gpm_ommd - session 12

8/8/2019 Gpm_ommd - Session 12

1/12

Optimization Models for ManagerialDecisionSession 12 Transshipment + Assignment


2/12

Dipankar Bose - XLRI

Case Designing Optimal CapacityPlanning Strategies Issues

Cost at each cell Option 1 Transportation cost only

Option 2 Transportation cost + Production cost

Which option will you take in the following cases? Total demand = Total Supply

Total Demand > Total Supply

Total Demand < Total Supply

In which case Dummy is required?


3/12


Transshipment Example 1

XYZ manufactures Product M at two factories, one inLocation A and one in Location B. Location A factory can

produce 150 units, and Location B factory can produce

200 units per day. Product M is shipped by air to

customers in Location C and Location D. The customers ineach city require 130 units per day.

Because of the deregulation of airfares, XYZ believes that it

may be cheaper first fly some widgets to Location E or F

and then fly them to their final destinations. The cost of flying a unit is shown next.

XYZ wants to minimize the total cost of shipping.


4/12


Transshipment Example 1 Continued

Determine Supply and Demand values Solve the problem using Transportation Simplex

method

What happens if

Location A becomes additional Transshipment point Location D becomes additional Transshipment point

Both Location A & D become additional Transshipment

point

E F C D

A 8 13 25 28B 15 12 26 25

E 0 6 16 17

F 6 0 14 16


5/12


Transshipment Example 21 3

2 4

5

6

150

250

100

300

14

32

1 3

6

5

8

1

Determine Supply and Demand values

Solve the problem using Transportation Simplex

method


6/12


Assignment Problem 1

ABC has four jobs to be completed. Each machine must beassigned to complete one and only one job. The time

required to setup each machine for completing each job is

shown in the table below. ABC wants to minimize the total

setup time needed to complete the four jobs.Time (Hours)Job1 Job2 Job3 Job4

Machine 1 8 26 17 11Machine 2 13 28 4 26Machine 3 38 19 18 15Machine 4 19 26 14 10


7/12


Assignment Problem Continued

What happens if We increase Row 1 each value by 1 unit?

We increase Column 1 each value by 1 unit?

If all cij values 0

The feasible solution with Z = 0 is optimal solution


8/12


Assignment Problem 2

ABC has five jobs to be completed. Each machine must beassigned to complete one and only one job. The time

required to setup each machine for completing each job is

shown in the table below. ABC wants to minimize the total

setup time needed to complete the five jobs.Time (Hours)Job1 Job2 Job3 Job4 Job5

Machine 1 5 5 7 4 8Machine 2 6 5 8 3 7Machine 3 6 8 9 5 10Machine 4 7 6 6 3 6Machine 5 6 7 10 6 11


9/12


Assignment Problem HungarianMethod Step 1: Find Opportunity Cost Matrix by reducing eachrow and then each column cell values so that

Each row has at least one zero

Each column has at least one zero

Step 2: Assigning the zeros Start from top row

Find the row with single zero, assign that zero and

cross all other zeros in that column

Leave the row if it has more than one zero After checking all the rows do the same for all columns

starting from left column

Repeat the steps until all the zeros assigned/ crossed


10/12


Assignment Problem HungarianMethod Continued

If assigned zero = number of row/column Optimal solution

Else Step 3: Use the method for Covering the Zeros Step 4: Reduced Cost Matrix

Find minimum of uncovered cells Subtract this value from all uncovered cells

Add this value to twice covered cells

Keep once covered cells unchanged

Follow Step 2 again If there is no row/column with single zero

Choose row/column with minimum zero

Assign any zero and follow Hungarian method


11/12


Hungarian Method Step 3 Covering the Zeros

Mark unassigned rows In the marked rows

Look for any zero and mark corresponding column

In the marked column

Look for assigned zeros and mark corresponding row Repeat the steps

Till no marking is possible

COVER unmarked rows and marked columns


12/12


Scheduling Use of AssignmentMethod Example

Departure Route Arrival Departure Route ArrivalPlace X Place Y Place X Place Y06:00 A 12:00 05:30 1 11:30

07:30 B 13:30 09:00 2 15:00

11:30 C 17:30 15:00 3 21:00

19:00 D 01:00 18:30 4 00:30

00:30 E 06:30 00:00 5 06:00

There are 5 drivers

Every driver should get rest of > 4 hours and maximum24 hours before their return trip

The journey can be started either from Place X or Place Y

Needs to minimize total waiting time

gpm_ommd - session 12

Documents