List of GATE Mechanical Engineering Books

 List of GATE Mechanical Books

(1) Operations Research by Hira & Gupta - Buy now

(2) Thermodynamics by Cengel & Boles - Buy now

(3) Heat & Mass Transfer by Incropera - Buy now

(4) Fluid Mechanics by Cengel & Cimbala- Buy now

(5) Manufacturing Engineering by  Kalpakjian- Buy now

(6) Theory of machines by SS rattan- Buy now

(7) Strength of materials by Timoshenko- Buy now

(8) Machine design by VB Bhandari- Buy now

(9) Engineering Mathematics by B.S. Grewal- Buy now

(PDF) Degeneracy in Transportation Problem Using Modi[u-v] method

Definition- A basic feasible solution in which the total number of non-negative allocations is less than m+n-1 is called degenerate basic feasible solution. 

In simple words, Degeneracy in transportation problem means number of allocations less than m+n-1. 

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Modi[u-v] method in Transportation Problem

Definition- Modi[u-v] or Modified distribution method is used to check optimality of the initial basic feasible solution determined by using any of the ibfs(initial basic feasible solution) methods viz. North-west corner rule, Least cost method and vogel's approximation method.

Procedure:

Step 1- Calculate ibfs(initial basic feasible solution).

Step 2- Check optimality conditions i.e. feasible solution has exactly m+n-1 number of allocations and all these allocations are at independent positions(it doesn't form a loop).

Step 3- Apply Modi method if above conditions satisfies. Calculate ui+vj=cij for occupied cells. Take u1=0 {u values along row and v values along column}.

Step 4- Calculate cij-(ui+vj) for unoccupied cells.

Step 5- If all values of cij-(ui+vj) ≥ 0 {Optimal solution}
Most -ve value enters (if not)

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(PDF) Comparison between Stepping stone method and Modi [u-v] method | Transportation Problem

Objective- The objective of both the methods viz. stepping stone method and modi [u-v] method is same i.e. to provide us an optimal solution(best possible solution) to the transportation problem. Either we use stepping stone or modi method, we are going to get the same optimal solution.

What is the difference?
In the stepping-stone method, we allocate +1 unit to each unoccupied cell and a closed loop is traced for each unoccupied cell. Net change in transportation cost T.C. (cell evaluations) are found for the loop. If all cell evaluations are  ≥ 0 our optimal solution has reached. if not the most -ve cell evaluation value enters.

On the other hand, In modi[u-v] method cell evaluations of all the unoccupied cells are calculated using cij-(ui+vj). if cij-(ui+vj)  ≥ 0 our optimal reaches. if not the most -ve cell evaluation enters and only one closed loop is traced for that particular cell. Thus it provides considerable time saving over the stepping-stone method. 

P.S. Cell evaluations are same in both the cases.

Example- Download the following notes in which i have taken an example and solved the problem using stepping-stone method and modi[u-v] method. You can also see cell evaluations are same in both cases only approach is different to solve the transportation problem.

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Vogel's Approximation Method(VAM) | Transportation Problem

Definition- Vogel's Approximation method(VAM) or Penalty Method is a heuristic method which is used to calculate initial basic feasible solution(ibfs) to the transportation problem. It doesn't provide us the optimal solution. VAM is an improved version of Least-cost method. The ibfs obtained from Vogel's approximation method is better as compared to other methods. I am optimistic that following video will help you to solve VAM method easily.

Procedure of VAM:

Step 1- Calculate penalties for each row and column. 
Penalty= 2nd Minimum-Minimum number

Step 2- Select maximum penalty and allocate as much as possible to the minimum cost for that penalty.

Step 3- In case of tie among the highest penalties, select row or column having minimum cost.

Step 4- In case of tie in the minimum cost also, select cell which can have maximum allocation.

Step 5- If there is tie in maximum allocation also, select the cell arbitrarily for allocation.


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Assignment problem minimization | Practice questions

Here you can practice assignment problem minimization questions. Minimization case in assignment problem means we have to minimize our cost or time. I hope the following video will help you conquer the fear of solving an assignment problem. If there is any problem you're facing while solving a question please let me know in the comments section.

                                      

                                

                                

Solve the following assignment problems for minimization:

(1) 

11	17	8	16	20
9	7	12	6	15
13	16	15	12	16
21	24	17	28	26
14	10	12	11	13

Ans-60

(2) 

10	5	9	18	11
13	9	6	12	14
3	2	4	4	5
18	9	12	17	15
11	6	14	19	10

Ans-39

 (3)

5	7	11	6
8	5	9	6
4	7	10	7
10	4	8	3

 Ans-23

(4) 

10	5	13	15	16
3	9	18	3	6
10	7	2	2	2
5	11	9	7	12
7	9	10	4	12

Ans-22 

(PDF) Stepping Stone Method | Transportation Problem

Definition- Stepping Stone Method is used to check optimality of the initial basic feasible solution determined by using any of the ibfs methods i.e. Northwest corner rule, Least cost method and Vogel's Approximation method.

Procedure:

Step 1- Calculate Ibfs(initial basic feasible solution)

Step 2- Check Optimality conditions i.e. If feasible solution has exactly m+n-1 Number of allocations and all these allocations are at independent positions(it doesn't form a loop).

Step 3- Apply stepping stone method if above conditions satisfies. Allocate +1 unit to all unoccupied cells(non basic variables) and find out net change in transportation cost(cell evaluation) for the loop.

Step 4- If all cell evaluations are  ≥ 0 our optimal solution has reached. if not the most -ve cell evaluation value enters.

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