heuristic solutions

E211- OPERATIONS PLANNING II

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

Transportation Problem

� How much should be shipped from several sources to several destinations

� Sources: supply centers such as factories, warehouses, etc.

� Destinations: receiving centers such as warehouses, stores, etc.


stores, etc.

� Transportation models

� Find the shipping arrangement with the lowest cost

� Used primarily for existing distribution systems

Transportation Problem (Example)

DesMoines

(100 unit

capacity)

Cleveland

(200 units required)

Albuquerque

Boston

(200 units

required)

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

Fort Lauderdale

(300 units capacity)

Evansville

(300 units

capacity)

Albuquerque

(300 units

required)

required)

Characteristics of Transportation Problems

� The Requirements Assumption � Each source has a fixed supply of units, where this entire supply

must be distributed to the destinations.� Each destination has a fixed demand for units, where this entire

demand must be received from the sources.

� The Feasible Solutions Property� A transportation problem will have feasible solutions if and only if

the sum of its supplies equals the sum of its demands.


� The Cost Assumption� The cost of distributing units from any particular source to any

particular destination is directly proportional to the number of units distributed.

� This cost is just the unit cost of distribution times the number of units distributed.

� The objective is to minimize the total cost of distributing the units.

Transportation Table

Cost per Unit Distributed

Destination 1 2 … n Supply

1 c11 c12 … c1n S1

2 c21 c2n S2


2 c21 c2n S2

Source … … … … … …

m cm1 cm2 cmn Sm

Demand d1 d2 … dn

Network Representation

[s1]

[d1]

[d2][s ]

c11

c12

c1n

c21

c22


[d2]

[sm]

[s2]

[dn]

c22

c2n

cm1

cm2

cmn

The Transportation Model

� Any problem (whether involving transportation or not) fits the model for a transportation problem if

� It can be described completely in terms of a transportation table that identifies all the sources, destinations, supplies, demands, and unit costs, and

satisfies both the requirements assumption and the


� satisfies both the requirements assumption and the cost assumption.

� The objective is to minimize the total cost of distributing the units.

Transportation Problem Solution Steps

� Define problem

� Set up transportation table (matrix) or network diagram which

� Summarizes all data

� Keeps track of computations


� Keeps track of computations

� Develop a heuristic solution

� North-West Corner Rule or Lowest Cost Method

� Or find an optimal solution

� Solve LP model using Solver

Today’s problem:Transportation Table (Data Setup)

From/To Institute of Tech.

Design Institute

Business Institute

Institute of Fine Arts

Warehouse Supply

Kuala Lumpur

21 16 17 22 790

Johor Bahru

13 24 10 18 425


Bahru

Ipoh 25 21 20 15 585

University Demand

350 475 465 510 -

Today’s problem: Network Representation

Kuala

Lumpur

Johor

Institute

of Tech.

Design

Institute

790

350

475

425

21

16

2217

1324


Johor

Bahru

Ipoh

Business

Institute

Institute of

Fine Arts

585

425

465

510

2410

18

252120

15

Today’s problem:Heuristic Solution – North-West Corner Rule

� Identify the cell which is at the northwest corner.

� Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment.


� Move to the next cell to the right or below, depending on whether the column or row has been crossed out.

� Repeat steps 2 & 3 until all units have been allocated.

Today’s problem:Heuristic Solution using North-West Corner Rule

To/

From

Kuala

Lumpur21 16 17 22 790

Destinations

Institute

of Tech.

Design

Institute

Business

Institute

Institute of

Fine Arts

Warehouse

Supply

350

1

2

440440

3 5


Johor

Bahru13 24 10 18 425

Ipoh 25 21 20 15 585

University

Demand350 475 465 510

Total Cost = 28280

Sources

35

35390

75

510

390

4

51075

Today’s problem:Heuristic Solution – Lowest Cost Method

� Identify the cell with the lowest cost. Arbitrarily break any ties for the lowest cost.

� Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment.


� Find the cell with the lowest cost from the remaining cells.

� Repeat steps 2 & 3 until all units have been allocated.

Today’s problem:Heuristic Solution using Lowest Cost Method

To/

From

Kuala

Lumpur21 16 17 22 790

Johor 13 24 10 18 425

Warehouse

Supply

Destinations

Institute

of Tech.

Design

Institute

Business

Institute

Institute of

Fine Arts

1


Johor

Bahru13 24 10 18 425

Ipoh 25 21 20 15 585

University

Demand350 475 465 510

Sources

1

40

425

Today’s problem: Heuristic Solution using Lowest Cost Method

To/

From

Kuala

Lumpur21 16 17 22 790

Johor 13 24 10 18 425

Destinations

Institute

of Tech.

Design

Institute

Business

Institute

Institute of

Fine Arts

Warehouse

Supply

1

23

315 275

4

5

6

275475

40


Johor

Bahru13 24 10 18 425

Ipoh 25 21 20 15 585

University

Demand350 475 465 510

Total cost 27830

Sources

1

40

75

75

6

425

75 510

The solution from the Lowest Cost Method incurs lowercost than the North-West Method. But is this solution optimal?

27830

Today’s problem: LP Formulation

Let xij be the number of laptops to be shipped from warehouse i to

university j (i = 1,2,3; j = 1,2,3,4) – Decision variables

Minimize cost = 21x11 + 16x12 + 17x13 + 22x14 + 13x21 + 24x22 + 10x23 + 18x24 +25x31 + 21x32 + 20x33 + 15x34 – Objective function

Subject to: - Constraints

Kuala Lumpur supply: x11 + x12 + x13 + x14 = 790


Kuala Lumpur supply: x11 + x12 + x13 + x14 = 790

Johor Bahru supply: x21 + x22 + x23 + x24 = 425

Ipoh supply: x31 + x32 + x33 + x34 = 585

Institute of Tech. demand: x11 + x21 + x31 = 350

Design Institute demand: x12 + x22 + x32 = 475

Business Institute demand: x13 + x23 + x33 = 465

Institute of Fine Arts demand: x14 + x24 + x34 = 510

Xij>= 0 (i = 1,2,3; j = 1,2,3,4) – Non-negativity constraint

Today’s problem: Integer Solutions Property

� As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables.

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


model that restrict these variables to only have integer values.

Today’s problem: Optimal Solution using Excel Solver

To/ Institute of Tech.

Design Institute

Business Institute

Institute of Fine Arts

Warehouse Supply

From

Kuala Lumpur 0 475 315 0 790

Johor Bahru 350 0 75 0 425

Ipoh 0 0 75 510 585


LP Total cost = $ 27,405

Compared to Minimum Cost Method ($27,830), difference of $425

Compared to North-West Corner Rule ($28280), difference of $875

Ipoh 0 0 75 510 585

University Demand

350 475 465 510

Today’s problem: Unbalanced Supply-Demand Formulation

� If total supply > total demand, it means that some laptops won’t be distributed. Constraints to be changed are:

Subject to: - Constraints

Kuala Lumpur supply: x11 + x12 + x13 + x14 ≤ 790

Johor Bahru supply: x21 + x22 + x23 + x24 ≤ 425

Ipoh supply: x31 + x32 + x33 + x34 ≤ 585

Institute of Tech. demand: x + x + x ≤ 350


� If total supply < total demand, it means some demand can’t be fulfilled. Constraints to be changed are:

Institute of Tech. demand: x11 + x21 + x31 ≤ 350

Design Institute demand: x12 + x22 + x32 ≤ 475

Business Institute demand: x13 + x23 + x33 ≤ 465

Institute of Fine Arts demand: x14 + x24 + x34 ≤ 510

Conclusion

� Transportation problems deal with the distribution of goods from several sources of supply to several destinations having the demand

� The objective is to schedule the shipment in a way to minimize total transportation costs

� Other areas of applications include facility location,


� Other areas of applications include facility location, production-inventory control, personnel assignment and equipment maintenance.

Learning Objectives

� Understand transportation problems that deal with the distribution of goods from multiple supply sources to multiple demand destinations.

� Set up transportation table (matrix) or network diagram;

� Develop heuristic solution using Northwest Corner Rule and Lowest Cost Method;


and Lowest Cost Method;

� Formulate the transportation problem as an LP problem;

� Use MS Excel Solver to find an optimal shipment schedule to minimize total transportation costs.

heuristic solutions

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