Network Flow Problems

 

Example of Network Flow problems:

• What shipping plan minimizes cost to ship from m warehouses to n customers?• How do you maximize efficiency of a machine shop through the assignment of jobs to a group of machines?• What is the maximum flow that can be obtained through a series of pipes?• How do you maximize the flow of traffic through a series of one-way streets?• What is the shortest route if a truck must make a “milk-run” through a series of stops (TSP)?• How do you schedule a series of project activities in order to minimize the total project duration time?

Network Flow Problems – Transportation Problem

 

1

3

2

1

3

2

4Warehouse

Customer

a – warehouse capacityb – customer demand

a1

a2

a3

b1

b2

b3

b4

c11

c12

Network Flow Problems – Transportation Problem

1 2 … n

1 c11 c12 … c1n

Warehouses 2 c21 c22 … c2n

… …

m cm1 cm2 … cmn

Customers

Given:Supply Vector: a = [a1 a2 … am ]Demand Vector: b = [b1 b2 … bn ]Transportation cost matrix: c

Objective: Find shipping plan that minimizes transportation cost that meets all customer demands while being constrained by supply capacities.

Network Flow Problems – Transportation Problem

01

1

1 1

ij

m

ijij

n

jiij

m

i

n

jijij

x

bx

ax

xcZMinimize

s.t.

Total Cost

(supply restriction)

i = 1…m

(demand requirement)

j = 1…n

Network Flow Problems –The Assignment Problem

 

Consider the problem of assigning n assignees to n tasks. Only one task can be assigned to an assignee, and each task must be assigned.

There is also a cost associated with assigning an assignee i to task j, cij.

The objective is to assign all tasks such that the total cost is minimized.

Network Flow Problems –The Assignment Problem

 

Examples:• Assign people to project assignments• Assign jobs to machines• Assign products to plants• Assign tasks to time slots

Network Flow Problems –The Assignment Problem

 

To fit the assignment problem definition, the following assumptions must be satisfied:

• The number of assignees and the number of tasks are the same (denoted by n).• Each assignee is to be assigned to exactly one task.• Each task is to be assigned to exactly one assignee.• There is a cost cij associated with assignee i performing task j.• The objective is to determine how all n assignments should be made to minimize the total cost.

Assignment Problem – Flow Diagram

 

1

3

2

1

3

2

n

assignees tasks

a – assigneet – tasks a1

a2

a3

t1

t2

t3

t4

c11

c12

n cnn

an

Assignment Problem –Cost Matrix

1 2 … n

1 c11 c12 … c1n

Assignees 2 c21 c22 … c2n

… …

n cn1 cn2 … cnn

Tasks

Let the following represent the standard assignment problem cost matrix, c:

Assignment Problem –Conversion to Standard Cost Matrix

1 2 3 41 10 9 8 7

Assignees 2 4 - 5 63 2 1 8 -

Tasks

Consider following cost matrix, how do you convert to satisfy the standard definition of the assignment problem?

1 2 3 41 10 9 8 7

Assignees 2 4 M 5 63 2 1 8 M4 0 0 0 0

Tasks

Add “big M” to avoid incompatible assignments, and add a dummy assignee (or task) to have equal assignees and tasks.

Assignment Problem – Math Formulation

) and allfor binary, is(

0

1

1

1

1

1 1

jix

x

x

x

xcZ

ij

ij

n

iij

n

jij

m

i

n

jijij

Minimize

s.t.

Total Cost

i

j

Does this formulation look familiar?Is this a Linear Program?

Network Flow Problems –Maximal Flow Problems

 

Consider the following flow network:

s1

2

3

nks1

k23

k13

k3n

k1n

ks2

k21

The objective is to ship the maximum quantity of a commodity from a source node s to some sink node n, through a series of arcs while being constrained by a capacity k on each arc.

Maximal Flow Problems

 

Examples:1. Maximize the flow through a company’s distribution

network from its factories to its customers.2. Maximize the flow through a company’s supply network

from its vendors to its factories.3. Maximize the flow of oil through a system of pipelines.4. Maximize the flow of water through a system of aqueducts.5. Maximize the flow of vehicles through a transportation

network.

Maximal Flow Problems

 

Definitions:Flow network – consists of nodes and arcsSource node – node where flow originatesSink node – node where flow terminateTransshipment points – intermediate nodesArc/Link – connects two nodesDirected arc – arc with direction of flow indicatedUndirected arc – arc where flow can occur in either directionCapacity(kij) – maximum flow possible for arc (i,j)Flow(f ij) – flow in arc (i,j). Forward arc – arcs with flow out of some nodeBackward arc – arc with flow into some nodePath – series of nodes and arcs between some originating and some terminating node Cycle – path whose beginning and ending nodes are the same

Maximal Flow Problems – LP Formulation

 

s1

2

3

nff

Objective: Maximize Flow (f)Constraints: 1) The flow on each arc, fij, is less than or equal to the capacity on each arc, kij. 2) Conservation of flow at each node. Flow in = flow out.

Maximal Flow Problems – LP Formulation

 

s1

2

3

nff

Max Z = fsts) fs1 + fs2 = f1) f13 + f1n = fs1 + f21 2) f21 + f23 = fs2

3) f3n = f13 + f23 n) f = f3n + f1n

0 <= fij <= kij

Objective: Maximize Flow (f)Constraints: • The flow on each arc, fij, is less than or equal to the capacity on each arc, kij. • Conservation of flow at each node. Flow in = flow out.

ji,

Maximal Flow Problems – Conversion to Standard Form

 

s1 1

2

3

n1

What if there are multiple sources and/or multiple sinks?

s2

n2

Maximal Flow Problems – Conversion to Standard Form

 

f

f

Create a “supersource” and “supersink” with arcs fromthe supersource to the original sources and from the original sinks to the supersink. What capacity should we assign tothese new arcs?

 

s1 1

2

3

n1

s2

n2

s

n

Maximal Flow Problems – Conversion to Standard Form

 

s1

2

3

nff

What if there is an undirected arc (flow can occur in either direction)? See arc (1,2).

k12

Maximal Flow Problems – Conversion to Standard Form

 

s1

2

3

nff

Create two directed arcs with the same capacity. Upon solving the problem and obtaining flows on each arc, replace the two directed arcs with a single arc with flow | fij – fji |, in the directionof the larger of the two flows.

k21k12

Project Management - PERT/CPM

Let each node represent a project event/milestone (node 1 isstart of project, node 11 is end of project).

Let each arc represent a project task/job.

Each arc is identified by a job letter and duration. Note thedummy jobs indicating precedence that jobs H and I must complete before K or L begins.

 

2A,5

1 3 4 5 6 9 10

7

8

11B,8 C,15

D,12

F,8

E,10G,11

H,5

I,4

J,2K,8

L,14

M,5

N,5

0

0

Project Management - PERT/CPM

What questions might project managers be interested in?

• How long will the project take?• Can I add manpower or tools to reduce the overall project length?• To which tasks should I add manpower?• What tasks are on the critical path?• Is the project on schedule?• When should materials and personnel be in place to begin a task?• Other?…

Project Management - Examples

• University Convocation Center• Windsor Engine Plant• Other major construction projects• Large defense contracts• NASA projects (space shuttle)• Maintenance planning of oil refineries, power plants, etc…• other…

Project Management – Minimum Completion Time

2A,3

1

3

4 5C,4

D,2B,1

E,5

0

LP Solution: Let ti be the time of event i.

Min Z = t5 – t1

s.t. t2 – t1 >= 3 t3 – t2 >= 0

t3 – t1 >= 1t4 – t2 >= 4t4 – t3 >= 2t5 – t4 >= 5 ti >= 0 for all i

CPM – Critical Path MethodCan normal task times be reduced?

Is there an increase in direct costs?• Additional manpower• Additional machines• Overtime, etc…

Can there be a reduction in indirect costs?• Less overhead costs• Less daily rental charges• Bonus for early completion• Avoid penalties for running late• Avoid cost of late startup

CPM addresses these cost trade-offs.

CPM – Critical Path Method

LP Approach:

Let tij – decision variable for time to complete task connecting events i and j. kij – normal completion time of task connecting events i and j. lij – minimum completion time of task connecting events i and j. Cij – incremental cost of reducing task connecting events i and j.

Model I: Given project must be complete by some time T, which tasksshould be reduced to minimize the total cost?

Min

s.t.

i j

ijijij tkCZ )(

01

i

n

ijijij

ijij

t

Ttt

ktl

ttt for all jobs (i,j)

for all jobs (i,j)

for all i

CPM – Critical Path Method

LP Approach:

Model II: Given an additional budget of $B for “crashing” tasks, what minimum project completion time can be obtained while staying within your budget?

Min

s.t.

1ttZ n

0

)(

i

iji j

ijij

ijijij

ijij

t

BtkC

ktl

ttt for all jobs (i,j)

for all jobs (i,j)

for all i