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
Structure of other monosaccharides and properties tutorial all along with the key concepts of Sugar acids, Deoxy sugars, Amino sugars, Properties of monosaccharides, Mutarotation, Glycoside formation, Ester formation
tutorsglobe.com osmosis assignment help-homework help by online absorption and movement tutors
tutorsglobe.com virulent or lytic cycle assignment help-homework help by online life cycle of a phage tutors
Population genetics tutorial all along with the key concepts of History of Population Genetics, study of allele frequency, emergence of modern evolutionary synthesis, Population genetics as a discipline
www.tutorsglobe.com offers Layering homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
Air Pollution tutorial all along with the key concepts of Air Pollution-Past, Present and Future, Major Air Pollutants-their sources, Ozone, Carbon monoxide, Airborne carcinogens, Hydrocarbons, Cigarette smoking
tutorsglobe.com haemocytometer assignment help-homework help by online medical lab methods tutors
tutorsglobe.com nitrogen fixation assignment help-homework help by online nitrogen cycle tutors
electron affinity tutorial all along with the key concepts of factors affecting electron affinity, periodicity in electron affinity, trends across periods, trend across groups, atomic radius, electronic configuration, effective nuclear charge
Mass Analyzers used in Mass Spectrometry tutorial all along with the key concepts of Sample Introduction, Methods of Sample Ionization, Electron Impact Ionization, Chemical Ionization, Fast Atom Bombardment, Electrospray Ionization, Desorption Techniques, Detection-Recording of Sample Ions
Physiology of behavior tutorial all along with the key concepts of Approaches to Animal Behavior, Causes of behavior, Timing of behavior, Examples of Innate behavior, Anthropomorphism, Maturation and Instinct Learning Interactions
Finding out topmost Political Theory Assignment Help service at reasonable rates? Avail it from qualified tutors!
tutorsglobe.com role of bacteria in industry assignment help-homework help by online beneficial activities of bacteria tutors
Avail Logic Assignment Help service and get top-notch solution document with 24x7 support of experts at budget-friendly prices.
Refraction at Plane Surfaces tutorial all along with the key concepts of Bending of Ray of Light when travels from Air to Water, Laws of Refraction, Snell's Law, Refractive Index, Critical Angle, Total Internal Reflection
1939091
Questions Asked
3689
Tutors
1453144
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!