190 Ch 10

*Def:* Shortest path between two nodes in a network

Shortest Route

*Def:* Maximum amount of flow that can enter and exit a network system during a given period of time

Maximal Flow

In the general linear programming model of the assignment problem, *a.* one agent can do parts of several tasks. *b.* one task can be done by several agents. *c.* each agent is assigned to its own best task. *d.* one agent is assigned to one and only one task

*d.* one agent is assigned to one and only one task.

The objective of the transportation problem is to *a.* identify one origin that can satisfy total demand at the destinations and at the same time minimize total shipping cost. *b.* minimize the number of origins used to satisfy total demand at the destinations. *c.* minimize the number of shipments necessary to satisfy total demand at the destinations. *d.* minimize the cost of shipping products from several origins to several destinations

*d.* minimize the cost of shipping products from several origins to several destinations

The assignment problem constraint x<sub>31 + x<sub>32 + x<sub>33 + x<sub>34 ≤ 2 means *a.* agent 3 can be assigned to 2 tasks. *b.* agent 2 can be assigned to 3 tasks. *c.* a mixture of agents 1, 2, 3, and 4 will be assigned to tasks. *d.* there is no feasible solution

*a.* agent 3 can be assigned to 2 tasks.

We assume in the maximal flow problem that *a.* the flow out of a node is equal to the flow into the node. *b.* the source and sink nodes are at opposite ends of the network. *c.* the number of arcs entering a node is equal to the number of arcs exiting the node. *d.* None of the alternatives is correct.

*a.* the flow out of a node is equal to the flow into the node.

The assignment problem is a special case of the *a.* transportation problem. *b.* transshipment problem. *c.* maximal flow problem. *d.* shortest-route problem

*a.* transportation problem

The number of units shipped from origin i to destination j is represented by *a.* x<sub>ij *b.* x<sub>ji *c.* c<sub>ij *d.* c<sub>ji

*a.* x_ij

If a transportation problem has four origins and five destinations, the LP formulation of the problem will have *a.* 5 constraints *b.* 9 constraints *c.* 18 constraints *d.* 20 constraints

*b.* 9 constraints

Which of the following is not true regarding an LP model of the assignment problem? *a.* Costs appear in the objective function only. *b.* All constraints are of the ≥ form. *c.* All constraint left-hand side coefficient values are 1 *d.* All decision variable values are either 0 or 1

*b.* All constraints are of the ≥ form.

The shortest-route problem finds the shortest-route *a.* from the source to the sink. *b.* from the source to any other node. *c.* from any node to any other node. *d.* from any node to the sink

*b.* from the source to any other node.

Constraints in a transshipment problem *a.* correspond to arcs. *b.* include a variable for every arc. *c.* require the sum of the shipments out of an origin node to equal supply. *d.* All of the alternatives are correct.

*b.* include a variable for every arc.

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, *a.* the nodes represent stoplights. *b.* the arcs represent one way streets. *c.* the nodes represent locations where speed limits change. *d.* None of the alternatives is correct.

*b.* the arcs represent one way streets.

The problem which deals with the distribution of goods from several sources to several destinations is the *a.* maximal flow problem *b.* transportation problem *c.* assignment problem *d.* shortest-route problem

*b.* transportation problem

Which of the following is not true regarding the linear programming formulation of a transportation problem? *a.* Costs appear only in the objective function. *b.* The number of variables is (number of origins) x (number of destinations). *c.* The number of constraints is (number of origins) x (number of destinations). *d.* The constraints' left-hand side coefficients are either 0 or 1.

*c.* The number of constraints is (number of origins) × (number of destinations)

The difference between the transportation and assignment problems is that *a.* total supply must equal total demand in the transportation problem *b.* the number of origins must equal the number of destinations in the transportation problem *c.* each supply and demand value is 1 in the assignment problem *d.* there are many differences between the transportation and assignment problem

*c.* each supply and demand value is 1 in the assignment problem

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.

*c.* indicate the direction of the flow.

Consider a shortest route problem in which a bank courier must travel between branches and the main operations center. When represented with a network, *a.* the branches are the arcs and the operations center is the node. *b.* the branches are the nodes and the operations center is the source. *c.* the branches and the operations center are all nodes and the streets are the arcs. *d.* the branches are the network and the operations center is the node.

*c.* the branches and the operations center are all nodes and the streets are the arcs.

The parts of a network that represent the origins are *a.* the capacities *b.* the flows *c.* the nodes *d.* the arc

*c.* the nodes

In a transshipment problem, shipments *a.* cannot occur between two origin nodes. *b.* cannot occur between an origin node and a destination node. *c.* cannot occur between a transshipment node and a destination node. *d.* can occur between any two nodes.

*d.* can occur between any two nodes.

*Def:* The lines connecting the nodes in a network

Arc(s)

*Def:* A network flow problem that often involves the assignment of agents to tasks; It can be formulated as a linear program and is a special case of the transportation problem

Assignment Problem

*LP Model* Minimize product of arc flows and cost, subject to (1) Sum of agents' Outflows less than or equal to (≤) 1 (2) Sum of tasks' Inflows equal to (=) 1 (x_ij≥0 for all i & j)

Assignment Problem

*Def:* A variation of the basic transportation problem in which some or all the arcs are subject to capacity restraints

Capacitated Transportation Problem

*Def:* A variation of the transshipment problem in which some or all of the arcs are subject to capacity restraints

Capacitated Transshipment Problem

*Def:* An origin added to a transportation problem to make the total supply equal total demand; The supply assigned to the dummy origin is the difference between total demand and total supply

Dummy Origin(s)

*(T/F)* A dummy origin in a transportation problem is used when supply exceeds demand (supply ≥ demand)

False

*(T/F)* A transportation problem with 3 sources and 4 destinations will have 7 decision variables.

False

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

False

*(T/F)* When the number of agents exceeds the number of tasks in an assignment problem, one or more dummy tasks must be introduced in the LP formulation or else the LP will not have a feasible solution.

False

*Def:* Maximum flow for an arc of the network; May not be equal in reverse directions despite equal lengths

Flow Capacity

*LP Model* Maximize arc from destination to origin, subject to (1) Each Arc less than or equal to (≤) their Capacity (2) Each node's Outflows minus (-) their Inflows, equal to (=) 0 (x_ij≥0 for all i and j)

Maximal Flow Problem

*Def:* A graphical representation of a problem consisting of numbered circles (nodes) interconnected by a series of lines (arcs); Arrowheads on the arcs show the direction of flow

Network

*Def:* Transportation, assignment, and transshipment problems

Network Flow Problems

*Def:* The intersection or junction points of a network

Node(s)

*ex:* Arc less than or equal to (≤) units; "Limited transportation space"

Route Capacity Constraint

*ex:* Arc greater than or equal to (≥) units; "Deliver at least x units"

Route Minimum Constraint

*LP Model* Minimize product of flow of units and cost, subject to (1) Origin Outflow less than or equal to (≤) Supply (2) Destination Inflow equal to (=) Demand* (x_ij≥0 for all i and j; *subtract outflows, if any)

Transportation Problem

*LP Model* Minimize product of arcs and distance, subject to (1) Sum of origin's Outflows equal to (=) 1 (2) Transshipment's Outflows minus (-) Inflows equal to (=) 0 (3) Sum of destination's Inflows equal to (=) 1 (x_ij≥0 for all i and j)

Shortest-Route Problem

*Def:* Set of all interconnected resources involved in producing and distributing a product

Supply Chain

*Def:* An extension of the transportation problem to distribution problems involving transfer points and possible shipments between any pair of nodes

Transhipment Problem

*Def:* A network flow problem that often involves minimizing the cost of shipping goods from a set of origins to a set of destinations; It can be formulated and solved as a linear program by including a variable for each arc and a constant for each node

Transportation Problem

*LP Model* Minimize product of arc flows and cost, subject to (1) Origin's Outflows minus (-) Inflows less than or equal to (≤) Supply (2) Transshipment's Outflows minus (-) Inflows equal to zero (= 0) (3) Destination's Outflows minus (-) Inflows equal to (=) Demand (x_ij≥0 for all i & j)

Transshipment Problem

*(T/F)* A transshipment constraint must contain a variable for every arc entering or leaving the node.

True

*(T/F)* Converting a transportation problem LP from cost minimization to profit maximization requires only changing the objective function; the conversion does not affect the constraints.

True

*(T/F)* The capacitated transportation problem includes constraints which reflect limited capacity on a route.

True

*(T/F)* The direction of flow in the shortest-route problem is always out of the origin node and into the destination node.

True

*(T/F)* The maximal flow problem can be formulated as a capacitated transshipment problem.

True

*(T/F)* The shortest-route problem is a special case of the transshipment problem

True

*(T/F)* Transshipment problem allows shipments both in and out of some nodes while transportation problems do not.

True

*(T/F)* When a route in a transportation problem is unacceptable, the corresponding variable can be removed from the LP formulation.

True

*(T/F)* Whenever total supply is less than total demand in a transportation problem, the LP model does not determine how the unsatisfied demand is handled.

True