PROBLEM SET

1. network figure the shipping routes from nodes I and 2 by way of nodes 3 and 4. Ihe unit shipping costs arc shown on the rcspectiSt arcs (a) Develop the corresponding transshipment moae(.

(b) Solve(he problcm, and show how the shipments are routed from the sources to the

1. In Problem l,suppose that source node 1 can be linked to source node 2 with a unit shir ping cost of Sl.lbe unit shipping cost from node 1 to node 3 is increased to $5. Formulate the problem as a transshipment model, and find the optimum shipping schedule.
2. network in figure 5.12 shows the routes for shipping cars from three plants (nodes 1.

2, and 3) to three dealers (nodes 6 to 8) by way of two distribution centers (nodes 4 and shipping costs per car (in $100) are shown on the arcs (a) Solve the problem as a transshipment model.

(b) Suppose that distribution center 4 can sell 240 cars directly to customers find the new optimum solution.

Network for Problem 1, Set 5.5a

1100

1000

FIGURE 5.12

Network for Problem 3, Set 5.5a 1200

1. Consider the transportation problem in which two factories supply three stores with a commodity. The number of supply units available at sources I and 2 is 200 and 300; .those demanded at stores 1, 2, and 3 are 100, 200,vnd 50, respectively. Units may be transshipped among the factories and the stores before reaching their final destination.

Find the optimal shipping schedule based on the unit costs in Table 5.45.

TABLE 5.45

Factory Store

1 2 1 2 3

$7 $1

$7 $8 $9 $4 $3 $5 $1 $1 $0 $4

1

Factory 2

1

Store 2

3

1. Consider the oil pipeline network shewn in Figure 5.13. The different nodes represent pumping and receiving stations Distances in miles between the stations are shown on the network. nie transportation cost per gallon between two nodes is directly proportional to the length of the pipeline. Develop the associated transshipment model, and find the optimum solution using TORA
2. Shortest-Route. Problem. Find the shortest route between nodes I and 7 of the network in figure 5.14 by formulating the problem as a transshipment model. ne distance between the different nodes in shown on the network. (Hint: Assume that node I has a net supply of I unit, and node 7 has a net demand also of I unit.)

FIGURE 5.13

Network for Problem 5, Set 5Sa

FIGURE 5.14

Network for Problem 6, Set 5.5a

1. In the transshipment model of Example 5.5-1, define x,) as the amount shipped from node i to node J. problem can be formulated as a linear program in which each node pro duces a constraint equation. Develop thc linear program, and show that the resulting formulation has thc characteristic that the constraint coefficients, of the variable are

1. For constraint i

-1, For constraint j

0, Otherwise

1. An employment agency must provide the following laborers over the next 5 months:

Month	1	2	3	4	5
No. of laborers	100	120	80	170	50

Because the cost of labor depends on the length of employment, it may be more economical to keep more laborers than needed during some months of the 5-month planning horizon.%e following table estimates the labor cost as a function of the length of

employment:

Months of employment	1	2	4	5
Cost per laborer (S)	100	130 180	220	250

Formulate the problem as a linear program.Then, using proper algebraic manipula tions of the constraint equations, show that the model can be converted to a transshipment model, and find the optimum solutioq to the Scheduling problem. (Hint: Use the transshipment characteristic in Problem 7 to convert the constraints of the scheduling problem into those of the transshipment

model.)