Transportation Algorithm for Minimization Problem (MODI Method)
Step 1
Make the transportation table entering the origin capacities ai, the cost cij and destination requirement bj
Step 2
Determine an initial basic feasible solution by vogel's technique or by any of the known method.
Step 3
For all the basic variables xij, solve the system of equations ui + vj = cij, for all i, j for which cell (i, j) is in the basis, starting at first with some ui = 0, compute the values of ui and vj on the transportation table
Step 4
Calculate the cost differences dij = cij - ( ui + vj ) for all the non-basic cells
Step 5
Apply optimality test by testing the sign of each dij
Step 6
Assume the variable xrs enter the basis. Assign an unknown quantity ? to the cell (r, s). Then make a loop that begins and ends at the cell (r, s) and links some of the basic cells. The amount ? is added to and subtracted from the transition cells of the loop in such a way that the availabilities and necessities remain fulfilled.
Step 7
Allocate the largest possible value to the ? in such a manner that the value of at least one basic variable comes out to be zero and the other basic variables remain non-negative. The basic cell whose allotment has been made zero will leave the basis.
Step 8
Now, go back to step 3 and repeat the process until an optimal solution is achieved.
Worked Examples
Illustration 1
Determine an optimal solution
W1
W2
W3
W4
Availability
F1
19
30
50
10
7
F2
70
40
60
9
F3
8
20
18
Requirement
5
14
Answer
1. Put vogel's approximation method for determining the initial basic feasible solution
Penalty
5(19)
(30)
(50)
2(10)
X
(70)
7(40)
2(60)
(40)
8(8)
10(20)
Minimum transportation cost comes out to be 5 (19) + 2 (10) + 7 (40) + 2 (60) + 8 (8) + 10 (20) = Rs. 779
2. Test for Non-degeneracy
The initial fundamental feasible solution has m + n - 1 i.e. 3 + 4 - 1 = 6 allocations in independent positions. Therefore optimality test is fulfilled.
3. Computation of ui and vj : - ui + vj = cij
ui
u1= -10
u2 = 40
u3 = 0
vj
v1 = 29
v2 = 8
v3 = 0
v4 = 20
Allocate a 'u' value to zero. (Convenient rule is to choose the ui, which has the maximum number of allocations in its row)
Assume u3 = 0, then
u3 + v4= 20 which means 0 + v4 = 20, so v4 = 20
u2 + v4= 60 which means u2 + 20 = 60, so u2 = 40
u1 + v4= 10 which means u1 + 20 = 10, so u1 = -10
u2 + v3= 40 which means 40 + v3 = 40, so v3 = 0
u3 + v2= 8 which means 0 + v2 = 8, so v2 = 8
u1 + v1= 19 which means -10 + v1= 19, so v1 = 29
4. Computation of cost differences for non basic cells dij = cij - ( ui + vj )
cij
ui + vj
-2
-10
69
48
29
0
dij = cij - ( ui + vj )
32
1
-18
11
5. Doing Optimality test
dij < 0 i.e. d22 = -18
so x22 is entering the basis
6. Creation of loop and allotment of unknown quantity ?
We assign ? to the cell (2, 2). Reallocation is done by transferring the maximum possible amount ? in the marked cell. The value of ? is achieved by equating to zero to the corners of the closed loop. That is min (8-?, 2-?) = 0 which gives ? = 2. Hence x24 is outgoing as it turns out to be zero.
5 (19)
2 (10)
2 (30)
7 (40)
6 (8)
12 (20)
Minimum transportation cost comes out to be 5 (19) + 2 (10) + 2 (30) + 7 (40) + 6 (8) + 12 (20) = Rs. 743
7. Enhanced Solution
u2 = 22
v3 = 18
(60)
51
42
52
As dij > 0, an optimal solution is achieved with minimal cost of Rs.743
Monocotyledons and Dicotyledons tutorial all along with the key concepts of Number of cotyledons, Pollen structure, Number of flower parts, Leaf veins, Stem vascular arrangement and Root development
theory and lecture notes of load power and internal resistance all along with the key concepts of theory of load power, maximum power transfer and internal resistance. tutorsglobe offers homework help, assignment help and tutor’s assistance on theory of load power and internal resistance.
theory and lecture notes of exponential and logarithmic functions all along with the key concepts of exponential functions and their graphs, exponential and logarithmic models, properties of logarithms. tutorsglobe offers homework help, assignment help and tutor’s assistance on exponential and logarithmic functions.
www.tutorsglobe.com offers coordination chemistry homework help, coordination chemistry assignment help, online tutoring assistance, inorganic chemistry solutions by online qualified tutor's help
Chemical Instrumentation in Environmental Sciences tutorial all along with the key concepts of Determination of Chloride, Determination of Sulphate, Determination of Fluoride, Determining Iron by Phenanthroline Method
looking for instant and first class set theory assignment help service? hire phd experts and get authentic papers with 24x7 support to score a++.
tutorsglobe.com five kingdom system of classification assignment help-homework help by online systematics tutors
tutorsglobe.com perianth assignment help-homework help by online position of flower tutors
Just the once the company has made its primary share issue to begin business (generally soon after the company is first created) it might decide to make additional issues of new shares.
Theory and lecture notes of Costs of Inflation all along with the key concepts of Costs of Moderate Expected Inflation, Costs of Moderate Unexpected Inflation, Hyperinflation and Its Costs, Inflation Tax. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Costs of Inflation.
Theory and lecture notes of LU Decomposition all along with the key concepts of lu decomposition, linear algebra, Using LU to solve equations. Tutorsglobe offers homework help, assignment help and tutor’s assistance on LU Decomposition.
tutorsglobe.com mesic habitats adaptations assignment help-homework help by online mesophytes tutors
tutorsglobe.com importance of the law assignment help-homework help by online equi-marginal utility tutors
tutorsglobe.com diffusion of charged particles assignment help-homework help by online passive transport tutors
Heat Engines tutorial all along with the key concepts of Basic Operation of Heat Engine, Thermodynamic Efficiency of Real Engines, Efficiency of Otto Engine, Efficiency of Stirling Engine, Heat Flow into Ideal Stirling Engine
1962783
Questions Asked
3689
Tutors
1483063
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!