Operations Research_Class1 to Class 3

8/3/2019 Operations Research_Class1 to Class 3

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Faculty: P.N. Ram Kumar

E-mail: [email protected]

Phone Extension: 426
mailto:[email protected]:[email protected]


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Operations research (OR) is an interdisciplinarymathematical science that focuses on the

effective useofresources/technologyby

organizations
http://en.wikipedia.org/wiki/Mathematical_sciencehttp://en.wikipedia.org/wiki/Mathematical_science


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As the name implies, OR involves research on operations

OR is applied to problems that concern how to conduct,

manage and coordinate the operations (activities) within an

organization in the best possible manner

Nature of the organization is essentially immaterial

OR has been applied extensively in diverse areas such as

manufacturing, transportation, construction,

telecommunications, financial planning, health care, military

and public services, to name just a few.


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Scarce resources Labor, materials, machinery, money

Quest for best combination of inputs

Minimizing costs or Maximizing profits

Linear Programming (LP) problems- Linear objective function

- Linear constraints

Why only LP?

- Majority of real-life problems can be approximated

- Efficient solution procedures

- Easy Sensitivity Analysis


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Linear Programming problems Formulation, Graphicalmethod - Sensitivity Analysis and Duality

Transportation, Transshipment and Assignment problems

Network optimization models

Integer programming problems

Project management

Simulation A brief overview

Waiting line models

Multi-objective decision problems

Dynamic Programming

Markovian Models


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A. Exams: Mid term: 25%; End term: 35%

B. Assignments + Case Presentation: 15%

C. Surprise Quizzes: 15%

D. Class participation: 10%


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Its only for Quants guys, not for us!

Its too mathematical!!

OR models can replace decision makers

A model should never be taken too literally

Beware of overselling a model

Verification and Validation


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Project management


Waiting line models


Dynamic Programming

Markovian Models


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Project management


Waiting line models


Dynamic Programming

Markovian Models


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A small factory makes three products, soap,

shampoo and detergent. The processing is done in

two stages at two plants, I and II, each of which

works for 40 hours every week.

The time required for processing each product at

each plant and the profit of each product are shown

in the table

Formulate a linear program to maximize the profits


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Plant I Plant II Profit / Kg

Soap 3 5 Rs.10Shampoo 4 4 Rs. 13Detergent 4 2 Rs. 12

Hours /week 40 hrs 40 hrs


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1. Decision Variables???

2. Let X1, X2, and X3 be the quantities of soap,

shampoo and liquid soap manufactured per

week.

3. Constraints???

4. Time available per week (40 hours)

5. Last, the objective function (Max Profit or

Minimize Cost)


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MaximizeZ = 10X1 +13X2 + 12X3

Subject to

3X1 + 4X2 + 4X3 < 40

5X1 + 4X2 + 2X3 < 40

X1> 0, X2> 0, X3> 0

X1, X2 ,X3

Objective Function

Constraints

Non-NegativityConstraints

Decision Variables


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A middle-aged woman, convalescing after a surgery,

has been advised by her doctor to plan her diet in

such a way as to ensure that she gets the prescribed

quantities of vitamins.

The choice of foods available to her, the amounts

of vitamins they contain, the cost of each unit and

the daily requirement of vitamins are shown in the

table


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Cost Vitamin A Vitamin B Vitamin D

Eggs Rs 1.50/unit 15 mg/unit 40 mg/unit 10 mg/unit

Milk Rs. 14.00/lit 30 mg/lit 25 mg/lit 20 mg/litCereals Rs. 18/unit

(kg) 10 mg/kg 55 mg/kg 30 mg/kgDaily

requirement 100 mg 250mg 120mg


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Formulate a linear programme to determine the

quantity of each food that the woman needs to buy in

order to minimize her total expenditure, ensuring at

the same time that she meets her daily requirements

of vitamins.

Decision Variables? X1, X2, X3 be the quantities of eggs, milk and cereals

bought

Constraints?

Dietary requirements are to be satisfied

Objective Function

Minimization of expenditure


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Let X1, X2, X3be the quantities of eggs, milk andcereals bought.

The problem is of the form


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(i) There is an objective function which describeswhat we desire

(ii) A set of constraints, usually in the form ofinequalities

(iii) Non-negativity constraints for the decisionvariables

(iv) The objective function and the constraints arelinear.


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Max / Min Z = C1 x1 + C2x2+..+ Cnxn

Subject to

a11

x1

+ a12

x2+ + a

1nx

n


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1

/

1.....

1.....

x 1.....

j j

j

ij j i

j

i

j

Maximize Minimize Z C x

Subject to

a x or b i m

b Unrestricted in sign i mUnrestricted in sign j n


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Proportionality

Additivity

Divisibility

Certainty


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A financial advisor who recently graduated from IIMK

received a call from a client who wanted to invest a

portion of a $ 150,000 inheritance

The client wanted to realize an annual income, but

also wanted to spend some of the money

After discussing the matter, the client and the

adviser agreed that a mutual fund, corporate bonds,

and a money market account would make suitable

investments.


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The client was willing to leave allocation of the funds among

these investment vehicles to the financial adviser, but with the

following conditions:

At least 25 percent of the amount invested should be in the money

market account

A maximum of only 35% should be invested in corporate bonds

The investment must produce at least $ 12,000 annually (ROI)

The un-invested portion should be as large as possible

The annual returns would be 11 percent for the mutual fund, 8

percent for the bonds, and 7 percent for the money market

Formulate an LP model that will achieve the clients requests.

Ignore transaction costs, the advisers fee and so on.


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Let X1, X2, X3be the amounts to be invested in mutual fund,

corporate bonds and money market

Non-negativity condition: X1, X2, X3 0

At least 25 percent of the amount invested should be in the

money market accountX3 / (X1+X2+X3) 0.25

A maximum of only 35% should be invested in corporate bonds

X2 / (X1+X2+X3) 0.35

The investment must produce at least $12,000 annually (ROI)

0.11X1 + 0.08X2 + 0.07X3 12000

Objective Function: Maximize Z = 150,000 (X1 + X2 + X3)


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A trucking company with Rs.20 crores to spend on newequipment is contemplating three types of vehicles

Vehicle A has a 10-ton payload and is expected to cover 55

km per hour. It costs Rs.40 lakhs. Vehicle B has a 20-ton payload and is expected to cover 50

km per hour. It costs Rs.65 lakhs.

Vehicle C is a modified form of B: it carries sleeping quarters

for one driver, and this reduces its capacity to 18-tons and

raises the cost to Rs.75 lakhs.


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Vehicle A requires a crew of one man, and, if driven on threeshifts per day, could be run for an average of 18 hours per day.

Vehicles B and C require a crew of two men each. Where, B would

be driven 18 hours per day with three shifts, C could be run for

an average 21 hours per day.

The company has 150 drivers available each day and would find

it very difficult to obtain further crews. Maintenance facilities are

such that the total number of vehicles must not exceed 30.

How many vehicles of each type should be purchased if the

company wishes to maximize its capacity in ton-km per day?


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A senior executive of a public sector company

recently quit his job under the VRS with a hefty

packet of Rs 1 Crore. Messrs. Dhana-chor Chit

Fund Company has offered the following

investment scheme for the benefit of such retired

people:


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"Invest a certain sum (in lakhs of rupees) in any

month, invest half of that amount in the next

month. In the subsequent month, one would get

twice the amount invested originally in the first

month

This scheme is available only for the next six

months (Encashment is possible on 181st day)

Returns received at the end of any month can be

used immediately for reinvesting either as a fresh

investment or as a follow-up investment


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Let Xt = Fresh investment on the 0th,

30th

, 60th

,90th, 120th day respectively (t = 1,2,3,4,5) (in

lakhs of Rs.)

Noting that, in any month, the cash outflow mustnot exceed the cash on hand, the following linear

programme is formulated


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10: 100Day X 1

2 130: 100

2

XDay X X

2 1

3 260: 1002 2

X X

Day X X

3 1 2

4 390: 100

2 2 2

X X XDay X X

34 1 2

5 4120: 100

2 2 2 2XX X XDay X X

5 31 2 4

5150: 100

2 2 2 2 2

X XX X XDay X


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On 181st day, the cash on hand must be maximum

Hence the objective function is


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A hospital administrator has the following minimal daily requirementsfor nursing personnel:

Nurses report to the hospital wards at the beginning of each period

and work for 8 consecutive hours. The hospital wants to determine

the minimal number of nurses to employ so that there will be a

sufficient number of personnel available for each period.

Period Clock Time Minimal No. of nurses1 6 AM to 10 AM 60

2 10 AM to 2 PM 70

3 2 PM to 6 PM 60

4 6 PM to 10 PM 50

5 10 PM to 2 AM 20

6 2 AM to 6 AM 30


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A brewery blends three raw materials, A, B and C invarying proportions to obtain three final products,

D, E and F. The salient data is given below.

FinalProduct D E FSelling

price/ Litre$15 $9 $12

Ingredient A B CCost/ Litre $10 $8 $7

QuantityAvailable

10,000 12,000 15,000


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Formulate a linear programme to maximize the netprofit of the brewery.

Final Product SpecificationsD At least 25% of ingredient A

Not more than 15% of C

E At least 50% of ingredient BNot more than 30% of C

F At least 65% of ingredient CNot more than 5% of A


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Let Xij be the no. of litres of the ith

raw material in jth

the product

Solution to Example Problem No.7

A

B

C


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A shipping company has to lift three types of

cargo whose details are given below.

Type ofCargo Quantity(tons) Volume perton Profit perton

A 300 1.2 m3 Rs. 10,000B 500 0.8 m3 Rs. 7,000C 400 1 m3 Rs. 8,000


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Weight in aft, center, forward must be in the

same proportion as the holding capacities

by weight, i.e., 100 : 200 : 75

Formulate a linear programme to maximize

the profits of the shipping company.


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Solution to Example Problem No.8


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Constraints on holding space


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A final product is assembled with 4 units of

component A and 3 units of component B.

The manufacturing shop runs three different

processes, each of which requires varying amounts ofraw materials and produce different amounts of A

and B.

Two types of raw materials are used. 100 units of raw

material I (RM I) and 200 units of raw material II (RM

II) are available to the shop each day


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ShopInput per run

(units) Output per run(units)RM I RM II A B

I 7 5 6 4II 4 8 5 8III 2 7 7 3

100 200Formulate a linear programme to maximize the number

of completed assemblies produced each day.


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Let X1, X2, X3 be the number of runs of each

process operated per day

Total quantity of A produced =

Total quantity of B produced =


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No. of units of finished product??


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A machine tool company conducts a job-training program

for machinists Trained machinists are used as teachers in the program at a

ratio of one for every ten trainees. The training program

lasts for one month (Assumption: No. of trainees multiple of 10)

From past experience it has been found that out of ten

trainees hired, only seven complete the program

successfully (the unsuccessful trainees are released)

Trained machinists are also needed for machining and the

companys requirements for the next three months are as

follows:

January 100

February 150

March 200


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In addition, the company requires 250 trained machinists by

April. There are 130 trained machinists available at the

beginning of the year. Payroll costs per month are:

Each Trainee $ 400

Each trained machinist $ 700

(machining or teaching)Each trained machinist idle $500

(Union contract forbids firing trained machinists)

Set up the linear programming problem that will produce theminimum cost hiring and training schedule and meet the

companys requirements.


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Every month, a trained machinist can do one of the

following:

(1) Work a machine, (2) Teach or (3) Stay idle

Since the number of trained machinists for machining is

fixed, the only decision variables are the number teaching

and the number idle for each month. Thus, the decision

variables are:

X1 trained machinists teaching in January

X2 trained machinists idle in January

X3 - trained machinists teaching in February

X4 trained machinists idle in February

X5 trained machinists teaching in March

X6 trained machinists idle in March


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Number machining + Number teaching + Number idle

= Total trained machinists available at the beginningof the month

For January : 100 + X1 + X2 = 130

For February : 150 + X3 + X4 = 130 + 7X1

For March : 200 + X5 + X6 = 130 + 7X1 + 7X3

Since the company requires 250 trained machinists by

April,

130 + 7X1 + 7X3 + 7X5 = 250Objective Function: Minimize Z = 400 (10X1 + 10X3 +

10X5) +700 (X1 + X3 + X5) +500 (X2 + X4 + X6)


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F1

F2

DC

W1

W2

50 unitsproduced

40 unitsproduced

$ 200/unit 10 units max

$ 900/unit

$

200/u

nit

$3

00/unit

30 unitsneeded

60 units needed

Operations Research_Class1 to Class 3

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