Initial basic solution is feasible if number of occupied cells = m + n – 1, where m = no. of rows and n = No. of columns
If no. of occupied cells ≠ m + n – 1 , then initial basic solution is degenerate
To remove such degeneracy a very small quantity Є (epsilon) is allocated to one or more least cost independent unoccupied cell ( such least cost cell should be independent means that a closed path cannot be made from this cell.). If the least cost cell is not independent then next least cost independent cell is chosen (Є can be added to unoccupied dummy cell )
Degeneracy may also arise at later stage. Thus after every improved solution degeneracy is to be checked (Є may have to be used in improved solution also)
Step 3 Check if initial basic feasible solution is optimal or not
It can be done by Modified distribution method (MODI) or Algebra method.
let the row wise costs are Ui &column wise costs are Vj, where i = 1,2,3 ….m & j = 1,2,3……n
Assign 0 value arbitrarily to a row or column variable Uj or Vj. (assign 0 to U1) Taking each cost cell (Uj+Vj) = Cij , calculate individually all values of Ui & Vj Calculate opportunity cost for each unallocated cell i.e. Ui + Vj – Cij = Δij
If all Δij values are 0 or –ve, solution is optimal, if any Δij value is +ve then solution is not optimal.
If any Δij value is 0 it means alternate optimal solution also exist
If solution is not optimal, the cell with largest +ve opportunity cost Δij should be selected. Form this cell a closed loop starting and ending at this cell should be made and reallocate the solution
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Q1 .Transportation cost (Rs/unit) Distribution centers
Plant X Y Z Availabilty (units)
A 50 30 20 20,000
B 15 27 40 18,000
C 25 25 45 12,000
Demand 10,000 15,000 25,000
Suggest optimal solution for the following transportation problem and indicate the total minimum transportation cost (no degeneracy, optimal sol)
If the company wants atleast 5,000 units to be transported from plant B to distribution center Z, what will be the transportation schedule and effective cost?
(restriction, no degeneracy, not optimal) Q2 A product is manufactured by four factories A, B, C and D. The Unit production costs are Rs.2,
Rs.3, Re.1 and Rs.5 respectively. Their daily production capacities are 50, 70, 30 and 50 units respectively. These factories supply the product to four P, Q, R and S. The demand made by these stores are 25, 35, 105 and 20 Units transportation cost in rupees from each factory to each store is given in the following table;
Stores
P Q R S
A 2 4 6 11
B 10 8 7 5
C 13 3 9 12
D 4 6 8 3
Determine the extent of deliveries from each of the factories to each of the stores so that the total cost (production and transportation together ) is minimum. (May 2002)
(only table imp, non degeneracy, optimal)
Q3 A compressed Natural Gas (CNG) company has three plants producing gas and four outlets. The cost of transporting gas from different production plants to the outlets, production capacity of each plant and requirement at different outlets is shown in the following cost-matrix table:
outlets
Plants A B C D capacity of production
X 4 6 8 6 700
Y 3 5 2 5 400
Z 3 9 6 5 600
Requirement 400 450 350 500 1700
Determine a transportation schedule so that the cost is minimized.
The cost in the cost-matrix is given in thousand of rupees. (Nov 2001) (Simple, non deg, optimal, multiple optimal)
PRAVINN MAHAJAN CLASSES 9871255244, 8800684854
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Q4 Solve the following problem using transportation method, obtaining the initial feasible solution by VAM.
The cell entries in the table are unit costs
To
1 2 3 4 5 supply
From
1 80 69 103 64 61 12
2 47 100 72 65 40 16
3 16 103 87 36 94 20
4 86 15 57 19 25 8
5 27 20 72 94 19 8
Demand 16 14 18 6 10
(Degeneracy, non optimal, balanced, difficult path)
Q5 A company has 4 terminals U,V,W and X. At the start of a particular day 10,4,6 and 5 trailers respectively are available at these terminals. During the previous night 13,10,6 and 6 trailers respectively were loaded at plants A,B,C,D. The company dispatcher has come up with the costs between the terminals and plants as follows
Plants
A B C D
U 20 36 10 28
Terminals V 40 20 45 20
W 75 35 45 50
X 30 35 40 25
Find the allocation of loaded trailers from plants to terminals in order to minimize transportation cost.
Q6 Garg and Garg a leading firm has 3 auditors. Each auditor can work upto 160 hours during the next month, during which time 3 projects must be completed. Project 1 will take 130 hours, Project 2 will take 140 hours and project 3 will take 130 hrs. The amount per hour billed for assigning the auditor a project is:
Project (amount in rupees)
1 2 3
Auditor
A 1200 1500 1900
B 1400 1300 1200
C 1600 1400 1500
Formulate this as a transportation problem and find the optimal solution. Also find out the maximum total billings during the next month
Q7 Consider the following transportation cost table. The costs are given in rupees, supply and demand are in units. Determine an optimal solution
Destination 1 2 3 4 5 Supply
Source
I 40 36 26 38 30 160
II 38 28 34 34 198 280
III 36 38 24 28 30 240
Demand 160 160 200 120 240 n88
(unbalanced, degeneracy, non optimal, difficult path, two e)
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Q8 A company has two factories at A, B, and C which supply warehouses at D, E, F and G. Monthly factory capacities are 160, 150, and 190 units respectively. Monthly warehouse requirements are 80, 90, 110 and 160 units respectively. Unit shipping costs (in rupees) are as follows:
To
D E F G
From
A 42 48 38 37
B 40 49 52 51
C 39 38 40 43
Determine the optimum distribution for this company to minimize shipping costs.
(degen, unbal, min, )
Q9 The cost per unit of transporting goods from factories X, Y, Z to destinations A, Band C and the quantities demanded and supplied are tabulated below. As the company is working out the optimium logistics, the company has announced a fall in the oil prices. The revised unit costs are exactly half the costs given in the table. You are required to evaluate the minimum transportation costs.
Destination
A B C Supply
Factories
X 15 9 6 10
Y 21 12 6 10
Z 6 18 9 10
Demand 10 10 10 30 J09
(Min,bal,deg,2e)
Q10 The initial allocation of a transportation problem along with the unit cost of transportation from each origin to the destination is given below. You are required to arrive at the minimum transportation cost by the Vogel’s approximation method and check for optimality (Hint take u1 = 0 at Row 1 for initial cell valuation)
Requirement 18
10 8 4
Availability 12 8 8 8 4 40 M07 (degeneracy, optimal)
11 2 8 6 2
9 9 12 9 6
7 6 3 7 7
9 3 5 6 11
8 6 4
10
2
8
2
PRAVINN MAHAJAN CLASSES 9871255244, 8800684854
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Q11 Alpha Company has 3 plants and 3 warehouses. The cost of sending a unit from different plants to the warehouses, production at different plants and demand at different warehouses are shown in the following matrix:
Determine the transportation schedule so that cost is minimized. Assume that cost in the cost matrix is
given in thousands of rupees. M01
(Min, unbal,tie,no deg, optimal,multi)
Q12 Solve the following transportation problem for minimum cost:
Destination/origin A B C D Requirement
Q13 Home building construction company is interested in taking loans from banks for its projects- P,Q,R,S,T.
The rates of interest and the lending capacity differ from bank to bank. All these projects are to be completed. The relevant details are provided below .
Assuming the role of a consultant, advise the company as to how it should take the loans so that the total interest payable is least. Find out the alternate optimal solutions, if any. N90 …(very multi, min, bal, no deg) Q14 Solve the following transportation problem and state whether the solution derived by you is unique.
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