows that Avery doesn’t have the experience to be scheduled for client B. If this consulting assignment is not permitted, what impact does it have on total billings? What is the revised schedule?

Problem 10-07 (Algorithmic)

Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation’s major residential markets, the annual demand in each market (in megawatts or MWs), and the cost to supply electricity to each market from each power generation plant (prices are in \$/MW).

1. If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem? Which cities should be supplied by which power plants? What is the total annual power distribution cost for this solution? If required, round your answers to two decimal places.
The optimal solution is to produce fill in the blank 1 MWs in Los Angeles, fill in the blank 2 MWs in Tulsa, and fill in the blank 3 MWs in Seattle. The total distribution cost of this solution is \$  fill in the blank 4.
2. If at most 3800 MWs of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation? If required, round your answers to two decimal places.
The optimal solution is to produce fill in the blank 5 MWs in Los Angeles, fill in the blank 6 MWs in Tulsa, and fill in the blank 7 MWs in Seattle. The total distribution cost of this solution is \$  fill in the blank 8. The increase in cost associated with the additional constraints is \$  fill in the blank 9.

Problem 10-25

Cleveland Area Rapid Delivery (CARD) operates a delivery service in the Cleveland metropolitan area. Most of CARD’s business involves rapid delivery of documents and parcels between offices during the business day. CARD promotes its ability to make fast and on-time deliveries anywhere in the metropolitan area. When a customer calls with a delivery request, CARD quotes a guaranteed delivery time. The following network shows the street routes available. The numbers above each arc indicate the travel time in minutes between the two locations.

Develop a linear programming model that can be used to find the minimum time required to make a delivery from location 1 to location 6. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.

1. How long does it take to make a delivery from location 1 to location 6?
fill in the blank 46 minutes
2. Assume that it is now 1:00 P.M. and that CARD just received a request for a pickup at location 1. The closest CARD courier is 8 minutes away from location 1. If CARD provides a 20% safety margin in guaranteeing a delivery time, what is the guaranteed delivery time if the package picked up at location 1 is to be delivered to location 6?
fill in the blank 47 p.m.

Problem 10-31

A long-distance telephone company uses a fiber-optic network to transmit phone calls and other information between locations. Calls are carried through cable lines and switching nodes. A portion of the company’s transmission network is shown here. The numbers above each arc show the capacity in thousands of messages that can be transmitted over that branch of the network.

To keep up with the volume of information transmitted between origin and destination points, use the network to determine the maximum number of messages that may be sent from a city located at node 1 to a city located at node 7.

Maximum number of messages = fill in the blank 1

Problem 10-03

Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:

1. Choose the correct network representation of this problem.

Model______?________

1. Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.
Let xij = amount shipped from supply node i to demand node j.
1. Describe the distribution plan and show the total distribution cost.
1. Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity?
AmountCostSouthern – Hamiltonfill in the blank 38\$fill in the blank 39Southern – Butlerfill in the blank 40fill in the blank 41Southern – Clermontfill in the blank 42fill in the blank 43Northwest – Hamiltonfill in the blank 44fill in the blank 45Northwest – Butlerfill in the blank 46fill in the blank 47Northwest – Clermontfill in the blank 48fill in the blank 49Total Cost\$fill in the blank 50

2. From the new solution we see that Tri-County should contract with ________?_________ Gas for the additional 100 units.

Problem 10-13

Sports of All Sorts produces, distributes, and sells high-quality skateboards. Its supply chain consists of three factories (located in Detroit, Los Angeles, and Austin) that produce skateboards. The Detroit and Los Angeles facilities can produce 350 skateboards per week, but the Austin plant is larger and can produce up to 700 skateboards per week. Skateboards must be shipped from the factories to one of four distribution centers, or DCs (located in Iowa, Maryland, Idaho, and Arkansas). Each distribution center can process (repackage, mark for sale, and ship) at most 500 skateboards per week.

Skateboards are then shipped from the distribution centers to retailers. Sports of All Sorts supplies three major U.S. retailers: Just Sports, Sports ’N Stuff, and The Sports Dude. The weekly demands are 200 skateboards at Just Sports, 500 skateboards at Sports ’N Stuff, and 650 skateboards at The Sports Dude. The following tables display the per-unit costs for shipping skateboards between the factories and DCs and for shipping between the DCs and the retailers.

1. Choose the correct network representation of this problem.
1. Build a model to minimize the transportation cost of a logistics system that will deliver skateboards from the factories to the distribution centers and from the distribution centers to the retailers. What is the optimal production strategy and shipping pattern for Sports of All Sorts? What is the minimum attainable transportation cost? If required, round your answers to two decimal places.
Let xij = units shipped from node i to node j.Minfill in the blank 2x1,4fill in the blank 3x1,5fill in the blank 4x1,6fill in the blank 5x1,7fill in the blank 6x2,4fill in the blank 7x2,5fill in the blank 8x2,6fill in the blank 9x2,7fill in the blank 10x3,4fill in the blank 11x3,5fill in the blank 12x3,6fill in the blank 13x3,7fill in the blank 14x4,8fill in the blank 15x4,9fill in the blank 16x4,10fill in the blank 17x5,8fill in the blank 18x5,9fill in the blank 19x5,10fill in the blank 20x6,8fill in the blank 21x6,9fill in the blank 22x6,10fill in the blank 23x7,8fill in the blank 24x7,9fill in the blank 25x7,10

for all i and j.
Solving the formulation above, the optimal cost is \$  fill in the blank 96 per week.

1. Sports of All Sorts is considering expansion of the Iowa DC capacity to 800 units per week. The annual amortized cost of expansion is \$40,000. Should the company expand the Iowa DC capacity so that it can process 800 skateboards per week? (Assume 50 operating weeks per year.)
yes or no
Explain.
The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Problem 10-11

The distribution system for the Herman Company consists of three plants, two warehouses, and four customers. Plant capacities and shipping costs per unit (in \$) from each plant to each warehouse are as follows:

Customer demand and shipping costs per unit (in \$) from each warehouse to each customer are as follows:

1. Choose the correct network representation of this problem.

Mosel____?_________

1. Formulate a linear programming model of the problem. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.
Let Xij represents relation between plants to warehouses or relation between warehouses to customers.MINX14+X15+X24+X25X34+X35+X46+X47+X48+X49+X56+X57+X58+X59
2. S.T.1)fill in the blank 16X14fill in the blank 17X15<fill in the blank 182)fill in the blank 19X24fill in the blank 20X25<fill in the blank 213)fill in the blank 22X34fill in the blank 23X35<fill in the blank 244)fill in the blank 25X46fill in the blank 26X47fill in the blank 27X48fill in the blank 28X49fill in the blank 29X14fill in the blank 30X24fill in the blank 31X34fill in the blank 325)fill in the blank 33X56fill in the blank 34X57fill in the blank 35X58fill in the blank 36X59fill in the blank 37X15fill in the blank 38X25fill in the blank 39X35fill in the blank 406)fill in the blank 41X46fill in the blank 42X56fill in the blank 437)fill in the blank 44X47fill in the blank 45X57fill in the blank 468)fill in the blank 47X48fill in the blank 48X58fill in the blank 499)fill in the blank 50X49fill in the blank 51X59fill in the blank 52
3. Solve the linear program to determine the optimal shipping plan.
Objective Function Value = fill in the blank 53VariableValueReduced CostsX14fill in the blank 54fill in the blank 55X15fill in the blank 56fill in the blank 57X24fill in the blank 58fill in the blank 59X25fill in the blank 60fill in the blank 61X34fill in the blank 62fill in the blank 63X35fill in the blank 64fill in the blank 65X46fill in the blank 66fill in the blank 67X47fill in the blank 68fill in the blank 69X48fill in the blank 70fill in the blank 71X49fill in the blank 72fill in the blank 73X56fill in the blank 74fill in the blank 75X57fill in the blank 76fill in the blank 77X58fill in the blank 78fill in the blank 79X59fill in the blank 80fill in the blank 81

4. There is an excess capacity of  units at .Plant 1, Plant 2 or Plant 3?

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