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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]8/3/2019 Operations Research_Class1 to Class 3
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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_science8/3/2019 Operations Research_Class1 to Class 3
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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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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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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 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