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Exam 10: Distribution and Network Models

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Question 21

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Arcs in a transshipment problem

A) must connect every node to a transshipment node.
B) represent the cost of shipments.
C) indicate the direction of the flow.
D) All of the alternatives are correct.

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Question 22

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Transshipment problem allows shipments both in and out of some nodes while transportation problems do not.

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Question 23

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A beer distributor needs to plan how to make deliveries from its warehouse (Node 1)to a supermarket (Node 7),as shown in the network below.Develop the LP formulation for finding the shortest route from the warehouse to the supermarket.

A transportation problem with 3 sources and 4 destinations will have 7 variables in the objective function.

A professor has been contacted by four not-for-profit agencies that are willing to work with student consulting teams.The agencies need help with such things as budgeting,information systems,coordinating volunteers,and forecasting.Although each of the four student teams could work with any of the agencies,the professor feels that there is a difference in the amount of time it would take each group to solve each problem.The professor's estimate of the time,in days,is given in the table below.Use the computer solution to see which team works with which project. $\begin{array} { r | c c c c } & &\quad\quad\quad\quad\quad\quad\quad\quad{ \text { Projects } } \\\text { Team } & \text { Budgeting } & \text { Information } & \text { Volunteers } & \text { Forecasting } \\\hline \text { A } & 32 & 35 & 15 & 27 \\\text { B } & 38 & 40 & 18 & 35 \\\text { C } & 41 & 42 & 25 & 38 \\\text { D } & 45 & 45 & 30 & 42\end{array}$ ASSIGNMENT PROBLEM ************************ OBJECTIVE: MINIMIZATION SUMMARY OF UNIT COST OR REVENUE DATA ********************************************* $\begin{array} { c c c c c } & & { \text { TASK } } \\\text { AGENT } & 1 & 2 & 3 & 4 \\\hline1 & 32 & 35 & 15 & 27 \\2 & 38 & 40 & 18 & 35 \\3 & 41 & 42 & 25 & 38 \\4 & 45 & 45 & 30 & 42\end{array}$ OPTIMAL ASSIGNMENTS COST/REVENUE ************************ *************** ASSIGN AGENT 3 TO TASK 1 41 ASSIGN AGENT 4 TO TASK 2 45 ASSIGN AGENT 2 TO TASK 3 18 ASSIGN AGENT 1 TO TASK 4 27 ------------------------------------------- ----- TOTAL COST/REVENUE 131

The parts of a network that represent the origins are

The capacitated transportation problem includes constraints which reflect limited capacity on a route.

The network flows into and out of demand nodes are what makes the production and inventory application modeled in the textbook a

Draw the network for this assignment problem. $\begin{array} { l l } \text { Min } & 10 x _ { 1 \mathrm {~A} } + 12 x _ { 1 \mathrm {~B} } + 15 x _ { 1 \mathrm { C } } + 25 x _ { 1 \mathrm { D } } + 11 x _ { 2 \mathrm {~A} } + 14 x _ { 2 \mathrm {~B} } + 19 x _ { 2 \mathrm { C } } + 32 x _ { 2 \mathrm { D } } \\& + 18 x _ { 3 \mathrm {~A} } + 21 x _ { 3 \mathrm {~B} } + 23 x _ { 3 \mathrm { C } } + 29 x _ { 3 \mathrm { D } } + 15 x _ { 4 \mathrm {~A} } + 20 x _ { 4 \mathrm {~B} } + 26 x _ { 4 \mathrm { C } } + 28 x _ { 4 \mathrm { D } } \\\text { s.t. } \quad & x _ { 1 \mathrm {~A} } + x _ { 1 \mathrm {~B} } + x _ { 1 \mathrm { C } } + x _ { 1 \mathrm { D } } = 1 \\& x _ { 2 \mathrm {~A} } + x _ { 2 \mathrm {~B} } + x _ { 2 \mathrm { C } } + x _ { 2 \mathrm { D } } = 1 \\& x _ { 3 \mathrm {~A} } + x _ { 3 \mathrm {~B} } + x _ { 3 \mathrm { C } } + x _ { 3 \mathrm { D } } = 1 \\& x _ { 4 \mathrm {~A} } + x _ { 4 \mathrm {~B} } + x _ { 4 \mathrm { C } } + x _ { 4 \mathrm { D } } = 1 \\& x _ { 1 \mathrm {~A} } + x _ { 2 \mathrm {~A} } + x _ { 3 \mathrm {~A} } + x _ { 4 \mathrm {~A} } = 1 \\& x _ { 1 \mathrm {~B} } + x _ { 2 \mathrm {~B} } + x _ { 3 \mathrm {~B} } + x _ { 4 \mathrm {~B} } = 1 \\& x _ { 1 \mathrm { C } } + x _ { 2 \mathrm { C } } + x _ { 3 \mathrm { C } } + x _ { 4 \mathrm { C } } = 1 \\& x _ { 1 \mathrm { D } } + x _ { 2 \mathrm { D } } + x _ { 3 \mathrm { D } } + x _ { 4 \mathrm { D } } = 1\end{array}$

Write the linear program for this transshipment problem.

Define the variables and constraints necessary in the LP formulation of the transshipment problem.

Constraints in a transshipment problem

Consider a maximal flow problem in which vehicle traffic entering a city is routed among several routes before eventually leaving the city.When represented with a network,

The number of units shipped from origin i to destination j is represented by

A transshipment constraint must contain a variable for every arc entering or leaving the node.

In the LP formulation of a maximal flow problem,a conservation-of-flow constraint ensures that an arc's flow capacity is not exceeded.

A foreman is trying to assign crews to produce the maximum number of parts per hour of a certain product.He has three crews and four possible work centers.The estimated number of parts per hour for each crew at each work center is summarized below.Solve for the optimal assignment of crews to work centers. $\begin{array}{lcccc} &{\text { Work Center }} \\& \text { WC1 } & \text { WC2 } & \text { WC3 } & \text { WC4 } \\ \text { Crew A } & 15 & 20 & 18 & 30 \\\text { Crew B } & 20 & 22 & 26 & 30 \\\text { Crew C } & 25 & 26 & 27 & 30\end{array}$

Is it a coincidence to obtain integer solutions to network problems? Explain.

If a transportation problem has four origins and five destinations,the LP formulation of the problem will have

The problem which deals with the distribution of goods from several sources to several destinations is the