TRANSPORTATION
Compiled by – Sapna Bhupendra jain, 9811255704
Outline the basic idea in a Transportation Application.
Transportation applications relate to a LPP where goods are to be transported from “m”
production locations (factories) to “n” sales locations (warehouses).
The objectives are -
To meet the differing availability and requirements of these locations and
To minimize the total transportation costs.
The Transportation application can be solved in three stages –
Preliminary Check
Initial Basic Feasible Solution (IBFS)
Optimality Test
What are the different methods of finding the Initial Basic Feasible Solution to a transportation
problem?
IBFS can be determined using any of the following methods -
1. Northwest Corner Rule
2. Least Cost Cell Method
3. Vogel’s Approximation Method (VAM)
Highlight the stages involved in determining the solution to the Transportation Problem.
Stage 1: Preliminary check involves the following-
Verify Objective = Minimisation. In case of Profit Matrix, convert the same into an
opportunity Loss Matrix, by subtracting each number from the highest number in the
matrix.
Verify Nature of Data = Balanced. Data is said to be balanced if Total Availability = Total
Requirement. In case of Unbalanced Data, a Dummy Column or Row should be
introduced with Zero Transportation costs.
Stage 2: IBFS can be determined using any of the following methods_
(a) Northwest corner Rule.
(b) Least Cost Cell Method.
(c) Vogel’s Approximation method (VAM)
Stage 3: Optimality test : It consists of the following steps -
Computation of margin number, ‘U’ and ‘V’ for all rows and columns such that
U+V = Cost of allocated Cells.
Computation of U+V for unallocated cells.
Computation of Cost Less (U+V) for unallocated cells, i.e. Step 1 minus Step 2 above.
Procedure under North West Corner Rule
Ensure Availability = Requirement, by inserting dummy row or column, if required.
2. - Go to Top left-hand corner cell of the matrix
- Compare availability and requirement.
- Allocated availability or requirement, whichever is less, to that cell.
- Cancel the row or column where availability or requirement is exhausted.
3. Go to top left hand corner cell of the resultant matrix (after cancellation of row or column in step 2)
Repeat Step 2 procedure till all the row availability and column requirements are satisfied.
Compiled by – Sapna Bhupendra jain, 9811255704
Procedure under least Cost Cell Method
1. Ensure Availability = Requirement, by inserting dummy row or column , if required.
2. Identify the cell with the lowest Cost. In case of a tie, arbitrary selection may be made.
Compare availability and requirement.
Allocated availability or requirement, whichever is less, to that cell.
Cancel the row or column where availability or requirements exhausted.
3. Identify the next lowest cost cell in the matrix.
Repeat Step 2 procedure till all the row availability and column requirements are satisfied.
Advantage: The cost associated with each route is taken into consideration. So, this method leads to a better
allocation than North West Corner Rule Method.
Procedure under Vogel’s Approximation Method (VAM)
Compute Cost Differences for each row and column.
Cost Difference is the difference between the least cost and the next least cost in that row\column.
In case of tie in least cost, Cost Difference 0.
Ascertain the maximum of cost differences and select that row or column for allocation.
Choose the least cost cell in the selected row or column for allocation.
Compare availability and requirement for that cell.
Allocate availability or requirement, whichever is less, to that cell.
Cancel the row or column where availability or requirement is exhausted.
Compute Cost Difference for the resultant matrix and repeat the above procedure till all row availability and column
requirements are satisfied.
Steps involved in Optimality Test
Optimality Test involves the following steps-
1. Table 1: Ui +Vj for allocated cells:
Select the row / column with maximum number of allocations.
For that row / column, Ui + Vj is equal to zero. [Ui for Rows; Vj for columns]
The other set of numbers Ui /Vj are computed in such a way that Ui + Vj = Cost of allocated cells.
Note: The Ui +Vj table can be completed only if the IBFS is non-degenerate.
IBFS is said to be degenerate if number of allocations <(No. of rows + No. of column-1).
In case of degeneracy, a dummy allocation “e” (a number very close to Zero) is made in the least cost unallocated cell
falling in non-dummy row or column.
2. Table 2: Ui + Vj for unallocated Cells:
Draw a matrix for the given rows and columns
Block out the allocated cells.
Compute Ui + Vj (total of margin numbers) for all unallocated cells.
3. Table 3: Net Evaluation Table = ∆ij
Draw a matrix for the given rows and columns.
Block out the allocated cells.
Compute Cost Difference for all unallocated cells. Cost Difference = umber in Table (1) Less Number in
Table (2).
DECISION:
If all numbers in the Net Evaluation Table are non - negative (i.e.>0), IBFS is optimal and unique.
If all numbers are positive and one of the cells contains zero, IBFS is optimal but not unique. Alternative solution
exists. Total number of solutions = Number of Zeroes + 1.
Compiled by – Sapna Bhupendra jain, 9811255704
If any one number in Table 3 is negative, solution is not optimal. Reallocation should be made to determine alternative
Basic Feasible Solution (ABFS).
ABFS is tested for optimality by adopting the same procedure.
“Loop” Diagram in Transportation
Identify the worst negative in Table 3. If there is a tie, choose the least cost cell.
Draw a loop with the following principle-
Loop should commence from and end in the selected worst negative cell.
It should have only allocated cells as its other corners.
Only horizontal and vertical lines (not diagonal) shall be permitted.
Loop should result in an even sided figure.
Identify the selected cell as having scope for allocation. It is marked with plus (+) sign.
Other corners of the loop are identified with (-) and (+) signs alternatively.
ABFS is determined by reference to reallocation as specified by the corners of the loop.
Reallocation is done as under-
Identify the allocations to the negative cells (Corners marked with (-) sign.
Select the minimum out of the above allocations.
Add this minimum to (+) corners and subtract this minimum from (-) corners. Other allocations remain
unaffected.
This ABFS is tested for optimality using the procedure outlined in the earlier question. In case negatives still arise
in the Net Evaluation Table, the reallocation procedure should be continued further..
Questions
Q.1- Profit Maximisation – Unbalanced Data
A Company has four factories F1, F2, F3 and F4 manufacturing the same product. Production and raw material costs
differ from factory to factory and are given in the following table. The transportation costs from the factories to sales
depots S1, S2 and S3 are also given. The Sales price ad the total requirement at each depot as also the product capacity
at each factory is also stated. Determine the most profitable production and distribution schedule and the
corresponding profit. The Surplus production should be taken to yield zero profit.
Particulars F1 F2 F3 F4 Sale prices Requirements Per unit.
Per unit
Production Cost per unit 15 18 14 13
Raw Material Cost per unit 10 9 12 9
Transportation Cost per unit to
S1 3 9 5 4 34 80
S2 1 7 4 5 32 120
S3 5 8 3 6 31 150
Production capacity (units) 10 150 50 100
Compiled by – Sapna Bhupendra jain, 9811255704
Q. 2- Unbalanced Maximisation-Degeneracy
A Company has three factories and four customers. It furnishes the following schedule of profit per unit on
transportation of goods to customers in rupees. You are required to solve the transportation problem to maximize the
profit. Determine the resultant optimal profit.
Factory / Customer A B C D Supply
P 40 25 22 33 100
Q 44 35 30 30 30
R 38 38 28 30 70
Demand 40 20 60 30
Q. 3- Balanced Minimisation
The information on the available supply to each warehouse, requirement of each market and the unit transportation
cost from each warehouse to each market is given below:
Warehouse Market Supply
M1 M2 M3 M4
A 5 2 4 3 22
B 4 8 1 6 15
C 4 6 7 5 8
Demand 7 12 17 9
The shipping clerk has worked out the following schedule from his experience.
Unit 12 1 9 15 7 1
From-Warehouse A A A B C C
To-Market M2 M3 M4 M3 M1 M3
You required to-
Check and see if the clerk has the optimal schedule.
Find the optimal schedule and minimum total shipping cost and
If the check is approach by a carrier of route C to M2, who offers to reduce his rate in the hope of getting some
business, by how much should the rate be reduced before the clerk considers giving him an order.
Q. 4: Balanced Minimization -Multiple optimal solution.
Home Build 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 our alternate optimum solution, if any.
Sources Bank Interest rate in % for projects Max credit (in 000s)
P Q R S T
Private Bank 20 18 18 17 17 Any amount
Nationalized Bank 16 16 16 15 16 400
Compiled by – Sapna Bhupendra jain, 9811255704
Co-operative Bank 15 15 15 13 14 250
Amount required (in 000) 200 150 200 125 75
Q. 5:Unablanced Minimisation – Big M Cost-Production Planning
Timely and Co, a manufacturer must produce a product in sufficient quantity to meet contractual sales in next four
months. The production capacity and unit Cost of production vary from month to month. The production produced in
one month may be held for sale in later months but at an estimated storage cost of Re.1 per unit per month. No storage
cost is incurred for goods sold in the same month in which they are produced. There is no opening inventory and none
is desired at the end of four months. The necessary details are given below.
Month Contracted Sales Maximum production Unit cost of Production
1 20 40 14
2 30 50 16
3 50 30 15
4 40 50 17
How much should the manufacturer produce each month to minimise total Cost?
Q.6- Unbalanced Minimisation –production and supply scheduling
Alpha Co. 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:
Pant Warehouses Production
A B C
X 8 16 16 152
Y 32 48 32 164
Z 16 32 48 154
Demand 144 204 82
Determine the transportation schedule, so that the cost is minimised. Assume that cost in the cost matrix is given in
thousands of rupees.
Q.7 - Unbalanced Minimisation
Consider the following transportation cost table. The costs are given in rupees, supply and demand are in units
Determine an optimal solutions.
Source / Destination 1 2 3 4 5 Supply
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
Q. 8- Unbalanced Maximisation- Multiple Optimal Solutions
Somehow Achieve Profits (SAP) Ltd. has four production plants and four wholesale warehouse outlets. The
warehouses are situated away from the production plants. The production and transportation costs, the selling prices,
production capacities, and sales quantities are given below:
Production Plants Warehouses Production Capacity Cost of per unit(Rs.)
1 2 3 4 (Units) Material Labour
&OH
Compiled by – Sapna Bhupendra jain, 9811255704
A 10 14 7 10 140 4 6
B 8 12 5 10 100 5 8
C 3 7 11 8 150 4 9
D 9 12 6 13 160 3 8
Warehouse requirement in units 80 120 130 110
Selling price(ex warehouse)per unit 26 32 30 25
The cost of transporting a unit from a given plant to a warehouse is shown in the body of the matrix in rupees per unit.
Compute a plan for production and distribution that will achieve maximum profit for the company. Also, state the
profit achieved by such a plan.
Q. 9- Unbalanced Maximisation
Bishop Company has four terminals A, B, C and D. At the start of a particular day, 10,4,6 and 5 trailers respectively
are available at these terminals. During the previous night 13, 10,6and 6 trailers respectively were loaded at plant P, Q,
R and S. The Company dispatcher has come up with the following costs:
Terminals Plants
P Q R S
A 20 36 10 28
B 40 20 45 20
C 75 35 45 50
D 30 35 30 25
Find the allocation of loaded trailers from plants to terminals in order to minimise transportation costs.
Q. 10- Unbalanced Minimisation
A Company produces a small component for all industrial products and distributes it to five wholesalers at a fixed
price of Rs. 2.50 per unit. Sales forecasts indicate that monthly deliveries will be 3000,3000,10000,5000and 4000units
to wholesalers W1, W2, W3, W4and W5 respectively. The monthly production capabilities are 5000,10000,12500,at
plants P1, P2, and P3 respectively. The direct costs of production of each unit are Rs. 1.00,Rs.0.90, and Rs.0.80 at
plants P1, P2 and P3 respectively. The transportation costs of shipping a unit from a plant to a wholesaler is given
below:
Plant Warehouses
W1 W2 W3 W4 W5
P1 0.05 0.07 0.10 0.15 0.15
P2 0.08 0.06 0.09 0.12 0.14
P3 1.10 0.09 0.08 0.10 0.15
Find how many components each plant supplies to each wholesaler in order to maximize profit.
Compiled by – Sapna Bhupendra jain, 9811255704
Q. 11- Unbalanced Maximisation-Optimal Sales Distribution
A household product is manufactured in factories A, B, C and D and is sold at centres1,2and 3. The cost in Rupees of
the product per unit and capacity in kilograms per unit time of each plant as also the sale price in Rs. per unit and the
demand in Kg. Per unit time are given in the following table. You are required to find the optimal sales distribution.
Factory Cost per unit(Rs.) Capacity per unit (Kg.)
A 12 100
B 15 20
C 11 60
D 13 80
Sales Centre Sale price per unit(Rs.) Demand per unit (Kg)
1 15 120
2 14 140
3 16 60
Q. 12- Unbalanced Maximisation –Optimum Investment Strategy
High Yield Ltd. has provided the following data seeking your advice on the optimum investment strategy:
Investment made at the Net Return Data (in paise)of Selected investments Amount
Beginning of the year. P Q R S Available (Lacs)
1 95 80 70 60 70
2 75 65 60 50 40
3 70 45 50 40 90
4 60 40 40 30 30
Maximum investment(Lacs) -
40 50 60 60
The following additional information also provided:
P, Q, R and S represent the selected investment.
The company has decided to have four years investment plan.
The policy of the company is that amount invested in any year will remain so until the end of the fourth year horizon
(for e.g., a rupee invested in investment P at the beginning of year 1 will grow to Rs. 1.95 by the end of the fourth
year yielding a return of 95 paise).
Using the above information, determine the optimum investment strategy.
Q.13- Product Distribution –Multiple Optimal Solution
Solve the following transportation problem and state whether the solution derived by you is unique.
Godown 1 2 3 4 5 6 Stock Available
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Factory 1 7 5 7 7 5 3 60
Factory 2 9 11 6 11 - 5 20
Factory 3 11 10 6 2 2 8 90
Factory 4 9 10 9 6 9 12 50
Demand 60 20 40 20 40 40
Note: It is not possible to transport any quantity from Factory 2 to Godown 5.
Q.14- Advertisement Campaign decision.
The manufacturer of jeans is interested in developing an advertisement campaign that will reach four different age
groups. Advertising campaigns can be conducted through TV, Radio and Magazines. The following table gives the
estimated cost in paise per exposure for each age group according to the medium employed. In addition, maximum
exposure levels possible in each of the media, namely TV, Radio and Magazine are 40,30 and 20 million respectively.
Also the minimum desired exposures within each age group, namely 13-18,19-25, 26-35 and 36 and older are
30,25,15and 10 millions. The objective is to maximize the cost of attaining the minimum exposure level in each
group.
Age group 13-18 19-25 26-35 36 and older
TV 12 7 10 10
Radio 10 9 12 10
Magazines 14 12 9 12
(a) Formulate the above as a transportation problem, and find the optimum solution.
(b) Solve this problem if the policy is to provide at least 4 million exposures through TV in the 13-18 age group, and
at least 8 million exposures through TV in the age group 19-25.
Q. 15- Investment Strategy
A Company wishes to determine an investment strategy for each of the next four years. Five investment types have
been selected, investment capital has bee allocated for each of the coming four years, and maximum investment levels
have been established for each investment type. An assumption is that amounts invested in any year will remain
invested until the end of the planning horizon of four years. The following table summarizes the data for this problem.
The values in the body of the table represent net return on investment of one rupee upto the end of the planning
horizon. For example, a rupee invested in investment Type B at the beginning of Year 1 will grow to Rs. 1.90 by the
end of the fourth year, yielding a net return of Rs. 0.90.
Investment made at beginning Net Return Data from Investment Type Rupee Available
Of year A B C D E (000’s)
1 0.80 0.90 0.60 0.75 1.00 500
2 0.55 0.65 0.40 0.60 0.50 600
3 0.30 0.25 0.30 0.50 0.20 750
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4 0.15 0.12 0.25 0.35 0.10 800
Max. Rupee invt. (000’s) 750 600 500 800 1000
The objective in this problem is to determine the amount to be invested at the beginning of each year in an investment
type, so as to maximize the net rupee return for the four-year period. Solve the above transportation problem and get
an optimal solution. Also calculate the net return on investment for the 4 year planning period.
Q. 16- 2001-Dec MEC Ltd. has received an order for supply of100 units of a particular machine. Four
types of components (C1to C4) one each required per machine have to be fabricated. The firm has five
workshops ( W1to W5) each of which is capable of fabricating any component. However , for efficiency and
cost savings, a workshop will be assigned to fabricate at most one type of component. The costs( rupees per
unit) of fabrication vary and data are given below:
Find the optimum assignment of the components to the workshops and the overall costs.
If prior to executing the above plane, there is a breakdown in W1. How will this affect the above optimum
plan?
C1 C2 C3 C4
W1 21 23 29 17
W2 22 21 31 18
W3 24 20 30 19
W4 17 18 26 16
W5 23 21 29 18
Q. 17- 2002 Dec A Company has 4 Factories F1,F2, F3and F4 manufacturing the same product.
Production and raw material costs differ from factory to factory and are given in the table below in the first
two rows. The transportation costs from the factories to the sales depots S1, S2and S3 are also given. The
last two columns in the table below give the sales price and total requirement at each depot and the
production capacity of each factory is given in the last row.
F1 F2 F3 F4 Sales price /Unit(Rs.) Requirement
Production Cost / Unit (Rs) !5 18 14 13
Raw Material Cost / unit (Rs.) 10 9 12 9
Transportation Cost / Unit (Rs.)
S1 3 9 5 4 34 80
S2 1 7 4 5 32 120
S3 5 8 3 6 31 150
Production capacity 10 150 50 100
Determine the optimal solution and the associated profit by using the Vogel’s Approximation
Method(VAM).
Q. 18- 2003 June The Bombay Transport Company has trucks available at four different sites in the
following numbers:
Site A - 5 trucks
Site B – 10 trucks
Site C - 7 trucks
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Site D – 3 trucks
Customers – W, X and Y require trucks as shown below.
Customer W- 5 trucks
Customer X- 8 trucks
Customer Y- 10 trucks
Variable Costs of getting trucks to the Customers are given below.
From A to W- Rs. 7, to X- Rs. 3, to Y – Rs. 6
From B to W – Rs. 4, to X- Rs. 6, to Y- Rs. 8
From C to W- Rs .5, to X- Rs. 8 to Y- Rs. 4
From D to W – Rs. 8, to X- Rs. 4, to Y- Rs. 3
Solve the above transportation problem.
Q. 19- 2004 June; The following table shows all the necessary information on the availability of supply to
each factory of Best Industries Ltd., the requirement of each destination and the unit transport cost( in Rs.)
from each factory to each destination:
Factory Destination Supply
I II III
A 5 1 7 10
B 6 4 6 80
C 3 2 5 15
Demand 75 20 50
Since there is not enough supply, some of the demands at the three destinations may not be satisfied.
For the unsatisfied demands, let the penalty costs be Re. 1, Rs. 2 and Rs. 3 respectively.
Find the optimal allocation that minimize the transportation and penalty costs by using the Vogel’s
Approximation Method (VAM).
Q20. Unbalanced Maximaisation-Maximum Profits
Fashion Store wishes to purchase following types of dresses.
Dress Type SS S M L XL
Quantity 150 100 75 250 200
Tenders are submitted by four different manufacturers who undertake to supply not more than the quantities
below (all dress types combined).
Manufacture W X Y Z
Quantity 300 250 150 200
The store estimates that its profit per dress will vary with the manufacturer as show below:
Manufacturer / Dress Type SS S M L XL
W 11 14 17 9 6
X 12 13 18 7 4
Y 10 14 19 8 5
Z 13 11 16 10 7
Compiled by – Sapna Bhupendra jain, 9811255704
Use transportation techniques to determine how the orders should be placed. What is the maximum profit?
Compiled by – Sapna Bhupendra jain, 9811255704
