modified distribution method
DESCRIPTION
A Linear Programming Method to optimize delivery of goods from different sources to different destinations.TRANSCRIPT
Alvin G. NiereMBA-1Misamis University
The Modified Distribution Method
The modified distribution method, MODI for short , is an improvement over the stepping stone method for testing and finding optimal solutions.
The Modified Distribution Method
1. Find a basic solution by any standard method. If supply and demand are equal then it is a balanced transportation problem.
2. Test for optimality. The number of occupied cells should equal to m + n -1. If the initial basic feasible solution does not satisfy this rule, then optimal solution cannot be obtained. Such solution is a degenerate solution.
Steps Involved in MODI Method
3. Set up a cost matrix for allocated cells only.4. Determine a set of number Ui for each row and
a set of number Vj on the bottom of the matrix.
5. Compute the value of Ui and Vj with the formula Ui + Vj = Cij to all basic(occupied) cells.
6. Calculate the water value of of non-basic ( unoccupied) cells using the relation Ui+Vj=Cij.
Steps Involved in MODI Method
7. Compute the penalties for each unoccupied cell by using the formula Dij=Pij=Ui+Vj-Cij.
8. Examine whether all Pij ≤ 0. If all Pij < 0, then the solution is optimal and
unique. If all Pij ≤ 0, then the solution is optimal and
an alternative solution exists. If at least one Pij > 0, then the solution is not
optimal.
Steps Involved in MODI Method
9. If the solution is not optimal, identify and introduce +O in the non-basic cell which has the maximum penalty ( Pij ) and construct a loop starting from this +O cell and passing through basic cells. The +O and –O sign are alternatively assigned in the basic cells of the closed loop. +O is added and –O is subtracted with respective cells on the closed loop.
Steps Involved in MODI Method
10. With the new resulting allocation table, go to step 3, repeat until optimum allocation is made.
11. At the end, prepare the optimum solution table and calculate the optimum/minimum transportation cost.
Steps Involved in MODI Method
PROBLEM: DETERMINE THE OPTIMUM SOLUTION FOR THE COMPANY OF TRASPORTATION PROBLEM(USING NWCM AND MODI METHOD)
$8 $8 $15
$15 $10 $17
$3 $9 $10
REQUIREMENT 150 80 50
120
80
80
CAPACITY
F1
F2
F3
W1 W2 W3 WAREHOUSE
FACTORY
W1 W2 W3
F1
F2
F3
$8 $8 $15
$15
$3
$10 $17
$9 $10
120
30 50
30 50
150 80 50
120
80
80
IBFS WITH NWCM
IBFS= 120(8)+30(15)+50(10)+30(9)+50(10)
IBFS=960+450+500+270+500=$2680
OCCUPIED MATRIX UNOCCUPIED MATRIX
8
15 10
9 10
Vj V1=8 V2=3 V3=4
Ui
U1=0
U2=7
U3=6
Vj 8 3 4
Ui Ui
0
7
6
-5 -11
-6
11
3 4
11
14
-8 -15
-3
-17
Pij=(Ui+Vj)-CijUi + Vj = Cij
1588
15 10 17
3 9 10
10
120
30 50
30 50+
+_
_
8
15
3
8 15
10 17
9 10
120 E
80
3050
STONE SEQUARE=RIM REQUIREMENTm+n-1=5
DEGENERACY OCCUAR
Value of O is equal to the minimum of the existing allocation among the signed cells on the loop.
LOOP CONSTRUCT
120
80
80
Dj 150 80 50 280
Si
OPTIMUM SOLUTIONOPTIMUM SOLUTION TABLE
$8 $8 $15
$15 $10 $17
$3 $9 $10
OPTIMUM COST• F1 W1 8*120 =960
• F1 W2 8*E = _
• F2 W2 10*80 =800
• F3 W1 3*30= 90
• F3 W3 10*50 =500
____________
$2350
120
30
E
80
50
