#1. We need to ship several tons of cargo overnight from Boston to Dallas. The shipping capacity (in tons) on each of the routes that our cargo planes fly each night is shown in the table:

Maximum	From	To
31	Boston	Baltimore
28	Boston	Pittsburgh
35	Boston	Cincinnati
12	Baltimore	Atlanta
8	Baltimore	Cincinnati
15	Pittsburgh	Atlanta
20	Pittsburgh	Chicago
25	Atlanta	Memphis
10	Atlanta	Dallas
9	Cincinnati	Memphis
15	Cincinnati	Chicago
18	Memphis	Chicago
35	Memphis	Dallas
19	Chicago	Memphis
16	Chicago	Dallas

What is the maximum number of tons we can ship from Boston to Dallas in one night? Formulate an LP problem to model this situation, set it up in Excel, and solve. Here, please enter the maximum possible number of tons that can get from Boston to Dallas in one night.

#2

Suppose we will make Tea and Lemonade bottles to sell at a stand. We have 1,010 bottles, 400 dollars, and 600 minutes available. Each bottle of tea costs 1 bottle, 0.50 dollars, and 0.75 minutes. Each bottle of lemonade costs 1 bottle, 0.20 dollars, and 0.25 minutes.

The profit per tea bottle is 3 dollars for the first 600 we make. Each tea bottle made in excess of 600 has a profit of 3.50 dollars per bottle (due to bulk purchase discounts). The profit per lemonade bottle is 2 dollars (regardless of how many we make).

Formulate an ILP problem to model this situation and use a solver in Excel to find the maximum possible profit.

#3. Nodes for a wireless network can be installed at the following seven locations of a corporate campus for the indicated cost. The regions the node can reach are listed as well.

Node 1: Regions 1,2,5; Cost: $700 Node 2: Regions 3,6,7; Cost: $600 Node 3: Regions 2,3,7,9; Cost $900 Node 4: Regions 1,3,6,10; Cost $1250 Node 5: Regions 2,4,6,8; Cost $850 Node 6: Regions 4,5,8,10; Cost $1000 Node 7: Regions 1,5,7,8,9; Cost $100

If we install Nodes 1,2,3,4, and 5, then all ten regions are reached. The total cost is $4,300. Is there a cheaper way to reach all ten regions? What is it? Formulate the situation as an ILP problem and use a solver in Excel to find the cheapest way to reach all 10 regions with at least one node. Enter the cost of the cheapest way here:

#4 Three types of appliances, A, B, and C, can be manufactured on one of two machines, 1 and 2.

The variable costs (that is, per unit cost) of each appliance on each machine are

	A	B	C
1	$13	$8	$10
2	$11	$12	$7

There is an initial setup cost (fixed cost) to make each appliance on each machine. (Each fixed cost is incurred at most once, and it is incurred only if one or more of that appliance type is made on that machine):

	A	B	C
1	$52	$90	$60
2	$64	$60	$75

The time (in hours) required to make each appliance on each machine:

	A	B	C
1	0.4	1.1	0.9
2	0.5	1.2	1.3

We have to make 4 of appliance A, 9 of appliance B, and 5 of appliance C for a customer order. We can use Machine 1 for up to 10 hours and Machine 2 for up to 7 hours.

Formulate an ILP problem that models this situation. Set it up in Excel and use solver to find the optimal solution. What is the minimum cost to make the necessary numbers of appliances?

#5

#6Continued from previous.

What if we can send at most 1000 units along each arc? Modify your spreadsheet (from the previous problem) and find an optimal solution. What is the lowest possible shipment cost now?