E211- Operations Planning IIE211- Operations Planning II

SSCHOOL OF CHOOL OF EENGINEERINGNGINEERING EE211 211 –– OOPERATIONS PERATIONS PPLANNING IILANNING II

The Distribution Network Problem

� The distribution network problem

� Goods are shipped from known supply points (factories, plants) to known demand points (customers, retail outlets), possibly through intermediate points (regional and or/field warehouses) known as the transhipment points.the transhipment points.

� The overall objective is to find the best distribution plan, i.e., the amount to ship along each route from the supply points, through the intermediate points, to the demand points, where “best” is meant a plan that minimizes the total shipping cost.

� It is also necessary to satisfy certain constraints:

� Not shipping more than the specified capacity from each supply point;

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� Meeting known demand at the demand points;

� Shipping goods only along valid routes;

� Shipment not exceeding capacity limit imposed on certain routes.

The Minimum Cost Flow Model

� One favourable feature of the distribution network problem is that it can be precisely represented and solved by the Minimum Cost Flow Model.

The Minimum Cost Flow Model is a network optimization model � The Minimum Cost Flow Model is a network optimization model having the following features:

� Its objective is to find the cheapest possible way of sending a certain amount of flow from sources to destinations through a flow network;

� Like the maximum flow problem: it considers flow through a network with limited arc capacities;

� Like the shortest path problem: it considers a cost (or distance) for flow through an arc;

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through an arc;

� Like the transportation problem: it can consider multiple sources and multiple destinations for the flow, again with associated costs.

� All three problems above are special cases of the Minimum Cost Flow Problem.

Network Representation of the Minimum Cost Flow Model

[50] [-30]

Supply at Source A

Demand at Destination D

Cost of shipping one unit from A to D

[50] [-30]

A

B

C

D

E

2

9

4

3 1

2 3

(uA

B=

10

)

The objective is to minimize the total cost of sending the available supply through the network to satisfy the given demand

[0]

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An example of a network representation of the minimum cost flow problem

B

[40][-60]

Transhipment point

given demand

LP Formulation of the Minimum Cost Flow Problem

� The decision variables: represents the amount of flow through arc where i, j represents nodes in the network.

� The objective function:

ijx i j→

M inim ize 2 4 9 3 3 2A B A C A D B C C E D E E DZ x x x x x x x= + + + + + +

� The constraints:

� flow constraints (one for each node)

M inim ize 2 4 9 3 3 2A B A C A D B C C E D E E DZ x x x x x x x= + + + + + +

Node A: = 50 Node D: = 30

Node B: = 40 Node E: = 60

Node C: = 0

AB AC AD DE AD ED

AB BC CE DE ED

AC BC CE

x x x x x x

x x x x x

x x x

+ + − − −

− + − − + −

− − +

Amount shipped out – Amount shipped in = Net flow generated at the node

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� Upper-bound constraints

� Nonnegativity constraints: All

Node C: = 0AC BC CEx x x− − +

1 0; 8 0 .A B C Ex x≤ ≤

Flow trough an arc not exceeding the limited arc capacity

0ijx ≥

The Integer Solution Property of the Minimum Cost Flow Problem

� Like the transportation problem, the minimum cost flow problem also has the integer solution property.

� As long as every demand, supply and arc capacity have integer values, a minimum cost flow problem with feasible solutions is guaranteed to have an optimal solution with integer values for all the decision variables.

� Therefore, it is not necessary to add constraints to the model to restrict the decision variables to only have

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model to restrict the decision variables to only have integer values.

Today’s problem: Formulation as a minimum cost flow Problem

[0]

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Today’s problem: LP Formulation

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Today’s problem: Excel Solver solution

SSCHOOL OF CHOOL OF EENGINEERINGNGINEERING EE211 211 –– OOPERATIONS PERATIONS PPLANNING IILANNING II

LP Total cost = $ 655,000

Formulation of the shortest path problem as a minimum cost flow Problem

Net flow at intermediate

node

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Demand at Destination 5

Travelling distance

between nodes 2 and 5

LP formulation for the shortest path problem (with mandatory pass of each node)

1 arc is on the shortest pathi j→1 arc is on the shortest path

0 otherwiseij

i jX

→=

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We ignored those decision variables going into source node and coming out of destination node.

LP formulation for the shortest path problem (with mandatory pass of each node)

�Net flow constraints already imposed mandatory pass on source node 1 and destination node 5.

�Mandatory pass constraints are required only for intermediate (transhipment) nodes 2, 3 and 4.

�Due to the ‘integer solution’ property of the minimum cost flow problem, we do not need to specify the decision variables to be binary.

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be binary.

For the shortest path problem with no mandatory pass requirement, we can ignore this set of constraints.

For the shortest path problem with no mandatory pass requirement, we can ignore this set of constraints.

� The minimum cost flow problem holds a central position in the network optimization models.

� The transportation problem, shortest path problem, and maximum flow problem are all special cases of the minimum cost

Conclusion

maximum flow problem are all special cases of the minimum cost flow problem.

� One special application of the minimum cost flow problem is the distribution network problem.

� The distribution network problem is to find the minimum cost shipment arrangements distributing products from ‘sources’ to ‘destinations’ considering both transshipment points and flow

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‘destinations’ considering both transshipment points and flow capacity in a transportation network.

� Flow networks can also be used to model liquids flowing through pipes, parts through assembly lines, current though electrical networks, information through communication networks.

� Finding the minimum cost shipment arrangements distributing products from ‘sources’ to ‘destinations’ considering both transshipment points and flow capacity in a transportation network.

Learning Objectives

in a transportation network.

� Understanding of transshipment, flow capacity, mandatory pass modeling.

� Formulate LP model and use MS Excel Solver to find the optimal solution.

SSCHOOL OF CHOOL OF EENGINEERINGNGINEERING EE211 211 –– OOPERATIONS PERATIONS PPLANNING IILANNING II

� Appreciate that transportation problem, maximum flow problem, and shortest path problem are all special cases of the minimum cost flow problem.