h a c 11 transportation problems e r - amazon...

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CA. B L Chakravarti (FCA, CPA, CAIIB) 1.1

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Transportation Problems

Past Examinations Coverage:

Year M-13 N-12 M-12 N-11 M-11 N-10 N-10* M-10 M-10* N-09

Q No. - - 1b+3b 1d+4d 6 (a) 5 (a) 6 (b) 4 (b) 2 (b)

Marks - - 5+3# 5+4# 8 10 4# 4# 4#

Year N-09* J-09 J-09* N-08 N-08* M-08 N-07 M-07 N-06 M-06

Q No. 4 (c) 4 (a) 4 (a) 6 (a) 1 (b)

Marks 4# 4# 3# 10 6

Year N-05 M-05 N-04 M-04 N-03 M-03 N-02 M-02 N-01 M-01

Q No. 5(c) 2 (b) 3 (b) 3 (c) 3 (c)

Marks 8 3# 10 10 10

Year N-00 M-00 N-99 M-99 N-98 M-98 N-97 M-97 N-96 M-96

Q No. 3 (b) 4 (b) 2 (c) 3 (b) 4 (b) 5 (c) 2 (b) 4 (b) 2 (b) 2 (b)

Marks 10 10 10 10 10 10 10 10 10 10

* means questions from old syllabus

# means theory questions

Learning Objectives:

Understanding the feature of Assignment Problems.

Formulate an Assignment problem.

Hungarian Method

Unbalanced Assignment Problems

Profit Maximization Problems

Advance Management Accounting

1. Email: @icai.org Page 11.2

Transportation:

The transportation problem is concerned with the allocation of items between suppliers (called origins) and

consumers (called destinations) so that the total cost of the allocation is minimized. The objective of the

transportation problems is to determine the quantity to be shipped from each source to each destinations so as

the transportation cost is lowest to maintain the supply and demand requirements.

Applications:

(i) Minimize shipping costs from factories to warehouses (or from warehouses to retail outlets)

(ii) Determine lowest cost location for new factory; warehouse; office or other facility.

(iii) Find minimum cost production schedule that satisfies firm’s demand and production limitations

(called ‘Production Smoothing’)

Conditions: To use the transportation algorithm; the following conditions must be satisfied.

(i) The cost per item of each combination of origin and destination must be satisfied.

(ii) The supply of items at each origin must be known.

(iii) The requirement of items at each destination must be known.

(iv) The total supply must equal the total demand.

Stages in Transportation Problems: The transportation algorithm has four stages.

Stages -1: Arrange the data in table format and set up the transportation table to find any feasible allocation. A

feasible allocation is one in which all demand at the destinations is satisfied and all supply at the origin is

allocated.

Stages -2: Do preliminary check consist the following:

(i) Verify Objective = Minimization. In case of Maximization or profit matrix, convert the same in to

opportunity loss matrix by subtracting each element from the highest element of the matrix.

(ii) Verify that the matrix is balanced i.e. Total Availability = Total Requirement. In case of

unbalanced matrix, a dummy column or dummy row should be introduced with nil transportation

cost.

Stages -3: Develop an initial basic feasible solution (IBFS) by using any of the following methods.

(i) Least Cost method.

(ii) North-West corner Method.

(iii) Vogel’s Approximation Method.

Stages -4: Test the IBFS solution for optimality. Modified Distribution (MODI) is the most common method and

widely used for test the optimality.

Stages -5: In case the IBFS solution is not optimal, develop the improved solutions.

The Transportation Problems

CA. B L Chakravarti (FCA, CPA, CAIIB) Page 11.3

Method 1:– North-West corner method: The practical steps in the North-West Corner cell are given below:

Step 1: Make maximum possible allocation to the upper-left corner cell (known as North-West corner cell) in

the first row depending upon the availability to supply for that row and demand requirement for the

column containing that cell.

Step 2: Move to the next cell of the first row depending upon remaining supply for that Row and the demand

requirement for the next column. Go on till the Row totally is exhausted.

Step 3: Move to the next row and make allocation to the cell below the cell of the preceding row in which the

last allocation was made and follow Step-1 and Step-2.

Step 4: Repeat the Step-1 to Step-3 till all requirements are exhausted i.e. entire demand and supply is

exhausted.

Method 2:- Least Cost Method : The practical steps involved in the Least Cost are given below:

Step 1: Make maximum possible allocation to the least cost cell depending upon the demand/supply for the

column/row containing that cell. In case of tie in the least cost cell, make allocation to the cell by which

maximum demand or capacity is exhausted.

Step 2: Make allocation to the second least cost cell depending upon the remaining demand/supply for the

column/row containing that cell.

Step 3: Repeat the Step-1 and Step-2 till all requirements are exhausted i.e. entire demand and supply is

exhausted.

Method 3:- Vogel’s Approximation Method: This methods uses penalty costs. For each row and column the

penalty cost is the difference between the cheapest available route and the next cheapest.

Step 1: To calculate the penalty cost for each row and column, we look at the least cost cell and next least cell.

For each row and column subtract the least cost cell from the next least cell. This gives the penalty cost

of not allocating into the cell with cheapest cost. Write the penalty cost of each row on the right-hand

side of the concerned row and penalty cost of each column on the below of the concerned column.

Step 2: We choose the row or column with the largest penalty cost and allocate as much as possible to the least

cost cell in that row or column depending upon the quantity available. In this way, the highest penalty

costs; are avoided as far as possible.

In case of tie between the penalty cost, select the row or column having least cost cell however in case of

tie even in the case of least cost cell; make allocation to that cell by which maximum requirement are

exhausted.

Step 3: Shade the row/column whose availability or requirement is exhausted so that it shall not be considered

for further allocation.

Step 4: Repeat the Step-1 to Step-3 till all requirements are exhausted i.e. entire demand and supply is

exhausted.



Optimality Test: It can be applied to a transportation table if it satisfies the following conditions:

a) It contains exactly m+n-1 allocations, where m & n represent the no. of rows & columns of the table. {Note: (i) In case degeneracy occurs (i.e. allocations are less than m+n-1), infinitely small allocation(s) i.e. α

is/are introduced in least cost & independent cell to non-degenerate the solution. If least cost cell(s) is/are not independent, then cell(s) with next lower cost(s) is/are chosen.

(ii) Degeneracy can occur at initial basic feasible solution or at a later stage. It may be removed once the purpose is over.}

b) These allocations are independent i.e. loop can not be performed by them. {Note: (i) Closed path or loop should have even no. of turns & is formed with an allocation on each corner,

which in turn is a join of horizontal & vertical (not diagonal) lines. A loop may not involve all the allocations. (ii) Generally, IBFS obtained by Northwest Corner rule or VAM are always in independent positions though they may be m+n-1 or less than m+n-1 in no.}

The Stepping-Stone Method

1. Start with an arbitrary empty cell (a cell without allocation a.k.a. non-basic variables) and allocated +1 unit to this cell.

2. In order to maintain conditions of requirement & supply (a.k.a. rim constraints), 1 unit will be deducted/added to basic variables (i.e. cells having allocations).

3. Calculate Net change (a.k.a. opportunity cost or net evaluation) in transportation cost as a result of this perturbation.

4. Calculation Net evaluation for every empty cell. 5. If any cell evaluation is -ve, the cost can be reduced so that the solution under consideration can be

improved i.e. it is not optimal. On the other hand if all net evaluation are +ve, the given solution is optimal one. If any net evaluation is Zero, the given solution is optimal but there exist other optimal solutions as well.

6. If solution is not optimal, the cell with largest –ve opportunity cost should be selected & allocate maximum units to this route (subject to rim constraints).

7. Iterate towards optimal solution, if further savings are possible. Modified Distribution Method (MODI)

1. For the current IBFS (Initial Basic Feasible Solution) with m+n-1 occupied cell, calculate index numbers

(dual variables) Ui (i= 1,2,3,…m) and Vj (j= 1,2,3,…n) for rows and columns, respectively.

For calculating values of Ui andVj , the following relationship (formula) for occupied cells is used.

C ij = (Ui +Vj ) for all I,J.

2. Calculation Opportunity Cost for unoccupied cells by using the following formula:

∆ij = C ij - (Ui +Vj )

3. Examine unoccupied cells evaluation for all ∆ij.

(a) If ∆ij >0; then the cost of transportation will increase. i.e. and optimal solution has been arrived at.

(b) If ∆ij =0; then the cost of transportation will remain unchanged. But there exists are alternative

solution.

(c) If ∆ij <0; then an improved solution can be obtained by introducing cell (i,j) in the basis and go to

next step.

4. Select an unoccupied cell with largest negative opportunity cost among all unoccupied cells.



5. Construct a closed path for the unoccupied cell determined in Step 7 and assign plus (+) and minus (-)

sign alternatively beginning with plus sign for the selected unoccupied cell in clock wise or the direction.

eg:-

ᶲ 7-ᶲ ᶲmax i.e the maximum Qnty to be transferred

from occupied cell to unoccupied cell =

10- ᶲ 5+ᶲ min of -ᶲ cells i.e. min of 7, 10 i.e. 7 units

6. Assign as many units as possible to the unoccupied cells satisfying rim conditions. The smallest

allocation in a cell with negative sign on the closed path indicated the number of units that can be

transported to the unoccupied cells. This quantity is added to all the occupied cells on the path marked

with plus sign and subtract from those cells on the path marked with minus sign.

7. Go to Step-2 and repeat procedure until all ∆ij >0 i.e. an optimal solution is reached. Calculate the

associated total transportation cost.



Theoretical Questions

Q 1: Will the initial solution for a minimization problem obtained by Vogel’s Approximation Method and the

Least Cost Method be the same? Why? [M 11,4d, 4M]

Q 2: What do you mean by Degeneracy in transportation problem? How this can be solved?

[M 10, 6b, 4M]

Q 3: In an unbalanced minimization transportation problem, with positive unit transport costs from 3

factories to 4 destinations, it is necessary to introduce a dummy destination to make it a balanced

transportation problem. How will you find out if a given solution is optimal?

[M10, 4b, 4M]

[Ans.: No, Dummy destination won’t be introduced if total units available at factories = total units

required at destinations.]

[Nov-09 New]

Q 4: How do you know whether an alternative solution exists for a transportation problem?

[N 09, 2b, 4M]

Q 5: Explain the term degeneracy in a transportation problem. [N 09*, 4c, 4M]

Q 6: State the methods in which initial feasible solution can be arrived at in a transportation problem

[N 08 *, 4a, 4M]

Q 7: What are the common methods of obtaining initial feasible solution in a transportation problem?

[N-02, 2b, 3M]



Practical Questions

Q 1: [SA-1]

Find Initial basis feasible solution by (i) North-West Corner Method; (ii) Least Cost Method; (iii) Vogel’s

Approximation Method (VAM) for the following transportation problem in the following cases.

(i) When Cell entries represent the unit transportation cost

(ii) When Cell entries represent the Contribution

Store Factory

W1 W2 W3 W4 Supplies

F1 48 60 56 58 140 F2 45 55 53 60 260 F3 50 65 60 62 360 F4 52 64 55 61 220 Demand 200 320 250 210

Q 2: [SA-2]





Store Factory

W1 W2 W3 Supplies

F1 48 60 56 140 F2 45 55 53 260 F3 50 65 60 360 F4 52 64 55 220 Demand 200 320 250

Q 3: [SA-3]





Store Factory


F1 48 60 56 58 140 F2 45 55 53 60 260 F3 50 65 60 62 360 Demand 200 320 250 210



Q4: [SM-1]

A company has factories at A, B and C which supply warehouse 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. Units shipping costs (in Rs.) are as follows:

To From

D E F G

A 42 48 38 37 B 40 49 52 51 C 39 38 40 43

Q 5: [SM-2] [Multiple solution case]

Solve the following transportation problem

To From

D1 D2 D3 D4

Q1 5 3 6 2 19 Q2 4 7 9 1 37 Q3 3 4 7 5 34 16 18 31 25

Q 6: [SM-3] [MAXIMIZATION TRANSPORTATION PROBLEMS]

A company has 3 factories manufacturing the same product and 5 sole agencies in different parts of the country.

Production costs differ from factory to factory, and the sales prices from agency to agency. The shipping cost per

unit product from each factory to any agency is known. Given the following data, find the production and

distribution schedules most profitable to the company

Production Cost/Unit

Max Capacity No. of Units

Factory i

20 150 1 22 200 2 18 125 3

Shipping

Cost

Agency j

Demand to be Met

Sales Price

Q 7: [SM-4] [N-89] [Prohibited routes]

Solve the following transportation problem

Godown 1 Godown 2 Godown 3 Godown 4 Godown 5 Godown 6 Stock Availability

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. State whether the solution derived

by you is unique.

1 1 5 9 4

9 7 8 3 6

4 5 3 2 7

1 2 3 4 5

80 100 75 45 125

30 32 31 34 29



[Ans.: Total Cost Rs. 1120]

Q 8: [SM- 5]

The Link manufacturing company has several plants, three of which manufacture two principal products,

standard card table and deluxe card table. A new deluxe card table will be introduced which must be considered

in term of selling price & costs. The selling prices are: Standard Rs. 14.95, Deluxe Rs. 18.95, and new Deluxe Rs.

21.95.

Requirements Variable costs Available plant

capacity

Model Quantity Plant A Plant B Plant C Plant Capacity

Standard 450 8.00 7.95 8.10 A 800

Deluxe 1050 8.50 8.60 8.45 B 600

New

Deluxe

600 9.25 9.20 9.30 C 700

Solve this problem by the transportation technique for the greatest contribution.

Q 9: [SM-6] [SA-35]

A company wishes to determine an investment strategy for each of the next four years. Five investment types

have been selected, investment capital has been 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 summaries

the data for this problem. The values in the body of the table represent net return on investment of one rupee up

to 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 the beginning of year

Investment Type Amount available (Rs. In ‘000)

A B C D E

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 4 0.15 0.12 0.25 0.35 0.10 800

Maximum Rupee Investment (Rs. In ‘000)

750

600

500

800

1000

The objective is 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 planning horizon for

four-year period.

[Ans: 1-E-₹ 500; 2-B-₹ 390; 3-D-₹ 375; 4-A-₹ 37.5; 4-C-₹ 125; 4-D-₹ 17.5; Total Return = ₹ 1445.]



Q 10: [SM-7] [N 96, 2b, 10M] [V2-E9] [Investment decision]

XYZ and co. has provided the following data seeking your advice on optimum investment strategy:

Beginning of the year Net Return Data (in paise) of Selected Investments Amount available

(Lacs)

P Q R S

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 are also provided:

P, Q, R, and S represent the selected investments. The company 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. The values (paise) in the table represent net return on investment of one Rupee till the end of the

planning horizon (for example, 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, determine the optimum investment strategy.

Q 11: [SM-8] [N 97, 2b, 10M] [SA-30]

A particular product is manufactured in factories A, B, C and D; and is sold at centres 1, 2, and 3. The cost in Rs.

of product per unit and capacity in kgs per unit time of each plant is given below:

Factory Cost (Rs.) per unit Capacity (kgs) per unit

A 12 100

B 15 20

C 11 60

D 13 80

The sale price in Rs. per unit and the demand in kgs per unit time are as follows:

Sales Centre Sale Price (Rs.) per unit Demand (kgs) per unit

1 15 120

2 14 140

3 16 60

Find the optimal sale distribution.

[Ans.:

From Factory To Sales Centre Quantity Total Profit

A 1 100 300

B 2 20 -20

C 3 60 300

D 1 20 40

D 2 60 60

Dummy 2 60 0

680


CA. B L Chakravarti (FCA, CPA, CAIIB) .11

Q 12: [SM-9] [M 98, 5c, 10M] [V2-3] [SA-24]

Consider the following data for the transportation problem:

Factory Destination Supply to be exhausted

1 2 3

A 5 1 7 10

B 6 4 5 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 rupees 1,2 and 3 for destinations (1), (2) and (3) respectively. Find

the optimal allocation that minimizes the transportation and penalty cost.

Q 13: [SM-10] [N 98, 4b, 10M] [SA-34]

ABC Enterprises is having three plants manufacturing dry-cells, located at different locations. Production cost

differs from plant to plant. There are five sales offices of the company located in different regions of the country.

The sales prices can differ from region to region. The shipping cost from each plant to each sales office and other

data are given by following table:

Production Data Table

Production cost per unit Max Capacity in no. of units Plant no.

20 150 1

22 200 2

18 125 3

Shipping Cost and Demand & Sales Prices table

Shipping Cost

Sales Office 1 Sales Office 2 Sales Office 3 Sales Office 4 Sales Office 5

Plant 1 1 1 5 9 4

Plant 2 9 7 8 3 6

Plant 3 4 5 3 2 7

Demand and Sales Price


Demand 80 100 75 45 125

Sales Price 30 32 31 34 29

Find the production and distribution schedule most profitable to the company.

[Ans.:

Plant Sales Office Units Total Profit (Rs.)

1 1 50 450

1 2 100 1100

2 4 25 225

2 5 125 125

2 Dummy 50 0

3 1 30 240

3 3 75 750

3 4 20 280

Total 3170



Q 14: [SA-19]

Find Initial basic feasible solution by Vogel’s Approximation Method. Is the number of allocation equal to m+n-

1’if no, how will you deal with this situation?

Store Factory


F1 1 2 4 4 6 F2 4 3 2 0 8 F3 0 2 2 1 10 Demand 4 6 8 6

Note: Cell entries represent the unit transportation cost

Q 15: [SA-20] [Number of allocation = m+n-1]

Test the following Initial basic solution for Optimality.

Store Factory


F1

48

60 60

30 56

50 58

140

F2

45

260 55

53

60

260

F3

200 50

65

60

160 62

360

F4

52

64

220 55

61

220

Demand

200 320 250 210


Q 16: [SA-21] [Number of allocation < m+n-1]

Test the following Initial basic solution for Optimality.

Store Factory


F1

e 1

6 2

4

4

6

F2

4

3

2 2

6 0

8

F3

4 0

2

6 2

1

10

Demand

4 6 8 6




Q 17: [SA-22]

Solve the following transportation problem with the restriction that nothing can be transported from F2 to W3.

Store Factory


F1

4

6

6

1 15

F2

7

3

2

4

F3

3

9

2

4 11

Required

10 5 10 5


Q 18: [SA-36]

Solve the following transportation problem to maximize profit and give criteria for optimality:

Store Origin

Profit (₹)/Unit Destination Supplies

1 2 3 4 A 40 25 22 33 100 B 44 35 30 30 30 C 36 38 28 30 70 Demand

40 20 60 30

Q 19: [SA-37]

Solve the following transportation problem:

Stock Available 1 2 3 4 5

1 73 40 9 79 20 8 2 62 93 96 8 13 7 3 96 65 80 50 65 9 4 57 58 29 12 87 3 5 56 23 87 18 12 5 Demand

6 8 10 4 4

Q 20: [SA-27]

Consider the following transportation cost table. The costs are given in Rupees, the supply and demand are in

units. Determine the optimal solution:

Source

Destination Supply

1 2 3 4 5 I 40 36 26 38 30 160 II 38 38 26 34 198 280 III 36 38 34 24 30 240 Demand

160 160 200 120 240



Q 21: [SA-25]

Solve the following transportation cost table for minimum cost:

Destinations

Origins Requirements

A B C D I 7 4 3 4 15 II 3 2 7 5 25 III 4 4 3 7 20 IV 9 7 5 3 40 Availabilities

12 8 35 25

Q 22: [SA-23]

Solve the following transportation cost table for minimum cost:

Factory

Godowns Stock Available 1 2 3 4 5 6

I 7 5 7 7 5 3 60 II 9 11 6 11 - 5 20 III 11 10 6 2 2 8 90 IV 9 10 9 6 9 12 50 Demand

60 20 40 20 40 40

Note: It is not possible to transport any quantity from factor 2 to Godown 5.

Required: State whether the solution derived by you is unique.

Q 23: [V2-E2] [SA-32]

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 first two rows of the following table. The Transportation

costs from the factories to sales depots S1, S2 and S3 are given in the next three rows of the table. The

production capacity of each factory is given in the last row.

The last two columns in the table given the sales price and the total requirement at each depot:

Item Per unit Factory Sales price Per unit Requirement

F1 F2 F3 F4

Production cost 15 18 14 13 - -

Raw material cost 10 9 12 9 - -

3 9 5 4 34 80

Transportation cost 1 7 4 5 32 120

5 8 3 6 31 150

Production capacity 10 150 50 100 - -

Determine the most profitable production and distribution schedule and the corresponding profit. The surplus

should be taken to yield zero profit.

[Ans:Profit associated with the optimum Program is Rs. 480.]

Q 24: [SA-28] [N 90]

Stronghold Construction Company is interested in taking loans from banks for some of its projects P, Q, R, S and

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 in the following table. Assuming the role of a consultant, advise this company



as to how it should take the loans so that the total interest payable will be the least. Are there alternative

optimum solutions? If so, indicate one such solution.

Bank

Interest Rate in Percentage for Project Max. Credit (₹’000) P Q R S T

Pvt. Bank 40 36 26 38 30 Any Amount Nationalized 38 38 26 34 198 400 Co-Operative 36 38 34 24 30 250 Amount Required (₹’000)

160 160 200 120 240

Q 25: [SA-29]

A Manufacturer of jeans is interested in developing an advertising 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 Magazines are 40, 30 and 20

million respectively. Also the minimum desired exposures within each age group namely 13-18, 19-25, 26-35, 36

and older are 30, 25, 15 and 10 million. The objective is to minimum exposure level in each age group.

Media Age Group 13-18 19-25 26-35 36 &

Older TV 12 7 10 10 Radio 10 9 12 10 Magazines 14 12 9 12

Required:

(i) Formulate the above as a transportation problem, and find the optimal solution.

(ii) 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.



Practical Questions – Previous Examinations Paper Q: [M 11, 1d, 5M]

The following matrix is a minimization problem for transportation cost. The unit transportation costs are given

at the right hand corners of the cells and the ∆ij values are encircled.

D1 D2 D3 D4 Supply

Find the optimum solution (s) and the minimum cost.

[Ans:

(i) F1-D1-300 ₹ 900; F1-D2-100 ₹ 400; F1-D3-100 ₹ 400; F2-D2-300 ₹ 1800; F3-D3-200 ₹ 1000; Total Cost = ₹ 4500.

(ii) F1-D1-100 ₹ 300; F1-D2-100 ₹ 400; F1-D3-300 ₹ 1200; F2-D2-300 ₹ 1800; F3-D3-200 ₹ 800; Total Cost = ₹ 4500

Q: [N 10, 6a, 8M]

A company has three plants located at A, B and C. The production of these plants is absorbed by four distribution

centres located at X, Y, W and Z. the transportation cost per unit has been shown in small cells in the following

table:

Distribution Centre X Y W Z Supply

Factories (Units)

Find the optimum solution of the transportation problem by applying Vogel’s Approximation Method.

[Ans: Total Cost ₹ 129500]

F1

3

4

4

500

F2

9

8

6

300

7

2

300

F3 4

0

6

2

5

200

200

Demand 300 400 300 1000

A 6 9 13 7

6000

B 6 10 11 5

6000

C 4 7 14 8

6000

Demand 4000 4000 4500 5000

18000

17500



Q: [N 10*, 5a,10M]

A company has 3 factories F1, F2 and F3 which supply the same product to 5 agencies, A1, A2, A3, A4 and A5.

Unit production costs, shipping costs and selling prices differ among the different sources and destinations and

are given below:

F1 F2 F3

Production Cost (`/Unit) 28 35 29

Production Capacity (No. of units) 110 240 125

Agencies A1 A2 A3 A4 A5

Selling Price (`/Unit) 40 48 42 45 41

Demand (No. of units) 80 100 75 45 125

Shipping Costs/Unit

A1 A2 A3 A4 A5

F1 3 9 8 12 8

F2 6 10 6 2 5

F3 3 10 3 6 8

(i) Set up the initial transportation matrix for minimization.

(ii) After doing (i) above, you are given the following additional information:

(a) 40 units must be transported from F2 to A2 as per an earlier agreement made by F2 with A2’s customer. This quantity is included in the figures given for total production and demand at these locations.

(b) Not more than 30 units may be sent from F1 to A1, since the transporter’s vehicle lacks space in this route.

Incorporating conditions (a) and (b) above, obtain the initial solution by Vogel’s Approximation Method.

(Do not attempt to continue for the full and final solution.)

Q: [M 09, 4a, 6M] [V2-10][ Minimization-balanced-degeneracy]

The cost per unit of transporting goods from factories X, Y, Z to destinations A, B and C, and the quantities

demanded and supplied are tabulated below. As the company is working out the optimum logistics, the Govt. has

announced a fall in oil prices. The revised unit costs are exactly half the costs given in the table. You are required

to evaluate the minimum transportation cost.

Factories\Destinations A B C Supply

X 15 9 6 10

Y 21 12 6 10

Z 6 18 9 10

Demand 10 10 10 30

[Ans.: Min. transportation cost is ₹. 105]

Q: [M 08, 6a, 10M] [V2-8] [SA-27]

Goods manufactured at 3 plants, A, B and C are required to be transported to sales outlets X, Y and Z. The unit

costs of transporting the goods from the plants to the outlets are given below:

Sales outlets\Plants A B C Total Demand

X 3 9 6 20

Y 4 4 6 40

Z 8 3 5 60

Total supply 40 50 30 120



You are required to:

i. Compute the initial allocation by North-West Corner Rule. ii. Compute the initial allocation by Vogel’s approximation method and check whether it is optional.

iii. State your analysis on the optionality of allocation under North-West corner Rule and Vogel’s Approximation method.

[Ans: (i) A-X-20 ₹ 60; A-Y-20 ₹ 80; B-Y-20 ₹80; B-Z-30 ₹ 90; C-Z-30 ₹ 150; Total Cost = ₹ 460

(ii) A-X-20 ₹ 60; A-Y-20 ₹ 80; B-Z-50 ₹80; C-Y-20 ₹ 120; C-Z-10 ₹ 100; Total Cost = ₹ 460

(iii)The solution under VAM is optimal with a zero in R2C2 which means that the cell C2R2 which means that the cell C2R2 can come into solution, which will be another optimal solution. Under NWC rule the initial allocation had C2R2 and the total cost was the same Rs. 460 as the total cost under optimal VAM solution. Thus, in this problem, both methods have yielded the optimal solution under the 1st allocation. If we do an optimality test for the solution, we will get a zero for ∆ij in C3R2 indicating the other optimal solution which was obtained under VAM.]

Q: [M 07, 1b, 6M] [V2-7]

The initial allocation of a transportation problem, along with the unit cost of transportation from each origin to

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: Candidates may consider u1 = 0 at Row 1 for initial cell evaluation)

Requirement

[Ans: Minimum Transportation Cost ₹ 204]

Q: [M-03, 5c, 10M][ Maximization-unbalanced]

A company has three factories and four customers. The company furnishes the following schedule of profit per

unit on transportation of its goods to the customers in rupees:

Customers

Factory 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

11

8

2

8

6

6

4

2

18

10

9

9

12

9

6

10

7

6

8

3

7

7

8

2

9

3

5

2

6

11

4

Availability

12

8

8

8

4

40



You are required to solve the transportation problem to maximize the profit and determine the resultant of

optimal profit.

[Ans.: Maximum Profit ₹. 5130]

Q: [M-02, 3b, 10M] [V2 -1][ Minimization-Unbalanced]

A product is manufactured by four factories A, B, C & 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 stores P, Q, R, S. The demand made by these stores are 25, 35, 105 and 20 units respectively.

Unit transportation cost in rupees from each factory to each store is given in the following table:

Stores

Factories 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.

[Ans.: Total Cost ₹ 1465]

Q: [N 01, 3c, 10] [V2 -2] Multiple Optimal Solutions

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 &

requirement at different outlet 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.

[Ans.: Minimum costs ₹ 7350 thousands]

Q: [M 01, 3c, 10M]

A 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 cost matrix table:

Warehouses

Plants A B C Production

X 8 16 16 152

Y 32 48 32 164

Z 16 32 48 154

Demand 144 204 82

Determine a transportation schedule, so that the cost is minimized. Assume that the cost in the cost matrix is

given in thousand of rupees.

[Ans.: X-B-152 ₹ 2432; Y-B-42 ₹ 2016; Y-C-82 ₹ 2624; Y-dummy-40 ₹ 0; Z-A-144 ₹ 2304; Z-B-10 ₹ 320; Total Cost =

₹ 9696]



Q: [N 00, 3b, 10M] [V2-E4]

Following is the profit matrix based on four factories and three sales depots of the company:

S1 S2 S3 Availability

F1 6 6 1 10

Factories F2 -2 -2 -4 150

F3 3 2 2 50

F4 8 5 3 100

Requirement 80 120 150

Determine the most profitable distribution schedule and the corresponding profit, assuming no profit in case of

surplus production.

[Ans: Total Profit = ₹ 480]

Q: [M 00, 4b, 10M] [V2-E5]

A company produces a small component for all industrial products and distributes it to five wholesalers at a

fixed prices of Rs.2.50 per unit. Sales forecasts indicate that monthly deliveries will be 3,000, 3,000, 10,000,

5,000 and 4,000 units to wholesalers 1,2,3,4 and 5 respectively. The monthly production capabilities are 5,000,

10,000, 12,500 at plants 1, 2 and 3 respectively. The direct costs of production of each unit are Rs.1.00 and

Rs.0.80 at plants 1, 2 and 3 respectively. The transportation costs of shipping a unit from a plant to a wholesaler

are given below:

Wholesaler

1 2 3 4 5

1 0.05 0.07 0.10 0.15 0.15

Plant 2 0.08 0.06 0.09 0.12 0.14

3 0.10 0.09 0.08 0.10 0.15

Find how many components each plant supplies to each wholesaler in order to maximize profit.

[ Ans: Profit ₹ 32520]

Q: [N 99, 2c, 10M] [V2-E6]

The following table shows all the necessary information on the available supply to each warehouse, the

requirement of each market and the unit transportation cost from each warehouse to each market:

Market

I II III IV Supply

A 5 2 4 3 22

Warehouse B 4 8 1 6 15

C 4 6 7 5 8

Requirement 7 12 17 9

The shipping clerk has worked out the following schedule from his experience:

12 Units from A to II

1 Unit from A to III

9 Units from A to IV

15 Units from B to III

7 Units from C to I and

1 Unit from C to III

You are required to answer the following:

(i) Check and see if the clerk has the optimal schedule;

(ii) Find the optimal schedule and minimum total shipping cost; and



(iii) If the clerk is approached by a carrier of route C to II, who offers to reduce his rate in the hope of getting

some business, by how much should the rate be reduced before the clerk should consider giving him an order?

[Ans: (i) Clerk has not worked out the optimal schedule; (ii) Minimum total shipping cost ₹ 104; (iii) Total Shipping

Cost = Rs.103]

Q: [M 99, 3b, 10M] [V2-E7]

A company has three warehouses W1, W2 and W3. It is required to deliver a product from these warehouses to

three customers A, B and C. There warehouses have the following units in stock.

Warehouse: W1 W2 W3

No. of units: 65 42 43

and customer requirements are:

Customer: A B C

No. of units: 70 30 50

The table below shows the costs of transporting one unit from warehouse to the customer:

Warehouse

W1 W2 W3

A 5 7 8

Customer B 4 4 6

C 6 7 7

Find the optimal transportation route.

Answer Total Cost = Rs. 830

[Ans.: W1-A-65 ₹ 325; W2-A-5 ₹ 35; W2-B-30 ₹ 120; W2-C-7 ₹ 49; W3-C-43 ₹ 301; Total Cost = ₹ 680]

Q: [N 98, 4b, 10M] [SM-10] [SA-34]

ABC Enterprises is having three plants manufacturing dry-cells, located at different locations. Production cost

differs from plant to plant. There are five sales offices of the company located in different regions of the country.

The sales prices can differ from region to region. The shipping cost from each plant to each sales office and other

data are given by following table:

Production Data Table

Production cost per unit Max Capacity in no. of units Plant no.

20 150 1

22 200 2

18 125 3

Shipping Cost and Demand & Sales Prices table

Shipping Cost


Plant 1 1 1 5 9 4

Plant 2 9 7 8 3 6

Plant 3 4 5 3 2 7

Demand and Sales Price


Demand 80 100 75 45 125

Sales Price 30 32 31 34 29

Find the production and distribution schedule most profitable to the company.

[Ans.: 1-1-50 ₹ 450; 1-2-100 ₹ 1100; 2-4-25 ₹ 225; 2-5-125 ₹ 125; 2-dummy-50 ₹0; 3-1-30 ₹ 240; 3-3-75 ₹ 750; 3-

4-20 ₹ 280; Total Profit = ₹ 3170]



Q: [M 98, 5c, 10M] [SM-9] [V2-3] [SA-24]

Consider the following data for the transportation problem:

Factory Destination Supply to be exhausted

1 2 3

A 5 1 7 10

B 6 4 5 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 rupees 1,2 and 3 for destinations (1), (2) and (3) respectively. Find

the optimal allocation that minimizes the transportation and penalty cost.

[Ans.: A-2-10 ₹ 10; B-1-20 ₹ 120; B-2-10 ₹ 40; B-3-50 ₹ 300; C-1-15 ₹45; Dummy-1-40 ₹ 40; Total Cost = ₹ 555]

Q: [N 97, 2b, 10M] [SM-8] [SA-30]

A particular product is manufactured in factories A, B, C and D; and is sold at centres 1, 2, and 3. The cost in Rs.

of product per unit and capacity in kgs per unit time of each plant is given below:

Factory Cost (Rs.) per unit Capacity (kgs) per unit

A 12 100

B 15 20

C 11 60

D 13 80

The sale price in Rs. per unit and the demand in kgs per unit time are as follows:

Sales Centre Sale Price (Rs.) per unit Demand (kgs) per unit

1 15 120

2 14 140

3 16 60

Find the optimal sale distribution.

[Ans.: A-1-100 ₹ 300; B- 2-20 ₹ -20; C-3-60 ₹ 300; D-1-20 ₹ 40; D-1-60 ₹60; Dummy-2-60 ₹ 0;Total Profit = ₹ 680]

Q: [M 97,4b, 10M] [V2-E10][ Minimization-balanced-degeneracy]

A company has four 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, and D. The company dispatcher has come up with the costs between the terminals and plants as

follows:

Plants

Terminals A B C D

U 20 36 10 28

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.

[Ans: U-A ₹ 80; U- C₹ 60; V-B ₹ 60; V-D ₹ 20; W-B ₹210; X-D ₹ 125;Terminal Plant Cost = ₹555]



Q: [N 96, 2b, 10M] [SM-7] [V2-E9] [Investment decision]

XYZ and co. has provided the following data seeking your advice on optimum investment strategy:

Beginning of the year Net Return Data (in paise) of Selected Investments Amount

Available (Lacs)

P Q R S

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 are also provided:

P, Q, R, and S represent the selected investments. The company 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. The values (paise) in the table represent net return on investment of one Rupee till the end of the

planning horizon (for example, 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, determine the optimum investment strategy.

[Ans: Net Return ₹ 130.00 lacs]

Q: [M 96, 2b, 10M] [V2 –E8] [SA-33]

A company has four factories situated in four different locations in the country and four sales agencies located in

four other locations in the country. The cost of production (Rs. Per unit), the sales price (Rs. per unit), and

shipping cost (Rs. Per unit) in the case of matrix, monthly capacities and monthly requirements are given below:

Factory Sales Agency Monthly

Capacity

(Units)

Cost of

Production

1 2 3 4

A 7 5 6 4 10 10

B 3 5 4 2 15 15

C 4 6 4 5 20 16

D 8 7 6 5 15 15

Monthly

Requirement

8 12 18 22

Sale Price 20 22 25 18

Find the monthly production and distribution schedule which will maximize profit.

[Total Profit = ₹ 155]



Q: [May-95] [SA-31]

A leading firm has three auditors. Each auditor can work up to 160 hours during the next month, during which

time three projects must be completed. Project 1 will 130 hours, project 2 will take 140 hours, and project 3 will

take 160 hours. The amount per hour that can be billed for assigning each auditor to each project is given in

Table 1:

Table1

Project

Auditor 1 2 3

Rs. Rs. Rs.

1 1200 1500 1900

2 1400 1300 1200

3 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.

[Ans: 1-3 ₹ 304000; 2- 2 ₹ 143000; 3-1 ₹ 208000; 3-2 ₹ 4 2000; Total Billing= ₹ 697000]

Important Notes

h a c 11 transportation problems e r - amazon...

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