distribution and network models (1/2) chapter 6 mangt 521 (b): quantitative management

DISTRIBUTION AND NETWORK MODELS (1/2)

Chapter 6

MANGT 521 (B): Quantitative Management

Chapter 6 (1/2)-2

Chapter 6Distribution and Network Models

1. Transportation Problem• Network Representation• General LP Formulation

2. Assignment Problem• Network Representation• General LP Formulation

3. Transshipment Problem• Network Representation• General LP Formulation

4. Shortest-Route Problem

Chapter 6 (1/2)-3

Transportation, Assignment, and Transshipment Problems

A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

Examples of network problems:Transportation, assignment, transshipment, shortest-route, and maximal flow problems of this chapter as well as the minimal spanning tree and PERT/CPM problems (in Project Management courses).

Chapter 6 (1/2)-4

Transportation, Assignment, and Transshipment Problems

Each of the four problems of this chapter can be formulated as linear programs and solved by general purpose LP computer package.

For each of the four problems, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables.

However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.

Chapter 6 (1/2)-5

1. Transportation Problem

The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.

The # of constraints in a transportation LP formulation = (# of origins) + (# of destinations) = m + n

The network representation for a transportation problem with two sources and three destinations is given on the next slide.

Chapter 6 (1/2)-6


Network Representation

22

c1

1c12

c13

c21

c22c23

d1

d2

d3

s1

s2

Sources Destinations

33

22

11

11

Chapter 6 (1/2)-7


A General LP Model

Using the notation: xij = number of units shipped from

origin i to destination j cij = cost per unit of shipping from

origin i to destination j si = supply or capacity in units at origin i

dj = demand in units at destination jcontinued

Chapter 6 (1/2)-8


A General LP Model (continued)

To obtain a feasible solution in a transportation problem, “total supply ≥ total demand”

1 1

Min m n

ij iji j

c x

1

1,2, , Supplyn

ij ij

x s i m

1

1,2, , Demandm

ij ji

x d j n

xij > 0 for all i and j

Chapter 6 (1/2)-9

Transportation Problem Example:Foster Generations

Foster Generators operates plants in Cleveland, Ohio; Bedford, Indiana; and York, Pennsylvania. Production capacities over the next three-month planning period for one particular type of generator are as follows:

Chapter 6 (1/2)-10


The firm distributes its generators through four regional distribution centers located in Boston, Chicago, St. Louis, and Lexington; the three-month forecast of demand for the distribution centers is as follows:

Chapter 6 (1/2)-11


The cost for each unit shipped on each route is also given as follows:

Management would like to determine how much of its production should be shipped from each plant to each distribution center.

Chapter 6 (1/2)-12



Chapter 6 (1/2)-13


LP Formulation

• The objective of the transportation problem is to minimize the total transportation cost:

• Therefore, the objective function is:

Chapter 6 (1/2)-14


• Consider supply constraints• Total # of units shipped from Cleveland:

• Total # of units shipped from Bedford:

• Total # of units shipped from York:

Chapter 6 (1/2)-15


• Consider demand constraints• Four demand constraints are needed to

ensure that destination demands will be satisfied:

Chapter 6 (1/2)-16


Combining the objective function and constraints into one model provides a 12-variable, 7-constraint LP formulation of the Foster Generators’ transportation problem:

Chapter 6 (1/2)-17


Solution Summary

Chapter 6 (1/2)-18


Network Representation of Optimal Solution

Chapter 6 (1/2)-19

Transportation Problem Variations

Variations of the basic transportation model may involve one or more of the following situations:1) Total supply not equal to total demand2) Route capacities or route minimums3) Unacceptable routes

Can be easily accommodated with slight modifications

Chapter 6 (1/2)-20


1) Total supply not equal to total demand “Total supply > total demand”• No modification in the LP formulation is

necessary.• Excess supply will appear as slack (i.e.

unused supply or amount not shipped from the origin).

Chapter 6 (1/2)-21


1) Total supply not equal to total demand (cont’d) “Total supply < total demand”

• The LP Model of a transportation problem will NOT have a feasible solution.

• Add a dummy origin with supply equal to the shortage amount.

• Assign a zero (0) shipping cost per unit to the dummy origin.

• The amount “shipped” from the dummy origin (in the solution) will not actually be shipped.

• The destination(s) showing shipments being received from the dummy origin will be the destinations experiencing a shortfall, or unsatisfied demand.

Chapter 6 (1/2)-22


2) Route capacities or route minimums Also can accommodate capacities or minimum

quantities for one or more of the routes (“capacitated transportation problem”)

Maximum route capacity from i to j: xij < Lij

Fosters Generators example (Example 1):• If the York-Boston route (from origin 3 to

destination 1) had a capacity of 1,000 units because of limited space availability on its normal mode of transportation, the following route capacity constraint should be added to the existing LP model:

x31 < 1,000

Chapter 6 (1/2)-23


2) Route capacities or route minimums (cont’d) Minimum shipping guarantee from i to j:

xij > Mij

Fosters Generators example (Example 1):• If the Bedford-Chicago route (from origin 2 to

destination 2) had a previously committed order of at least 2,000 units, the following route minimum constraint should be added to the existing LP model:

x22 > 2,000

Chapter 6 (1/2)-24


3) Unacceptable routes Establishing a route from every origin to every

destination may not be possible. Simply drop the corresponding arc from the

network and remove the corresponding variable from the LP formulation.

Fosters Generators example:• If the Cleveland–St. Louis route (from origin 1 to

destination 3) were unacceptable or unusable, the arc from Cleveland to St. Louis (x13) could be removed from the LP formulation.

Chapter 6 (1/2)-25

2. Assignment Problem

Typical assignment problems involve:• Assigning jobs to machines, agents to tasks,

sales personnel to sales, territories, contracts to bidders, etc.

A special case of the transportation problem in which all supply and demand values equal to 1, and the amount shipped over each arc is either 0 or 1; hence assignment problems may be solved as linear programs.

It assumes all workers are assigned and each job is performed.

Chapter 6 (1/2)-26


An assignment problem seeks to minimize the total cost, minimize time, or maximize profit assignment of m workers to n jobs, given that the cost of worker i performing job j is cij.

The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

Chapter 6 (1/2)-27



22

33

11

22

33

11c11

c12

c13

c21 c22

c23

c31 c32

c33

Agents Tasks

Chapter 6 (1/2)-28

A General LP Model

Using the notation:

xij = 1 if agent i is assigned to task j

0 otherwise

cij = cost of assigning agent i to task j


Chapter 6 (1/2)-29

A General LP Model (continued)


1 1

Min m n

ij iji j

c x

1

1 1,2, , Agentsn

ijj

x i m

1

1 1,2, , Tasksm

iji

x j n

xij > 0 for all i and j

If an agent is permitted to work for multiple (t) tasks at the same time:

1

1,2, , Agentsn

ijj

x t i m

Chapter 6 (1/2)-30

Assignment Problem: Example #1Fowle Marketing Research

Fowle Marketing Research has just received requests for market research studies from three new clients. The company faces the task of assigning a project leader (agent) to each client (task). Currently, three individuals have no other commitments and are available for the project leader assignments. Fowle’s management realizes, however, that the time required to complete each study will depend on the experience and ability of the project leader assigned. The three projects have approximately the same priority, and management wants to assign project leaders to minimize the total number of days required to complete all three projects. If a project leader is to be assigned to one client only, what assignments should be made?

Chapter 6 (1/2)-31

Management must first consider all possible project leader–client assignments and then estimate the corresponding project completion times.

Estimated completion times (in days)


Chapter 6 (1/2)-32



Chapter 6 (1/2)-33


LP Formulation Using the notation:

xij = 1 if project leader i is assigned to client j 0 otherwise

where i = 1, 2, 3, and j = 1, 2, 3 The completion times for three project leaders:

Thus, the objective function is:

Chapter 6 (1/2)-34


LP Formulation (cont’d) Constraints reflect the conditions that each project

leader can be assigned to at most one client and that each client must have one assigned project leader. Thus:

• “# of project leaders = # of clients”: All the constraints could be written as “=“

• When “# of project leaders ≥ # of clients”: All the project leader constraints must be written as “≤“

Chapter 6 (1/2)-35


Combining the objective function and constraints into one model provides a 9-variable, 6-constraint LP formulation of the Fowle Marketing Research’s assignment problem:

Chapter 6 (1/2)-36

Computer Solution Output


Value of the optimal solution

Optimal solutionThe change in the optimal value of the solution per unit increase in the RHS of the constraint.

Terry is assigned to client 2 (x12 = 1), Carle is assigned to client 3 (x23 = 1), and McClymonds is assigned to client 1 (x31 = 1). The total completion time required is 26 days.

Chapter 6 (1/2)-37


Solution Summary

Chapter 6 (1/2)-38

Assignment Problem Variations

Similar to transportation problem1) Total # of agents not equal to the total

number of tasks 2) Unacceptable assignments

Can be easily accommodated with slight modifications

Chapter 6 (1/2)-39


1) Total # of agents not equal to the total number of tasks “# of agents > # of tasks”• No modification in the LP formulation is

necessary.• Extra agents will appear as slack (i.e.

unassigned agents).

Chapter 6 (1/2)-40

An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.

ProjectsSubcontractor A B C Westside 50 36 16

Federated 28 30 18 Goliath 35 32 20

Universal 25 25 14

How should the contractors be assigned so that totalmileage is minimized?

Assignment Problem: Example #2

Chapter 6 (1/2)-41

Network Representation50

36

16

2830

18

35 32

2025 25

14

West.West.

CC

BB

AA

Univ.Univ.

Gol.Gol.

Fed. Fed.

ProjectsSubcontractors


Chapter 6 (1/2)-42

Linear Programming Formulation

Min 50x11+36x12+16x13+28x21+30x22+18x23

+35x31+32x32+20x33+25x41+25x42+14x43

s.t. x11+x12+x13 < 1

x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j

Agents

Tasks


Chapter 6 (1/2)-43

The optimal assignment is:

Subcontractor Project Distance Westside C 16

Federated A 28Goliath (unassigned) Universal B 25

Total Distance = 69 miles


Chapter 6 (1/2)-44


1) Total # of agents not equal to the total number of tasks (cont’d) “# of agents < # of tasks”

• The LP Model of a assignment problem will NOT have a feasible solution.

• Add enough dummy agents to equalize the number of tasks.

• The objective function coefficients for the dummy agents would be zero (0).

• No assignments will actually be made to the clients receiving dummy project leaders.

Chapter 6 (1/2)-45


2) Unacceptable assignments When an agent does not have the experience

necessary for one or more of the tasks Simply remove the corresponding decision

variable from the LP formulation.

distribution and network models (1/2) chapter 6 mangt 521 (b): quantitative management

Documents

transportation problem

transportation lp formulation

total transportation

examples of network

pertcpm problems

foster generationsthe

network structure

total supply total demand