We have seen that an initial feasible solution to an m resources/origins and n destination problem consists of (m + n 1) basic variables which is the same as the number of occupied cells. However, if the number of occupied cells: is less than (m + n 1) at any stage of the solution, then the transportation problem is said to have a degenerate solution. Degeneracy as it is called can occur at two stages i.e., at the initial solution or during the testing of the optimal solution. Let us discuss both the cases.
DEGENERACY AT THE INITIAL SOLUTION STAGE
If degeneracy occurs at the initial solution stage, we introduce a very small quantity ε (Greek letter pronounced as epsilon) in one or more of the unoccupied cells to make the number of occupied cells equal to (m+n1). ε is so small a quantity that its introduction does not change the supply (sources) and demand (destinations) constraints or the rim conditions. E is placed in the unoccupied cell which has the least transportation cost and once e is allotted to it, it is supposed to have been occupied. ε stays in the solution till degeneracy is removed or the final solution is achieved. The value of ε is zero when used in the problems with movement of goods from one cell to another.
The use of ε and degeneracy can be explained with the help of an example.
Example 5.13. A forged parts manufacturing company forges the automobile parts of a particular passenger vehicle at its three plants P1, P2 and P3. The demand of these parts is from distributions located at five location DI, D2, D3, D4 and D5. Monthly capacity of the three plants is 100, 60, 80 respectively. Monthly requirement of the auto parts by the five distributors are 50, 50, 60, 40, 80 units respectively. Unit transport costs in Rs. is provided in the following table:
To
From 
D_{1} 
D_{2} 
D_{3} 
D_{4} 
D_{5} 
P1 P2 P3 
6 4 10 
10 7 4 
8 8 6 
8 8 8 
4 6 3 
REVIEW
 1. What is a transportation problem? How is it useful in business and industry?
 2. Explain the use of transportation problem is business and industry giving suitable examples.
 3. What do you understand by
(a) Feasible solution
(b) Northwest solution
(c) Vogel’s approximation method (YAM).
 4. Discuss various steps involved in finding initial feasible solution of a transportation problem.
 5. Discuss any two methods of solving a transportation problem. State the advantages and disadvantages of these methods.
 6. How can an unbalanced transportation problem be balanced? How do you interpret the optimal solution of an unbalanced transportation problem?
 7. Explain the differences and similarities between the MODI method and stepping stone method used for solving transportation problems.
 8. What is a transportation method. Explain its objectives. How can we used this model for solving a multiplesite facility beaten problem?
 9. Which method of solving transportation problems gives a more optimal solution? How will you know when you have achieve the least cost allocation of products between origins and destinations? Explain with examples.
 10. Formulate a cost minimisation model for the allocation of facilities to locations in problem.
ABC Ltd. is considering the layout of one of its plants divided into three different working areas. There, are three different production facilities and each one has one of them. Assume the data not available.
 11. An automobile manufacturing company has three factors say F_{1}, F_{2} and F_{3} which are feeding 5 different zones north, south, eastern, western and central. The monthly demands of these zones thousand are 25, 40, 30, 25, 20 thousand units respectively. The cost of transporting one unit from the factories F_{1}. F_{2} and F_{3} to each of the zones and factory capacity as shown in the table below:
Zone (Transportation costs)
Factory 
Capacity (units) 
Northern 
Southern 
Eastern 
Western 
Central 
F_{1} 
50,000 
8 
4 
6 
10 
12 
F_{2} 
40,000 
15 
10 
6 
8 
8 
F_{3} 
30,000 
12 
8 
8 
10 
10 
Perpetrate a transportation model and a LP model.
 1. XYZ Ltd. has three manufacturing plants P_{1}; P_{2} and P_{3} which are stepping the production to three warehouses W_{1}, W_{2} and W_{3}. The following data is available:
Plant 
Production (units) 
Warehouse 
Requirement (units) 
P_{1} 
150,000 
W_{1} 
160,000 
P_{2} 
120,000 
W_{2} 
130,000 
P_{3} 
130,000 
W_{3} 
80,00 
The rate of freight charges/unit is as shown below.
To
W_{1} 
W_{2} 
W_{2} 

P_{1} 
1.50 
1.60 
1.80 

From 
P_{2} 
2.0 
1.80 
2.50 
P_{3} 
1.60 
1.40 
3.00 
Determine the initial basic feasible solution using northwest comer method and YAM.
 1. Plant location of a firm manufacturing a single product has three plants located at A, Band C. Their production during week has been 60, 40 and 50 units respectively. The company has firm commitment orders for 25, 40, 20, 20 and 30 units of the product to customers C1, C2, C3, C4 and C.5
respectively. Unit cost of transporting from the three plants to the five customers is given in the table below:
C1 
C2 
C3 
C4 
C5 

Plant location 
A 
6 
1 
3 
4 
6 
B 
4 
4 
3 
3 
2 

C 
2 
6 
4 
4 
6 
Use VAM to determine the cost of shipping the product from plant locations to the customers.
 2. Solve the following transportation problem. Availability at each plant, requirements at each warehouse and the cost matrix is as shown below:
Warehouse 

W_{1} 
W_{2} 
W_{3} 
W_{4} 
Availability 

P_{1} 
200 
400 
600 
200 
80 

Plant 
P_{2} 
800 
400 
400 
500 
100 
P_{3} 
400 
200 
500 
400 
190 

Requirement 
60 
80 
80 
120 
 3. There are four supply points P1, P2, P3 and P4 5/ destination A, B, C, D and E. The following table gives in cost of transportation of materials from supply points to demand statements in rupees.
To 

A 
B 
C 
D 
E 

Plant 
P1 
10 
12 
15 
16 
18 
P2 
12 
10 
12 
10 
10 

P3 
15 
20 
6 
12 
16 

P4 
12 
18 
10 
12 
12 
The present allocation is as follows
P1 to A 100, P 1 to B 20, P 2 to B160, P3 to B10, P3 to C60, P3 to E120, P4 to 0200, P4 to E100. Find an optimal solution for allocations. If we reduce the cost from any supply point to any destination, what do you think will be the impact. Select any case and discuss the outcome.
 1. Anand Lamps India the (AUL) operates three manufacturing plants and four warehouse. Capacity of the plants and forecast demand of warehouses is as follows :
Factory at 
Capacity (tons) 
Warehouse at 
Forecast demand (tons) 
P1 
16 
A 
6 
P2 
8 
B 
7 
P3 
8 
C 
6 
D 
12 
The transportation cost per ton in Rs. is as given below.
From/To 
A 
B 
C 
D 
P1 
100 
120 
80 
20 
P2 
120 
200 
60 
60 
P3 
160 
200 
100 
80 
ALIL wishes to minimise its transport costs.
 2. The recruitment policy of the personnel department of a company is such that three types of workers managers, supervisors and workers are required 20, 40, 150 from four placement service centres A, B, C, D and E. The hourly rates in Rs. are given by the matrix below:
Placement service centre 

A 
B 
C 
D 
Requirement 

Managers 
16 
10 
8 
12 
20 

Category 
Supervisors 
20 
24 
30 
20 
40 
Workers 
200 
180 
60 
160 
150 

Availability 
50 
60 
200 
80 
390 
Determine the requirement pattern at the lowest cost.
 3. Find the initial basic feasible solution to the following transportation problem by
(a) Least cost method.
(b) Northwest corner rule.
State which of the methods is better
Table
TO 
Supply 

2 
7 
4 
5 

From 
3 
3 
1 
8 
5 
4 
7 
7 

1 
6 
2 
14 

Demand 
7 
9 
18 
 1. Solve the following transportation problem by VAM.
Consumers
A 
B 
C 
Available 

Suppliers I 
6 
7 
4 
14 
II 
4 
3 
1 
12 
III 
1 
4 
7 
5 
Required 
6 
10 
15 
31 
Use V AM to find an initial BFS.
 1. The apex company is the distributor for television receivers. It owns three warehouses with stocking capacity as follows.
Warehouse location 
Sets in stock 
A 
100 
B 
25 
C 
75 
It has the following order for set deliveries:
Market location 
Orders 
X 
80 
Y 
30 
Z 
90 
Delivery costs for warehouses to each customers are largely a function of mileage or distance. The per unit cost have been determined to be:
X 
Y 
Z 

A B C 
5 3 5 
10 7 8 
2 5 4 
The deliveries could be made in many ways but the distributor would like to deliver the T.V. sets in a way that would minimize the delivery cost. Give the distribution schedule. Use VAM only.
 2. A company has four factories situated in different locations and five warehouses in different cities. The matrix of transportation cost is given below.
Factories
Warehouse 
I 
II 
III 
IV 
Requiremet 
A 
4 
8 
7 
6 
150 
B 
9 
5 
8 
8 
50 
C 
6 
5 
8 
7 
40 
D 
5 
8 
6 
3 
60 
E 
7 
6 
5 
8 
200 
Capacity (units) 
100 
80 
120 
100 
Find the optimum transportation schedule using VAM only.
 1. Obtain the initial BFS to the following transportation problem by VAM.
D1 
D2 
D3 
D4 
D5 
ai 

A1 
5 
7 
10 
5 
3 
5 
A2 
8 
6 
9 
12 
14 
10 
A3 
10 
9 
8 
10 
15 
10 
Bj 
3 
3 
10 
5 
4 
25 
 2. A steel company has three furnaces and five rolling mills. Transportation cost (rupees per quintal) for sending steel from furnaces to rolling mills are given in the following table :
Furnaces 
M1 
M2 
M3 
M4 
M5 
Availability (Q) 
A B C 
4 5 6 
2 4 5 
3 5 4 
2 2 7 
6 1 3 
8 12 14 
Requirement (Quintal) 
4 
4 
10 
8 
8 
How should they meet the requirement? Use YAM.
 3. A cement factory manager is considering the best way to transport cement from his three and manufacturing centres P, Q and R to depot A, B, C, D and E. The weekly production and demand along with transportation costs per ton are given below:

A 
B 
C 
D 
E 
Tons 
P 
4 
1 
3 
4 
4 
60 
Q 
2 
3 
2 
4 
3 
35 
R 
3 
5 
2 
2 
4 
40 
Requirement (Quintal) 
22 
45 
20 
18 
30 
135 
What should be the distribution programme?
 4. Solve the following problem and test its optimality
Project A 
Project B 
Project C 
Plant 

Pant W 
4 
8 
8 
56 
Pant X 
16 
24 
16 
82 
Pant Y 
8 
16 
24 
77 
Project requirement 
72 
92 
41 
215 205 
 5. The relevant data on demand, supply and profit per unit of a product manufactured and sold by a company are given below:
Factory 
1 
2 
3 
4 
5 
Supply 
P 
5 
8 
14 
7 
8 
100 
Q 
2 
6 
7 
8 
7 
20 
R 
3 
4 
5 
9 
8 
60 
4 
10 
7 
8 
6 
20 

Demand 
45 
65 
70 
35 
15 
135 
Given the transportation from A to 3, and D to 2 are not allowed due to certain reasons, Find out using VAM optimal method of transportation from factories to marketing centers.
 1. Solve the following problem in which cell entries represent unit costs:
D1 
D2 
D3 
Available 

Q1 
2 
7 
4 
5 
Q2 
3 
3 
1 
8 
Q3 
5 
4 
7 
7 
Q4 
1 
6 
2 
14 
Required 
7 
9 
18 
44 
Apply MODI method to test optimality,
 2. Solve the following transportation problem
To From 
D1 
D2 
D3 
D4 
Available 
S1 
4 
3 
1 
2 
80 
S2 
5 
2 
3 
4 
60 
S3 
3 
5 
6 
3 
40 
Requirement 
50 
60 
20 
50 
180 Total 
Apply MODI method to test its optimality.
 3. Solve the following transportation problem.
1 
2 
3 
4 
5 
6 
Stock available 

F 
1 
7 
5 
7 
7 
5 
3 
60 
A 

C 
2 
9 
11 
6 
11 
– 
5 
20 
T 

O 
3 
11 
10 
6 
2 
2 
8 
90 
R 

Y 
4 
9 
10 
9 
6 
9 
12 
50 
Demand 
60 
20 
40 
20 
40 
40 
 4. 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 or 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 millions respectively. Also the minimum desired exposures with each age group, namely 1318, 1925, 2635, 36 and older, are 30, 25, 15 and 10 millions. The objective is to minimize the cost of attaining the minimum exposure level in each age group.
Media 
Age groups 

1318 
1925 
2635 
36 an 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 optimal solution.
(b) Solve the problem if the policy is to provide at least 4 million exposures through TV in the age groups and at least 8 million exposures through TV in the age group 1925.
 1. STRONGHOLD Construction Company is interested in taking loans from banks for some of 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 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 may be the least. Are there alternative optimum solutions? If so, indicate one such solution.
Bank 
Interest rate in percentage for project 
Max. credit (in thousands) 

P 
Q 
R 
S 
T 

Private Bank 
20 
18 
18 
17 
17 
Any amount 400 250 
Nationalized Bank 
16 
16 
16 
16 
16 

Cooperative Bank 
15 
15 
15 
14 
14 

Amount required (in thousands) 
200 
150 
200 
75 
75 
 2. A company has four terminals u, v, w and x. At the start of 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, S, C and D. The company dispatcher has come up with the costs between the terminals and plants as follows:
 1. The cost conscious company requires for the next month 300, 260 and 180 tonnes of stone chips for its three constructions, C_{1}, C_{2 }and C_{3} respectively. Stone chips are produced by the company at three, mineral fields taken on short lease. All the available boulders must be curshed into chips. Any excess
; chips over the demands at sites C_{1}, C_{2} and C_{3} will be sold exfields. The fields M1, M2 and M3 will yield 250, 320 and 280 tonnes chips respectively. Transportation costs from mineral fields to construction sites vary according to distances, which are given below in monetary unit (MU).
To From 
C1  C2 
C3 
M1 M2 M3 
85
7 
74
5 
69
5 
(a) Determine the optimal economic transportation plan for the company and the overall transportation cost in MU.
(b) What are the quantities to be sold from M1, M2 and M3 respectively?
 2. A company has 4 different factories in 4 different locations in the country and four sales agencies in four other locations in the country. The cost of productions, the sale price, shipping cost in the cell of matrix, monthly capacities and monthly requirements are given below:
Factory 
1 
2 
3 
4 
Capacity 
Cost of production 
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 

Selling price 
20 
22 
25 
18 
Find the monthly production and distribution schedule, which will maximize profits.
 1. A company wishes to determine an investment strategy for each of the next four years. Five investment types have been selected, investment capital 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 summarises 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 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 0.90.
Investment made at the beginning of year 
Investment type 
Rupees available in (000’s) 

A 
B 
C 
D 
E 

NET RETURN DATA 

1 
0.90 
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 rupees investment (in 000’s) 
750 
600 
500 
1,000 
The object 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 fouryear period.
Solve the above transportation problem and get an optimal solution. Also calculate the net return on investment for the planning horizon of fouryear period.
 1.
Investment made 
Net return rate 
Available 

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 
4 
60 
40 
40 
30 
30 
Maximum Investment (in lacs) 
40 
50 
60 
60 
Solve the given problem so as to give maximum return.
 2. Following is the profit matrix based on four factories and three sales depots of the company.
Sales depots
S1 
S2 
S3 
S4 

F1 
6 
6 
1 
10 
Factories F2 
2 
2 
4 
150 
F3 
3 
2 
2 
50 
F4 
8 
5 
3 
100 
Requirements 
80 
120 
150 
Determine the most profitable distribution schedule and the corresponding profit, assuming no profit in case of surplus production.
 1. Determine the optimal solution to the problem given below and find the minimum cost of transportation.
To From 
E 
F 
G 
H 
I 
ai 
A 
4 
7 
3 
8 
2 
4 
B 
1 
4 
7 
3 
8 
7 
C 
7 
2 
4 
7 
7 
9 
D 
4 
8 
2 
4 
2 
2 
bi 
8 
3 
7 
2 
2 
 2. A transport corporation has trucks at 3 garages A, Band C in a city. The number of trucks available in each garage, the number of trucks required by each customer and the distance (Km) from garage to customer’s locations are given below:
1 
2 
3 
4 
Availability 

Garages 

A 
7 
6 
9 
12 
12 
B 
8 
6 
7 
10 
8 
C 
10 
7 
8 
12 
10 
Requirements 
4 
3 
5 
8 
Just before sending the trucks, it is known that road from B to customers 2 is closed for traffic due to road block. How should the trucks be assigned to the customer is order to minimize the total distance to be covered to the road block.
 3. 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 take 130 hours.
 4. Determine an initial basic feasible solution to the following transportation problem using northcorner rule:
To 
Available 

From 
3 
4 
6 
8 
9 
20 
2 
10 
1 
5 
8 
30 

7 
11 
20 
40 
3 
15 

2 
1 
9 
14 
16 
13 

Demand 
40 
6 
8 
18 
6 
 5. Determine an initial basic feasible solution to the following transportation problem using row minima method:
To 
Available 

From 
5 
2 
4 
2 
22 
4 
8 
1 
6 
15 

4 
6 
7 
5 
8 

Demand 
7 
12 
17 
9 
 6. Find the initial basic feasible solution of the following transportation problem by approximate method :
W_{1} 
W_{2} 
W_{3} 
W_{4} 
Capacity 

F_{1} 
19 
30 
50 
10 
7 

Factory 
F_{2} 
70 
30 
40 
60 
9 
F_{3} 
40 
8 
70 
20 
18 

Requirement 
5 
8 
7 
14 
34 (Total) 
 7. Determine an initial basic feasible solution to the following L.P. using:
(a) Northwest comer rule,
(b) Vogel’s approximation method.
A_{1} 
B_{1} 
C_{1} 
D_{1} 
E_{1} 
Supply 

A 
2 
11 
10 
3 
7 
4 

Origin 
B 
1 
4 
7 
2 
1 
8 
C 
3 
9 
4 
8 
12 
9 

Demand 
3 
3 
4 
5 
6 
 8. Solve the transportation problem for which the cost, origin availabilities and destination requirements are given below:
D_{1} 
D_{2} 
D_{3} 
D_{4} 
D_{5} 
D_{6} 
a_{1} 

O_{1} 
1 
2 
1 
4 
5 
2 
30 
O_{2} 
3 
3 
2 
1 
4 
3 
50 
O_{3} 
4 
2 
5 
9 
6 
2 
75 
O_{4} 
3 
1 
7 
3 
4 
6 
20 
b_{1} 
20 
40 
30 
10 
50 
25 
175 (Total) 
 9. Give a mathematical formulation of the transportation and simplex methods. What are the differences in the nature of problems that can be solved by these methods.
 1. Given below is the unit costs array with supplies a_{i} = 1,2,3 and demands b_{1j}= 1,2,3 and 4.
1 
2 
3 
4 
a_{1} 

1 
8 
10 
7 
6 
50 

Source 
2 
12 
9 
4 
7 
40 
3 
9 
11 
10 
8 
30 

b_{1} 
25 
32 
40 
23 
120 (Total) 
Find the optimal solution to the above Hitchcock problem.
 2. Consider four bases of operations B and three tartets T. The tons of bombs per aircraft from any base that can be delivered to any target are given in the following table:
1 
2 
3 

1 
8 
6 
5 

Base (B) 
2 
6 
6 
6 
3 
10 
8 
4 

4 
8 
6 
4 
The daily sortie capability of each of the four bases is ISO sorties per day. The daily requirement in sorties over each individual target is 200. Find the allocation of sorties from each base to each target which maximizes the total tonnage over all the three targets explaining each step.
 3. General Electrodes is a big electrode manufacturing company. It has two factories and three main distribution centers in three cities. The supply and demand transportation. How should the trips be scheduled so that the cost of transportation is minimized?
The present cost of transportation is around Rs. 3,100 per month. What can be the maximum savings by proper scheduling?
Centers  A  B  C 
Requirement  50  50  150 
Cost per trip from X plant  25  35  10 
Cost per trip from Y plant  20  5  80 
Capacity of plant X  15 units of electrodes 
 1. A company has decided to manufacture some or all of five new products at three of its plants. production capacity of each of these three plants is as follows:
Plant No.  Production capacity in total number of units 
1  40 
2  60 
3  90 
Sales potential of the five products is as follows:
Product No.  1  2  3  4  5 
Market potential in units  30  40  70  40  60 
Plant No. cannot produce product No.5. The variable cost per unit for the respective plant and product combination is given below:
Product No.  1  2  3  4  5 
Plant No.1  20  19  14  21  16 
Plant No.2  15  20  13  9  16 
Plant No.3  18  15  18  20  … 
Based on above data, determine the optimum product to plant combination by programming.
 1. A fertilizer company has three plants A, Band C which supply to six major distribution centers 1, 2, 3, 4, 5 and 6. The table below gives the transportation costs per case, the plant annual capacities, and predicted annual demands at different centers in terms of thousands of cases. The variable production costs per case Rs. 8.50, Rs. 9.40 and Rs. 7 respectively at plants A, Band C. Determine the minimum cost production and transportation allocation.
Transportation cost Rs. per case
Major distribution centers
1 
2 
3 
4 
5 
6 
Annual production in thousands of cases 

A 
2.50 
3.50 
5.50 
4.50 
1.50 
4.00 
2,200 

Plants 
B 
4.60 
3.60 
2.60 
5.10 
3.10 
4.10 
3,400 
C 
5.30 
4.30 
4.80 
2.30 
3.30 
2.80 
1,800 

Annual demand in thousand of cases 
850 
750 
420 
580 
1,020920 
Prove that if the variable production costs are the same at every plant, once can obtain an optimal allocation by using transportations costs only.
 2. Describe the transportation problem. Give method of finding an initial feasible solution. Explain what is meant by an optimality test. Give the method of improving over the initial solution to reach the optimal feasible solution.
 3. The unit costs of transportation from site i to site j are given below. At site i= 1, 2, 3 stocks of 15O, 200 and 170 units respectively are available. 300 units are to be sent to site 4 and the rest to site 5. Find the cheapest way of doing this.
To
1 
2 
3 
4 
5 

1 
– 
3 
4 
10 
7 

2 
1 
– 
2 
16 
6 

From 
3 
7 
4 
– 
12 
13 
4 
8 
3 
9 
– 
5 

5 
2 
1 
7 
5 
– 
(Hint. In accordance with the restriction of supply and demand, table 3.142reduces to the following table:
To
4 
5 
Available 

1 
10 
7 
150 

From 
2 
16 
6 
200 
3 
12 
13 
170 

Required 
300 
220 
 1. Consider the following unbalanced problem:
To From 
1 
2 
3 
Supply 
1 
5 
1 
7 
10 
2 
6 
4 
6 
80 
3 
3 
2 
5 
15 
Demand 
75 
20 
50 
Since there is not enough supply, some of the demands at these destinations may not be satisfied. Suppose that there are penalty costs for every unsatisfied demand unit which are given by 5, 3 and 2 for destinations I, 2 and 3 respectively. Find the optimal solution.
[Hints. The balanced transportation table with dummy source and associated penalty costs in shown below.
To From 
1 
2 
3 
Supply 
1 
5 
1 
7 
10 
2 
6 
4 
6 
80 
3 
3 
2 
5 
15 
Dummy 
5 
3 
2 
40 
Demand 
75 
20 
50 
This table can now be solved by the usual MODI method.]
 2. A production control superintendent finds the following information on his desk. In departments A, B and C the number of surplus pallets is 18, 27 and 21 respectively. In departments G, H, I and J number of pallets required is 14, 12, 23 and 17 respectively. The time in minutes to move a pallet from
one department to another is given below.
To From 
G 
H 
I 
J 
A 
13 
25 
12 
21 
B 
18 
23 
14 
9 
C 
23 
15 
12 
16 
What is the optimal distribution plan to minimize the moving time?
 1. The following table gives the cost of transporting material from supply points A, B, C and D to demand points E, F, G, H and J.
To From 
E 
F 
G 
H 
J 
A 
8 
10 
12 
17 
15 
B 
15 
13 
18 
11 

C 
14 
20 
6 
10 
13 
D 
13 
19 
7 
6 
12 
The present allocation is as follows.
A to E 90, A to F 10, C to F 10, C to G 50, C to J 120, D to H 210, D to 170.
(a) Check if this allocation is optimum. If not, find an optimum schedule.
(b) If in the above problem the transportation cost from A to G is reduced to 10, What will be the new optimum schedule?
 2. 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 in rupees 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 working out the following schedule from 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 to 1 unit from C to III.
(a) Check and see if the clerk has made the optimal schedule.
(b) Find the optimum schedule and minimum total shipping cost.
(c) 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 must the rate be reduced before the clerk should consider giving him an order?
 3. A company has factories at A, Band C which supply warehouses at D, E, F and G. Monthly factory capacities are 250, 300 and 400 units respectively for regular production. If overtime production is utilised, factories A and B can produce 50 and 75 additional units respectively at overtime incremental costs of Rs. 4 and Rs. 5 respectively. The current warehouse requirements are 200, 225, 275 and 300 units respectively. Unit transportation costs in rupees from factories to warehouses are as follows:
To From 
D 
E 
F 
G 
A 
11 
13 
17 
14 
B 
16 
18 
14 
10 
C 
21 
24 
13 
10 
Determine the optimum distribution for this company to minimize costs.
[Hint. First table is made which takes into account the overtime production and the corresponding production costs.
D 
E 
F 
G 
Supply 

A 
11 15 
13 17 
17 21 
14 18 
250 50 
B 
16 21 
18 23 
14 19 
10 15 
300 75 
C 
21 
24 
13 
10 
400 
Demand 
200 
225 
275 
300 
In the above table, total supply = 1,075 units
total demand =1,000 units
Therefore, we add a dummy warehouse with demand of 75 units and cost coefficients zero in each of its cell.
D 
E 
F 
G 
Dummy 
Supply 

A 
11 15 
13 17 
17 21 
14 18 
0 0 
250 50 
B 
16 21 
18 23 
14 19 
10 15 
0 0 
300 75 
C 
21 
24 
13 
10 
0 
400 
Demand 
200 
225 
275 
300 
75 
The initial feasible solution can now be obtained and can be optimized using MODI method].
 1. A company has plants at A, B and C which have capacities to produce 300 kg, 200 kg and 500 kg respectively of a particular chemical per day. The production costs per kg in these plants are Re. 0.70, Re. 0.60 and Re. 66 respectively. Four buk consumers have placed orders for the product on the following basis:
kg required per day 
Price offered Rs. 1 kg 

I 
400 
1.00 

II 
250 
1.00 

Customer 
III 
350 
1.02 
IV 
150 
1.03 
Shipping costs (in paise per kg) from plants to consumers are given in the table below.
I 
II 
III 
IV 

A 
3 
5 
4 
6 

From 
B 
8 
11 
9 
12 
C 
4 
6 
2 
8 
Work out an optimal schedule for the above situation. Under what conditions would you change the schedule?
 1. ABC manufacturing company wishes to develop a monthly production schedule for the next months. Depending upon the sales commitments, the company can either keep the production constant, allowing fluctuations in inventory or inventories can be maintained at a constant level, with fluctuating production. Fluctuating production in working overtime, the cost of which is estimated to be double the normal production cost of Rs. 12 per unit. Fluctuating inventories result in inventory carrying cost of Rs. per unit. If the company fails to fulfill its sales commitment, it incurs a shortage cost of Rs. 4 per unit per month. The production capacities for the next three months are shown below.
Production capacity
Month 
Regular 
Overtime 
Sales 
1 
50 
30 
60 
2 
50 
0 
120 
3 
60 
50 
40 
Determine the optimal production schedule.
(Hint. Here regular and overtime production capacity is the source and the sales is the destination. The costs for different cells may be computed as follows:
(a) For items produced and sold in the same month, there will be no inventory carrying cost. Thus the costs for cells (1, 1), (2, 2), (3, 3) are Rs. 12 each and for cells (1, 1), (3, 3) are Rs. 24 each.
(b) For items produced in a particular month and sold in subsequent months, additional inventory cost of Rs, 2 per month will be incurred. Thus cells (1, 2), (1, 3), (2, 3), (1, 2) and (1, 3) will have costs of Rs. 14, Rs. 16, Rs. 14, Rs. 26 and Rs. 28 respectively.
(c) For items produced in a particular month to meet the backlog of sales during previous months, In addition to the production costs (normal or overtime), shortage costs of Rs. 4 per month will be incurred. Therefore, for cells (2, 1), (3, 2), (3,1), (3_{1}, 2) and (3_{1}, 1) the costs will be Rs. 16, Rs. 16, Rs. 20, Rs. 28 and Rs. 32 respectively. Thus the equivalent transportation table for the given problem will be one shown below:
1 
2 
3 
Dummy 
Production capacity 

1 
12 
14 
16 
0 
50 

2 
16 
12 
14 
0 
50 

Month 
3 
20 
16 
12 
0 
60 
1_{1} 
24 
26 
28 
0 
30 

3_{1} 
32 
28 
24 
0 
50 

Sales 
60 
120 
40 
20 
Review
 2. (i) Initial basic feasible solution
(ii) Northwest corner rule
(iii) Unbalanced transportation problem.
 3. (i) Initial basic feasible solution
(ii) Transportation problem
(iii) Northwest corner rule.
 4. (i) Degenerate solution
(ii) Prohibited routes in transportation problem,
 5. (i) Balanced transportation problem.
(ii) Initial Basic Feasible Solution of a transportation problem.
 6. Describe transportation problem with its general mathematical formation.
 7. Explain the various steps involved in solving a transportation problem by applying the N. West Corner Method.
 8. Describe transportation problem with its general mathematical formulation.
 9. Explain various steps in Vogel’s approximation method for finding initial
feasible solution of the transportation problem.  10. Explain various steps involved in solving a transportation problem by anyone of the Transportation Problem
 11. Write a brief note on Vogel’s approximation method to solving problem.
 12. Explain the various steps involved in solving a transportation problem by applying the North West Corner Method.
 13. Describe sequences of steps in MODI Method of solving a transportation problem.
 14. Explain the transportation problem giving examples.
 15. Explain
(a) NWCM,
(b) LCEM,
(c) VAM and Test of optimality by
(i) stepping stone method and
(ii) MODI method. Take suitable examples.
 1. Find the initial basic feasible solution of the following transportation problem with the help of Northwest corner method.
A 
B 
C 

X 
11 
21 
16 
14 

Plant 
Y 
07 
17 
13 
26 
Available at plant 
Z 
11 
23 
21 
36 

18 
28 
28 
Market requirement
 1. Find the initial basic feasible solution of the following transportation problem using least cost method:
W_{1} 
W_{2} 
W_{3} 
W_{4} 
Factory Capacity 

F_{1} 
30 
25 
40 
20 
100 
F_{2} 
29 
26 
35 
40 
250 
F_{3} 
31 
33 
37 
30 
150 
Warehouse requirement 
90 
160 
200 
50 
 2. A construction company needs 3, 3, 4 and 5 million cubic feet of fill at 4 dam sites. It can transfer the fill from three mounds A, Band C where 2, 6 and 7 million cubic feet of fill is available. Costs (in lakhs of Rs.) of transporting one million cubic feet of fill from the mounds to 4 dam sites are:
From/To 
I 
II 
III 
IV 
A 
15 
10 
17 
18 
B 
16 
13 
12 
13 
C 
12 
17 
20 
11 
Determine the optimum distribution for this company to minimize the total cost.
 3. Suggest an optimal transportation plan with a view to minimize cost from form the following information.
Source Destination 

Cost of shipping per unit 
Unit Demand 

F_{1} 
F_{2} 
F_{3} 

W_{1} 
Rs. 
0.9 
1 
1 
5 
W_{2} 
Rs. 
1 
1.4 
0.8 
20 
W_{3} 
Rs. 
1.3 
1 
0.8 
20 
Units Available 
20 
15 
10 
45 
 1. A distribution system has the following constraints:
Factory 
Capacity (units) 
Warehouse 
Demand (units) 
A 
45 
I 
25 
B 
15 
II 
55 
C 
40 
III 
20 
The transportation cost per unit (Rs.) associated with each route is as follows:
From 
I 
II 
III 
A 
10 
7 
8 
B 
15 
12 
9 
C 
7 
8 
12 
Find the optimum transportation schedule and the minimum total cost of transportation.
 2. Given the following information, compute optimal transportation cost using any method.
Project Requirement per week, Truck Available Per week.
Loads 
Plants 

A 
45 
X 
35 
B 
50 
Y 
40 
C 
20 
Z 
40 
Cost information 
To Project 

From 
A 
B 
C 
Plant X 
5 
10 
10 
Plant Y 
20 
30 
20 
Plant Z 
5 
8 
12 
 3. Determine the optimum solution to the following problem:
Cost Matrix
To


T_{1} 
T_{2} 
T_{3} 
T_{4} 
Availability 
F_{1} 
10 
20 
5 
7 
10 

F_{2} 
13 
9 
12 
8 
20 

From 
F_{3} 
4 
15 
7 
9 
30 
F_{4} 
14 
7 
9 
10 
40 

F_{5} 
3 
12 
6 
19 
40 

Demand 
60 
60 
20 
10 
 1. Solve the following transportation problem as
(a) maximization problem and
(b) minimization problem

D_{1} 
D_{2} 
D_{3} 
D_{4} 
D_{5} 
D_{6} 
Available 
Q_{1} Q_{2} Q_{3} Q_{4} 
2 3 3 4 
1 2 5 2 
3 2 4 2 
3 4 2 1 
2 3 4 2 
5 4 1 2 
60 40 60 30 
Required 
30 
50 
20 
40 
30 
10 
 2. Solve the following transportation cost problem:

A 
B 
C 
D 
E 
F 
Available 
Q R S T 
5 5 2 6 
3 6 1 10 
7 12 2 9 
3 5 4 5 
8 7 8 10 
5 11 2 9 
3 4 2 8 
Required 
3 
3 
6 
2 
1 
2 
 3. Find the initial basic feasible solution by at least three different methods for the following transportation problem:
From/To 
D_{1} 
D_{2} 
D_{3} 
D_{4} 
Available 
F_{1} F_{2} F_{3} 
10 1 7 
7 6 4 
3 7 5 
6 3 6 
3 5 7 
Demand 
3 
2 
6 
4 

 4. Solve the following cost minimising transportation problem.

Warehouses 
Capacity 

Factory 
95 115 195 
105 180 180 
80 40 95 
15 30 70 
12 7 1 
Demand 
5 
4 
4 
11 
 5. Describe transportation problem with its general mathematical formulation.
 6. Explain various steps in Vogel’s approximation method for finding initial basic feasible solution of the transportation problem.
 7. Explain various steps involved in solving a transportation problem by anyone of the method to solve it.
 8. Write a brief note on Vogel’s approximation method to solving transportation
problem.  9. Explain the various steps involved in solving a transportation problem by applying the North West Comer Method.
 10. Describe sequence of steps in MODI Method of solving a transportation problem.
 11. Discuss various methods of getting basic feasible solution of Transportation
problem. Which one would you prefer and why?  12. Describe transportation problem with its general mathematical formulation.
 13. Explain the various steps involved in solving a transportation problem by applying the North West Comer method.
 1. Find the initial basic feasible solution of the following transportation problem with the help of NorthWest Comer Method.
Market
A 
B 
C 

X 
11 
21 
16 
14 

Plant 
Y 
07 
17 
13 
26 
Available at plant 
Z 
11 
23 
21 
36 

18 
28 
25 

Market requirement 
 2. Find the initial basic feasible solution of the following transportation problem using least cost method;
W_{1} 
W_{2} 
W_{3} 
W_{4} 
Factory capacity 

F_{1} 
30 
25 
40 
20 
100 
F_{2} 
29 
26 
35 
40 
250 
F_{3} 
31 
33 
37 
30 
150 
Warehouse requirement 
90 
160 
200 
50 
 3. Suggest an optimal transportation plan with view to minimize following information.
Cost of shipping per unit
Source Destination 
F_{1} 
F_{2} 
F_{3} 
Units Demanded 
W_{1} W_{2} W_{3} 
0.9 1 1.3 
1 1.4 1 
1 0.8 0.8 
5 20 20 
Units Available 
20 
15 
10 
45 
 1. A construction company needs 3, 3, 4 and 5 million cubic feet of fill at 4 dam sites. It can transfer the fill from three mounds A, B and C where 2, 6 and 7 million cubic feet of fill is available. Costs (in lakhs of Rs.) of transporting one million cubic of fill from the mounds to 4 dam sites are:
From/To 
I 
II 
III 
IV 
A 
15 
10 
17 
18 
B 
16 
13 
12 
13 
C 
12 
17 
20 
11 
Determine the optimum distribution for this company to minimize the total cost.
 2. A distribution system has the following constraints:
Factory 
Capacity (units) 
Warehouse 
Demand (units) 
B B C 
45 15 40 
I II III 
25 55 20 
The transportation cost per unit (Rs.) associated with each route are as follows:
From 
I 
II 
III 
A B C 
10 15 7 
7 12 8 
8 9 12 
Find the optimum transportation schedule and the minimum total cost of
transportation.
 3. Given the following information, compute optimal transportation cost using any method.
Project 
Requirement per week, truck loads 
Plants 
Available per week, truck loads 
A B C 
45 50 20 
X Y Z 
35 40 40 
Cost information From 
To project 

A 
B 
C 

Plant X Plant Y Plant Z 
5 20 5 
10 30 8 
10 20 12 
 4. Solve the following transportation problem:
S_{1} 
S_{2} 
S_{3} 
S_{4} 

D_{1} D_{2} D_{3} D_{4} 
5 10 12 5 
7 12 10 7 
8 15 7 6 
9 18 9 9 
125 125 125 125 
25 
15 
30 
10 
 5. Solve the following transportation problem. The cost matrix is given below:
Requirement 
A B C 
Source 

1 
2 
3 
4 

7 
10 
12 
10 
40 

9 
12 
10 
10 
30 

12 
9 
14 
12 
20 

25 
15 
30 
10 
 6. Find the optimum cost of transportation for the following problem:
W 
X 
Y 
Z 
Availability 

I II III IV 
30 40 40 50 
30 50 40 20 
30 30 40 30 
60 50 60 70 
200 400 300 200 
Requirement 
350 
450 
200 
100 
 104. Find the optimum sol. to the following transportation problem in which the cells contain the transportation cost in rupees:

W_{1} 
W_{2} 
W_{3} 
W_{4} 
W_{5} 
Available 
F_{1} F_{2} F_{3} F_{4} 
7 8 6 5 
6 5 8 7 
4 6 9 7 
5 7 6 8 
9 8 5 6 
40 30 20 10 
Required 
30 
30 
15 
20 
5 
100 
 105. The table below gives the information regarding the quantity required by 4 markets and supply capacity of 3 warehouses. The unit transportation cost from warehouse market is also given below. Find the optimum allocation that minimizes the total shipping cost.
Market
Warehouse 
1 
2 
3 
4 
Supply 
A B C Requirement 
5 4 4 7 
2 8 6 12 
4 1 7 17 
3 6 5 9 
22 15 8 
 106. Find the basic feasible sol. of the following transportation problem by VAM. Also find the optimal transportation plan

1 
2 
3 
4 
5 
Available 
A B C D 
4 5 3 2 
3 2 5 4 
1 3 6 4 
2 4 3 5 
6 5 2 3 
80 60 40 20 
Required 
60 
60 
30 
40 
10 
200 (Total) 
 107. Solve the following transportation problem
To
From 
A 
B 
C 
Available 
P1 P2 P3 
4 16 8 
8 24 16 
8 16 24 
56 82 77 
Required 
72 
102 
41 

 108. (23) Solve the following transportation problem for maximum profit
Market
Warehouse 
A 
B 
C 
D 
X Y Z 
12 8 14 
18 7 3 
6 10 11 
25 18 20 
Availability at Warehouse  Demand in Markets 
X : 200 Units
Y : 500 Units Z : 300 Units 
A : 180 Units
B : 320 Units C : 100 Units D : 400 Units 
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