1. A company that manufactures four products P1, P2, P3, and P4.

Each of these products

requires processing in three treatment plants T1, T2, and T3. The time required for
processing (in hours) of each unit of products P1, P2, P3, and P4 in the plants T1, T2,
and T3 is given in Table. Each of the treatment plants T1, T2, and T3 are available for
100 hours per week. The profit obtained by selling the products P1, P2, P3, and P4 is
50, 100, 30, and 27, respectively. Whereas there is enough demand for products P1,
P3, and P4, the company forecasts a maximum demand of 3 units for product P2. How
many units of the products P1, P2, P3, and P4 should be produced to maximize the
demand?

2. A company which is planning to advertise its newly launched product in two TV

channels, TV1 and TV2. The company has set aside a budget of 1,000,000 GMD16 for
this TV advertising campaign. An advertisement costs 100,000 GMD and 200,000
GMD per minute in channels TV1 and TV2 respectively. TV1 and TV2 have informed
that advertisement slots can last a maximum of 3 and 4 minutes respectively. The
company has found that there is a linear relationship between the time of advertising
and the number of viewers of the advertisements. The company has estimated that the
viewership is 2 and 3 million per minute of advertisements on TV1 and TV2
respectively. Thus if the advertisement is shown for 3 minutes, the viewership would
be 6 and 9 million for TV1 and TV2 respectively. The company wishes to determine
the number of minutes that the advertisement should be aired on TV1 and TV2 such
that the number of viewers is maximized.
3. A company which is planning to advertise its newly launched product in two TV
channels, TV1 and TV2. Here, the company has not set aside a budget for this TV
advertising. However it wishes to minimize its cost of advertising. An advertisement
costs 100,000 GMD and 200,000 GMD per minute in channels TV1 and TV2
respectively. TV1 and TV2 have informed that advertisement slots can last a
maximum of 3 and 4 minutes respectively. The company has found that there is a
linear relationship between the time of advertising and the number of viewers of the
advertisements. The company has estimated that the viewership is 2 and 3 million per
minute of advertisements on TV1 and TV2 respectively. Thus, if the advertisement is
shown for 3 minutes, the viewership would be 6 and 9 million for TV1 and TV2
respectively. The company wishes to determine the number of minutes that the
advertisement will be aired on TV1 and TV2 such that the number of viewers is more
than 10 million.

4. The problem of preparing a dog’s daily diet by mixing appropriate quantities of raw
meat, brown rice, and vegetables to meet the diet requirements at the minimum cost. A
80 pound dog requires a minimum of 2000 calories per day along with a diet in which
the protein content should range from 1–40% of diet by weight, carbohydrate content
should range from 5–55% of diet by weight, and vegetable content should range from
1.5–25% of diet content by weight. Raw meat contains 15% protein by weight and 60
calories per ounce. Brown rice contains 25% carbohydrates by weight and 30 calories
per ounce. Vegetables contain 10 calories per ounce. The cost of raw meat, brown rice,
and vegetables is 12, 80, and 150 per ounce, respectively. Let us choose decision
variables x1, x2 , and x3 as the weight (in ounces) of raw meat, brown rice, and
vegetables used for preparing the daily diet.
5. A firm manufactures 2 types of products A & B and sells them at a profit or Rs. 2 on
type A & Rs 3 on type B. Each product is processed on 2 machines G & H. Type a
requires 1 minute of processing time on G and 2 minutes on H. Type B requires one
minute on G & 1 minute on H. The machine G is available for not more than 6 hrs. 40
mins., while machine H is available for 10 hrs. during any working day. Formulate the
problem as LPP.
 6. A company makes two products (X and Y) using two machines (A and B). Each unit
of X that is produced requires 50 minutes processing time on machine A and 30
minutes processing time on machine B. Each unit of Y that is produced requires 24
minutes processing time on machine A and 33 minutes processing time on machine B.

At the start of the current week there are 30 units of X and 90 units of Y in stock. Available
processing time on machine A is forecast to be 40 hours and on machine B is forecast to be
35 hours.

The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95
units. Company policy is to maximise the combined sum of the units of X and the units of Y
in stock at the end of the week.

 Formulate the problem of deciding how much of each product to make in the current
week as a linear program.

7. A company is involved in the production of two items (X and Y). The resources need
to produce X and Y are twofold, namely machine time for automatic processing and
craftsman time for hand finishing. The table below gives the number of minutes
required for each item:

Craftsman time
Machine time

Item X 13 20

Item Y 19 29

The company has 40 hours of machine time available in the next working week but only 35
hours of craftsman time. Machine time is costed at £10 per hour worked and craftsman time
is costed at £2 per hour worked. Both machine and craftsman idle times incur no costs. The
revenue received for each item produced (all production is sold) is £20 for X and £30 for Y.
The company has a specific contract to produce 10 items of X per week for a particular
customer.

 Formulate the problem of deciding how much to produce per week as a linear
program.
 8. A company manufactures two products (A and B) and the profit per unit sold is £3 and
£5 respectively. Each product has to be assembled on a particular machine, each unit
of product A taking 12 minutes of assembly time and each unit of product B 25
minutes of assembly time. The company estimates that the machine used for assembly
has an effective working week of only 30 hours (due to maintenance/breakdown).

Technological constraints mean that for every five units of product A produced at least two
units of product B must be produced.

 Formulate the problem of how much of each product to produce as a linear program.
 The company has been offered the chance to hire an extra machine, thereby doubling
the effective assembly time available. What is the maximum amount you would be
prepared to pay (per week) for the hire of this machine and why?
9. A carpenter makes tables and chairs. Each table can be sold for a profit of £30 and
each chair for a profit of £10. The carpenter can afford to spend up to 40 hours per
week working and takes six hours to make a table and three hours to make a chair.
Customer demand requires that he makes at least three times as many chairs as tables.
Tables take up four times as much storage space as chairs and there is room for at most
four tables each week.
10. A company manufactures and sells two models of lamps, L1 and L2. To manufacture
each lamp, the manual work involved in model L1 is 20 minutes and for L2, 30
minutes. The mechanical (machine) work involved for L1 is 20 minutes and for L2, 10
minutes. The manual work available per month is 100 hours and the machine is limited
to only 80 hours per month. Knowing that the profit per unit is $15 and $10 for L1 and
L2, respectively, determine the quantities of each lamp that should be manufactured to
obtain the maximum benefit.
11. With the start of school approaching, a store is planning on having a sale on school
materials. They have 600 notebooks, 500 folders and 400 pens in stock, and they plan
on packing it in two different forms. In the first package, there will be 2 notebooks, 1
folder and 2 pens, and in the second one, 3 notebooks, 1 folder and 1 pen. The price of
each package will be $6.50 and $7.00 respectively. How many packages should they
put together of each type to obtain the maximum benefit?
12. On a chicken farm, the poultry is given a healthy diet to gain weight. The chickens
have to consume a minimum of 15 units of Substance A and another 15 units of
 Substance B. In the market there are only two classes of compounds: Type X, with a
composition of one unit of A to five units of B, and another type, Y, with a
composition of five units of A to one of B. The price of Type X is $10 and Type Y,
$30. What are the quantities of each type of compound that have to be purchased to
cover the needs of the diet with a minimal cost?
13. There is only 600 milograms of a certain drug that is needed to make both large and
small pills for small scale pharmaceutical distribution. The large tablets weigh 40
milograms and the small ones, 30 milograms. Consumer research determines that at
least twice the amount of the smaller tablets are needed than the large ones and there
needs to be least three large tablets made. Each large tablet is sold for a profit of $2
and the small tablet, $1. How many tablets of each type have to be prepared to obtain
the maximum profit?
14. The management of xyz corporation is currently faced with the problem of
determining its product mix for the coming period. Since, the corporation is one of the
few suppliers of transformers for laser cover units, only liberal sales ceilings are
anticipated. The corporation should not plan on selling transformers more than 200
units of A type, 100 units of B type and 180 units of C type. Contracts call for
production of at least 20 units of A type and 70 units of C type. Within these bounds,
management is free to establish units production schedules. These are subject to the
capacity of the plant to produce without overtime. The production times prevail.

Formulate this as an LPP so as to maximize the total profit.

15.
16. A company makes two products (say, P and Q) using two machines (say, A and B).
Each unit of P that is produced requires 50 minutes processing time on machine A and
30 minutes processing time on machine B. Each unit of Q that is produced requires 24
minutes processing time on machine A and 33 minutes processing time on machine B.
Machine A is going to be available for 40 hours and machine B is available for 35
hours. The profit per unit of P is $25 and the profit per unit of Q is $30. Company
policy is to determine the production quantity of each product in such a way as to
maximize the total profit given that the available resources should not be exceeded.
17. An operations manager is trying to determine a production plan for the next week.
There are three products (say, P, Q, and Q) to produce using four machines (say, A and
B, C, and D). Each of the four machines performs a unique process. There is one
machine of each type, and each machine is available for 2400 minutes per week. The
unit processing times for each machine is given in Table.
Machine Data

The unit revenues and maximum sales for the week are indicated in Table 2. Storage
from one week to the next is not permitted. The operating expenses associated with
the plant are $6000 per week, regardless of how many components and products are
made. The $6000 includes all expenses except for material costs.
Product Data

18. A company manufactures four variants of the same product and in the final part of the
manufacturing process there are assembly, polishing and packing operations. For each
variant the time required for these operations is shown below (in minutes) as is the
profit per unit sold.
 Assembly Polish Pack Profit (£)

Variant 1 2 3 2 1.50

Variant 2 4 2 3 2.50

Variant 3 3 2 3 3.00

Variant 4 7 4 5 4.50

Given the current state of the labour force the company estimate that, each year, they have
100000 minutes of assembly time, 50000 minutes of polishing time and 60000 minutes of
packing time available. How many of each variant should the company make per year and
what is the associated profit?

 Suppose now that the company is free to decide how much time to devote to each of
the three operations (assembly, polishing and packing) within the total allowable time
of 210000 (= 100000 + 50000 + 60000) minutes. How many of each variant should
the company make per year and what is the associated profit?

19. A company is involved in the production of two items (X and Y). The resources need
to produce X and Y are twofold, namely machine time for automatic processing and
craftsman time for hand finishing. The table below gives the number of minutes
required for each item:

Machine time Craftsman time

Item X 13 20

Item Y 19 29

The company has 40 hours of machine time available in the next working week but
only 35 hours of craftsman time. Machine time is costed at £10 per hour worked and
craftsman time is costed at £2 per hour worked. Both machine and craftsman idle times
incur no costs. The revenue received for each item produced (all production is sold) is
£20 for X and £30 for Y. The company has a specific contract to produce 10 items of
X per week for a particular customer.
 Formulate the problem of deciding how much to produce per week as a linear
program.
20. A canning company operates two canning plants. The growers are willing to supply
fresh fruits in the following amounts:

 S1: 200 tonnes at £11/tonne

 S2: 310 tonnes at £10/tonne
 S3: 420 tonnes at £9/tonne

Shipping costs in £ per tonne are:

To: Plant A Plant B

From: S1 3 3.5
S2 2 2.5
S3 6 4

Plant capacities and labour costs are:

Plant A Plant B
Capacity 460 tonnes 560 tonnes
Labour cost £26/tonne £21/tonne

The canned fruits are sold at £50/tonne to the distributors. The company can sell at this price
all they can produce.

The objective is to find the best mixture of the quantities supplied by the three growers to the
two plants so that the company maximises its profits.

21. Maçka Police Station employs 30 police officers. Each officer works for 5 days per
week. The crime rate fluctuates with the day of week, so the number of the police
officers required each day depends on which day of the week it is: Monday, 18;
Tuesday, 24; Wednesday, 25; Thursday, 16; Friday, 21; Saturday, 28; Sunday, 18. The
Police Station wants to schedule police officers to minimize the number whose days
off are not consecutive. Formulate an LP that will accomplish this goal.

22. Consider the following problem.

 Max 3x1 + 2x2

s.t. 3x1 + x2 ≤ 12

x1 + x2 ≤ 6

5x1 + 3x2 ≤ 27

x1≥ 0, x2 ≥ 0.

23. Consider the following LP model:

24. To maintain the water at a minimum level of softness and to meet a minimum in health
protection, experts have decided that sixty (60) and forty (40) pounds of the two
chemicals that na'se up each product must be added to the water daily. At a cost of $3
and $6 per unit, respectively, for Chemico's and American Chemical's products, what
is the optimal quantity of each product that should be used to meet the minimum level
of softness and a minimum health standard? Set up the problem and solve it.

25. A company makes three products and has available 4 workstations. The production
time (in minutes) per unit produced varies from workstation to workstation (due to
different manning levels) as shown below:

Workstation
1 2 3 4
Product 1 5 7 4 10
2 6 12 8 15
3 13 14 9 17

Similarly the profit (£) contribution (contribution to fixed costs) per unit varies from
workstation to workstation as below:
 Workstation
1 2 3 4
Product 1 10 8 6 9
2 18 20 15 17
3 15 16 13 17

If, one week, there are 35 working hours available at each workstation how much of each
product should be produced given that we need at least 100 units of product 1, 150 units of
product 2 and 100 units of product 3. Formulate this problem as an LP.

26. (Diet problem): A dietician wishes to mix two types of foods in such a way that
vitamin contents of the mixture contain atleast 8 units of vitamin A and 10 units of
vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food
‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg
to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as
a linear programming problem to minimise the cost of such a mixture
27. An advertising company wishes to plan advertising campaign in 3 different media,
television, radio and magazine. The purpose of advertising is to reach as many
potential consumers as possible. Results of a marketing study are given below:
Television Particulars Prime Day Prime Time Radio Magazines Cost of an Advertising
Unit (Rs.) 40,000 75,000 30,000 15,000 No. of Potential Customers Reached per Unit
4,00,000 9,00,000 5,00,000 2,00,000 No. of Women Customers Reached per Unit
3,00,000 4,00,000 2,00,000 1,00,000 The company doesn’t want to spend more than `
8,00,000 on advertising. It further requires that: a. At least 2 million exposures take
place among women. b. Advertising on television be limited to Rs. 5,00,000. c. At
least 3 advertising units can be bought on prime day and 2 units prime time. d. The
number of advertising units on radio and magazine should each be between 5 and 10.
Formulate the Linear Programming model in order to maximize the total number of
potential customers reached.
28. Dieticians tell us that a balanced diet must contain certain quantities of nutrients such
as proteins, minerals, vitamins, etc. Suppose that you are asked to find out the food
that should be recommended from a large number of alternative sources of these
nutrients, so, that the total cost of food satisfying minimum requirements of balanced
diet is the lowest. The medical experts and the dieticians tell us that it is necessary for
 an adult to consume at least 75 gms. of proteins, 85 gms. of fats and 300 gms. of
carbohydrates daily. The following table gives the different items (which are readily
available in the market); Item analysis and their respective costs.

29. A rubber company is engaged in producing 3 different kinds of tyres A, B and C.

These three different tyres are produced at the company’s 2 different plants with
different production capacities. In a normal 8 hrs working day plant 1 produces 50,
100 and 100 tyres of A, B and C respectively. Plant 2 produce 60, 60 and 200 tyres of
type A, B and C respectively. The monthly demand for tyre A, B and C is 2,500, 3,000
and 7,000 units respectively. The daily cost of operation of plant 1 and 2 is Rs. 2,500
and Rs. 3,500 respectively. Find the minimum number of days of operation per month
at 2 different plants to minimize the total costs while meeting the demand.
30. A firm that makes products x and y has a total production capacity of 9 tonnes per day,
x and y requiring the same production capacity. The firm has a permanent contract to
supply at least 2 tonnes of x and 3 tonnes of y per day to another company. Each one
of x requires 20 machine hrs. Production time and y requires 50 machine hrs
production time. The daily maximum possible number of machine hours available is
360. All the firm’s output can be sold, and the profit set is ` 80 per tonne of x and ` 120
per tonne of y. You are required to determine the production schedule to maximize the
firm’s profit.
31. A firm manufacturers headache pills in two sizes A and B. Size A contains I grains of
aspirin, 5 grains of bicarbonate and 1 grain of codeine. It is found by uses that it
requires at least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine
for providing immediate effect. It is required to determine the least number of pills a
patient should take to get immediate relief. Formulate the problem as a standard LPP.
32. Consider a small plant which makes 2 types of automobile parts say A and B. It buys
castings that are machined, bored and polished. The capacity of machining is 25 per
hour for A and 40 hours for B, capacity of boring is 28 per hours for A and 35 per hour
 for B, and the capacity of polishing is 35 per hour A and 25 hour of B. Casting for port
A costs Rs. 2 each and for part B they cost Rs. 3 each. They sell for Rs. 5 and Rs. 6
respectively. The three machines have running costs of Rs. 20, Rs. 14 and Rs. 17.50
per hour. Assuming that any combination of parts A and B can be sold, what product
mix maximizes profit?
33. A ship is to carry 3 types of liquid cargo – X, Y and Z. There are 3,000 litres of X
available, 2,000 litres of Y available and 1,500 litres of Z available. Each litre of X, Y
and Z sold fetches a profit of Rs. 30, Rs. 35 and Rs. 40 respectively. The ship has 3
cargo holds-A, B and C of capacities 2,000, 2,500 and 3,000 litres respectively. From
stability considerations, it is required that each hold be filled in the some proportion.
Formulate the problem of loading the ship as a linear programming problem. State
clearly what are the decision variables and constraints.
34. A company produces two types of pens, say A & B. Pen A is superior in quality while
pen B is of lower quality. Net profits on pen A and B are Rs. 5 and Rs. 3 respectively.
Raw material required for pen A is twice as that of pen B. The supply of raw material
is sufficient only for 1,000 pens of B per day. Pen A requires a special nib and only
400 such nibs are available in a day. For pen B, only 700 nibs are available in a day.
Find the daily product mix so that the company can make maximum profits.
35. G.L Breweries Ltd. has two bottling plants located at Pune and Bangalore. Each plant
produces three drinks; whisky, beer and brandy. The number of bottles produced in a
day are as follows – whisky 1,500, beer 3,000 and brandy 2,000 at Pune and Whisky
1,500 beer 1,000 and brandy 5,000 at Bangalore. A market survey indicates that during
November, there will be demand for 20,000 bottles of whisky, 40,000 bottles of beer
and 44,000 bottles of brandy. The operating cost per day for plants at Pune and
Bangalore are Rs. 600 and Rs. 400 respectively. For how many days each plant be run
in November so as to meet the demand at minimum cost?
36. Allocation problem) A cooperative society of farmers has 50 hectare of land to grow
two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs.
10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used
for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than
800 litres of herbicide should be used in order to protect fish and wild life using a pond
which collects drainage from this land. How much land should be allocated to each
crop so as to maximise the total profit of the society?
37. Let us assume that there is a leading global automotive supplier company named JIM.
JIM has it’s production plants in many countries and supplies products to all the top
automotive makers in the world. For instance, let’s consider that there are three plants
in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The
plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per
day. The cost of transport per unit per km in INR and the distance between each source
and destination in Kms are given in the tables below.

The objective is to determine the unknown while satisfying all the supply and demand
restrictions. The cost of shipping from a source to a destination is directly proportional to
the number of units shipped.

38. The ICARE Company has three plants located throughout a state with production
capacity 50, 75 and 25 gallons. Each day the firm must furnish its four retail shops R1,
R2, R3, & R4 with at least 20, 20, 50, and 60 gallons respectively. The transportation
costs (in Rs.) are given below. Company Retail Supply R1 R2 R3 R4 P1 3 5 7 6 50 P2
2 5 8 2 75 P3 3 6 9 2 25 Demand 20 20 50 60. The economic problem is to distribute
 the available product to different retail shops in such a way so that the total
transportation cost is minimum?
39. Consider the transportation problem presented in the following table:

40. Obtain an Initial BFS to the following Transportation problem

41. Find the optimum transportation schedule and minimum total cost of transportation.

42. Find the optimum Transportation schedule and minimum total cost of Transportation.

43. Solve the assignment problem:

44. Solve the transportation problem

 45. Solve the following transportation problem:

46. XYZ Inc. has two factories with weekly production rates of 40 and 20 widgets. These
widgets must be shipped to three warehouses that have a demand of 25, 10, and 25
units. The cost to ship between each location is known (see the grid below). Which
factory should produce and ship how many widgets to which warehouses to meet the
demand at each location with minimal cost?

The standard table format of this problem is:

($cost) Warehouse 1 Warehouse 2 Warehouse 3 Supply

Factory A 550 300 400 40

Factory B 350 300 100 20

Demand 25 10 25
The central values in the table represent the point-to-point per unit shipping cost, such that
it costs $550 to ship from Factory A to Warehouse 1.

47. For simplicity, generic locations will be used and the cars are assumed to be fungible
(any car can be substituted for any other car). There is also a storage cost to keep an
unwanted car at an origin location which must be accounted for. The base problem
given is: Sites A, B, and C have 8, 12, and 10 cars on site, while Destinations X, Y,
and Z require 9, 7, and 11 cars respectively. The storage cost for each site is 100, 100,
and 80. The cost to move the cars between sites is shown in the table:
 S
u
($cost) Dest X Dest Y Dest Z p
p
ly

Site A 460 350 640 8

1
Site B 510 420 690
2

1
Site C 650 680 490
0

Demand 9 7 11
48. To optimize the initial basic feasible solution.

49. To optimize the initial basic feasible solution.

 50. A cement factory manager is considering the best way to transport cement for his three
manufacturing centres P, q, R to depots A, B , C, D and E. The weekly production and
demands along with transportation costs per tonne are given below:

Supply
(Tonnes
A B C D E )
Manufacturing P 4 1 3 4 4 60
Q 2 3 2 2 3 35
Centre R 3 5 2 4 4 40
Demand (Tonnes) 22 45 20 18 30 135
What should be the distribution programme?
51. A Company has received a contract to supply gravel for three new construction
projects located in towns A, B, C. Construction engineers have estimated the required
amounts of gravel which will be needed at these construction projects.

Weekly Requirement (truck

Project Location loads)
A 144
B 204
C 82
The Company has gravel pits located in towns W, X, Y respectively. The gravel
required by the construction projects can be supplied by these three plants. The amount
of gravel which can be supplied by each pit is as follows:

Pit W X Y
Amount
Available (Truck
Loads) 152 164 154
The company has computed the deliver cost from each pit to each project site. These
costs (in rupees) are shown in the adjoining table:

Pit A B C
x 8 16 16
Y 32 48 32
Z 16 32 48
52. A manufacturing company has three plants X, Y, Z, which supply to the distributors
located at, B, C, D and E. Monthly plant capacities are 80, 50 and 90 units
respectively. Monthly requirements of distributors are 40, 40, 50, 40 and 80units
respectively. Unit transportation costs are given below in rupees:
 To
From A B C D E
X 5 8 6 6 3
Y 4 7 7 6 6
Z 8 4 6 6 3
Determine an optimal distribution for the company in order to minimize the total
transportation cost.
53. Consider the following transhipment problem involving 4 sources and 2 destinations.
The supply value of the sources S1, S2, S3 and S4 are 200 units, 250 units, 300 units
and 450 units respectively. The demand value for destinations D1 and D2 are 600
units and 600 units respectively. The transportation cost per unit between different
sources and destinations are summarized in the following table.
Solve the transhipment problem.
53. Find an optimum solution to the problem:

54. Two reservoirs are available to supply the water needs of three cities. Each reservoir
can supply up to 50 million gallons of water per day. Each city would like to receive
40 million gallons per day. For each million gallons per day of unmet demand, there is
a penalty. At city 1, the penalty is $20; at city 2, the penalty is $22; and at city 3, the
penalty is $23. The cost of transporting 1 million gallons of water from each reservoir
to each city is shown in Table. Formulate a balanced transportation problem that can
be used to minimize the sum of shortage and transport costs.

55. Powerco has three electric power plants that supply the needs of four cities.† Each
power plant can supply the following numbers of kilowatt-hours (kwh) of electricity:
plant 1— 35 million; plant 2—50 million; plant 3—40 million (see Table). The peak
power demands in these cities, which occur at the same time (2 P.M.), are as follows
(in kwh): city 1—45 million; city 2—20 million; city 3—30 million; city 4—30
million. The costs of sending 1 million kwh of electricity from plant to city depend on
 the distance the electricity must travel. Formulate an LP to minimize the cost of
meeting each city’s peak power demand.

56. Sailco Corporation must determine how many sailboats should be produced during
each of the next four quarters (one quarter is three months). Demand is as follows: first
quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth
quarter, 25 sailboats. Sailco must meet demand on time. At the beginning of the first
quarter, Sailco has an inventory of 10 sailboats. At the beginning of each quarter,
Sailco must decide how many sailboats should be produced during the current quarter.
For simplicity, we assume that sailboats manufactured during a quarter can be used to
meet demand for the current quarter. During each quarter, Sailco can produce up to 40
sailboats at a cost of $400 per sailboat. By having employees work overtime during a
quarter, Sailco can produce additional sailboats at a cost of $450 per sailboat. At the
end of each quarter (after production has occurred and the current quarter’s demand
has been satisfied), a carrying or holding cost of $20 per sailboat is incurred.
Formulate a balanced transportation problem to minimize the sum of production and
inventory costs during the next four quarters.
57. Machineco has four machines and four jobs to be completed. Each machine must be
assigned to complete one job. The time required to set up each machine for completing
each job is shown in Table. Machineco wants to minimize the total setup time needed
to complete the four jobs. Use linear programming to solve this problem.

58. General Ford produces cars at L.A. and Detroit and has a warehouse in Atlanta; the
company supplies cars to customers in Houston and Tampa. The cost of shipping a car
between points is given in Table (“—” means that a shipment is not allowed). L.A. can
produce as many as 1,100 cars, and Detroit can produce as many as 2,900 cars.
 Houston must receive 2,400 cars, and Tampa must receive 1,500 cars. a Formulate a
balanced transportation problem that can be used to minimize the shipping costs
incurred in meeting demands at Houston and Tampa. b Modify the answer to part (a) if
shipments between L.A. and Detroit are not allowed.

59. Sunco Oil produces oil at two wells. Well 1 can produce as many as 150,000 barrels
per day, and well 2 can produce as many as 200,000 barrels per day. It is possible to
ship oil directly from the wells to Sunco’s customers in Los Angeles and New York.
Alternatively, Sunco could transport oil to the ports of Mobile and Galveston and then
ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels
per day, and New York requires 140,000 barrels per day. The costs of shipping 1,000
barrels between two points are shown in Table. Formulate a transshipment model (and
equivalent transportation model) that could be used to minimize the transport costs in
meeting the oil demands of Los Angeles and New York.

60. General Ford has two plants, two warehouses, and three customers. The locations of
these are as follows: Plants: Detroit and Atlanta Warehouses: Denver and New York
Customers: Los Angeles, Chicago, and Philadelphia Cars are produced at plants, then
shipped to warehouses, and finally shipped to customers. Detroit can produce 150 cars
per week, and Atlanta can produce 100 cars per week. Los Angeles requires 80 cars
per week; Chicago, 70; and Philadelphia, 60. It costs $10,000 to produce a car at each
plant, and the cost of shipping a car between two cities is given in Table. Determine
how to meet General Ford’s weekly demands at minimum cost.
61. Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per
week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped
to three customers. The profit earned per tube depends on the site where the tube was
produced and on the customer who purchases the tube (see Table). Customer 1 is
willing to purchase as many as 80 tubes per week; customer 2, as many as 90; and
customer 3, as many as 100. Televco wants to find a shipping and production plan that
will maximize profits. a Formulate a balanced transportation problem that can be used
to maximize Televco’s profits.

62. A company must meet the following demands for a product: January, 30 units;
February, 30 units; March, 20 units. Demand may be backlogged at a cost of
$5/unit/month. All demand must be met by the end of March. Thus, if 1 unit of
January demand is met during March, a backlogging cost of 5(2) $10 is incurred.
Monthly production capacity and unit production cost during each month are given in
Table. A holding cost of $20/unit is assessed on the inventory at the end of each
month. Formulate a balanced transportation problem that could be used to determine
how to minimize the total cost (including backlogging, holding, and production costs)
of meeting demand. b Find a basic feasible solution.
63. Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce
500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per
day. Oil is sent from the fields to a refinery, either in Dallas or in Houston (assume that
each refinery has unlimited capacity). It costs $700 to refine 100,000 barrels of oil at
Dallas and $900 at Houston. Refined oil is shipped to customers in Chicago and New
York. Chicago customers require 400,000 barrels per day of refined oil; New York
customers require 300,000. The costs of shipping 100,000 barrels of oil (refined or
unrefined) between cities are given in Table. Formulate a balanced transportation
model of this situation

64. A firm producing a single product has three plants and four customers. The three
plants will produce 3,000, 5,000, and 5,000 units, respectively, during the next time
period. The firm has made a commitment to sell 4,000 units to customer 1, 3,000 units
to customer 2, and at least 3,000 units to customer 3. Both customers 3 and 4 also want
to buy as many of the remaining units as possible. The profit associated with shipping
a unit from plant i to customer j is given in Table. Formulate a balanced transportation
problem that can be used to maximize the company’s profit.

65. Check whether the given transportation problem shown in Table is a balanced one. If
not, convert the unbalanced problem into a balanced transportation problem.
Transportation model with supply exceeding Demand
66. Find out the initial feasible solution for transportation cost involved in the problem
shown through Table

67. Find the initial feasible solution of the transportation problem illustrated through Table

68. Find the initial basic feasible solution for the transportation problem given in
following table.

69. Solve the transportation problem

70. Find the initial transportation cost for the transportation matrix

71. Consider the following transportation problem (cost in rupees). Find the optimum
solution
72. Find the minimum cost for the following problem:

73. Solve the following assignment problem and find the minimum cost.

74. A departmental head has 4 subordinates and 4 tasks are to be performed. Subordinates
differ in efficiency and tasks differ in their intrinsic difficulty. Time each man would
take to perform each task is given in the effective matrix. How the tasks should be
allocated to each person so as to minimize the total man hours?

75. In a textile sales emporium, 4 sales girls Arpitha (A1), Archana (A2), Aradhana (A3)
and Aakansha (A4) are available to 4 sales counters M, N, O and P. Each sales girl can
handle any counter. The service of each sales counter [in hours] when carried out by
each sales girl is given below:

How to allocate the appropriate sales counters to sales girls so as to minimize the
service time?
76. A project consists of 4 major jobs for which 4 contractors have submitted their tenders.
The tender amount quoted in lakhs of rupees are given in the matrix below:
 Find the assignment which minimizes the total project cost [each contractor has to be
assigned at least one job].
77. A firm plans to begin production of three new products. They own three plants and
wish to assign one new plant. The unit cost of producing i at plant j is Cij as given by
the following matrix. Find the assignment that minimizes the total unit cost.

78. A company has 4 machines on which do 3 jobs. Each job can be assigned to one and
only one machine. The cost of each job on each machine is given in the following
table:

Determine the optimum assignment.

79. A machine tool company decides to make 4 sub-assemblies through four contractors.
Each contractor is to receive only one sub-assembly. The cost of each sub-assembly is
determined by the bids submitted by each contractor and is shown in table below (in
'000 `). Assign different assemblies to contractors so as minimize the total cost:

80. Find the minimum cost for the following problem:

81. Find the minimum cost for the following problem:

 82. Solve and find the maximum total expected sale.

83. A student has to select one and only one elective in each semester and the some
elective should not be selected in different semesters. Due to various reasons, the
expected grades in each subject, if selected in different semester, vary and they are
given below.

The grade points are; H = 10, A = 9, B = 8, C = 7, D = 6, E = 5 and F = 4. How will the

student select the electives in order to maximize the total expected point's and what will be
his maximum expected total points?

84. Find the minimum cost for the following problem:

85. A company has 4 machines to do 3 jobs. Each job can be assigned to 1 and only 1
machine. The cost of each job on a machine is given to the following table:

What are the job assignments which will minimize the cost?
86. Find the assignment of machines to jobs that will result in a maximum profit and
which is the job that should be declined from the following. The owner of a small
machine shop has 4 machines available to assign 2 jobs for the day, 5 jobs are offered
with the expected profit (`) per machinist on each job being as follows.

87. Solve the following travelling salesman problem with the following matrix.

88. A travelling sales man has to visit 5 cities. He wishes to start form a particular city
visit each city once and then return to his starting point. The travelling time (in hours)
for each city from a particular city is given below:

What should be the sequence of visit of the salesman, so that the total travelled time is
minimum?
89. A salesman has to visit 4 cities A, B, C, and D. The distance (100 kms) between 4
cities is:

If the salesman starts from city 'A' and comes back to city 'A', which route should he
select so that total distance travelled by him is minimum?
90. A manufacturer of garments plans to add 4 regional warehouses to meet increased
demand. The following bids in lakhs of rupees have been for construction of the
warehouses.
 Explain why each warehouse contract cannot simply be given to the contractor who
bids cheapest. How will you then determine the optimal contract?
91. 5 operators have to be assigned to 5 machines. The assignment costs are given in the
table.

Operator A cannot operate machine II and operator C cannot operate machine IV. Find
the optimal assignment schedule.
92. Determine an optimum assignment schedule for the following assignment problem.
The Notes cost matrix is –

If job C cannot be assigned to machine 6, will the optimum solution change?

93. A company has 6 jobs to be processed by 6 machines. The following table gives the
return in rupees when the ith job is assigned to the jth machine (I, J = 1…….6). How
should the jobs be assigned to the mechanics so as to maximize the overall return?

94. A company is faced with the problem of assigning 4 machines to 6 different jobs (one
machine to one job only). The profits are:
95. A process can be carried out an anyone of 6 machines. The average time taken by any
operator on any specific machine is tabulated in the given matrix. It is proposed to buy
a new machine to replace one of the existing ones for operations. To carry out the
process on this machine average times have been estimated and entered in the matrix.
Is it advantageous at this stage to use the new machine? If so, which of the original
machines should be replaced and how should the operators be allocated?

96. A section head has five stenotypists and five jobs to complete. The stenos differ in
their efficiency and the jobs differ in their intrinsic complexity. The estimate of the
time (in hours) each steno would take to perform the task is given in the effectiveness
matrix below. How should tasks be allocated, one to a person, so as to maximize the
total time taken to complete all the jobs?

97. A tourist car rental agency has a surplus car in each of the cities A, B, C, D, E and F,
and a deficit of one car in each of the cities AA, BB, CC, DD, EE, and FF. The
distances between cities with a surplus car and cities with a deficit car are given in the
following matrix. How should the cars be dispatched so as to maximize the total
distance covered?