Ni Rav Assignment

7/30/2019 Ni Rav Assignment

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Assignment problem

transportation problem in general are concerned with

distributing any commodity from any group of supply

centers, called sources, to any group of receiving

centers, called destinations, in such a way as to

minimize the total distribution cost.

Assumptions:

Requirement assumption: each source has a fixed

supply of units, where this entire supply must be

distributed to the destinations. similarly, each destinationhas a fixed demand for units, where this entire demand

must be received from the source.

ASSIGNMENT PROBLEM

Characteristics & assumptions of transportation problem


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Assignment problem

Cost assumption: the cost of distributing units

from any particular source to any particular

destination is directly proportional to that number

of units distributed.therefore,this cost is just theunit cost of distribution times the number of

units distributed.

The feasible solutions property: A

transportation problem will have feasiblesolutions if and only if the sum of its supplies

equals the sum of its demands.

ASSIGNMENT PROBLEM



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Assignment problem

The model:

The linear programming model hasconstraints for supply at each source and

demand at each destination. All constraints are equalities in a balancedtransportation model where supply equalsdemand.

Constraints contain inequalities inunbalanced models where supply does notequal demand.


ASSIGNMENT PROBLEM


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Assignment problem

Shipping goods

Assigning plants to product

Profit maximization problem Overtime production

Production scheduling

ASSIGNMENT PROBLEM

Application of transportation problem


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Assignment problem

ASSIGNMENT MODEL

ASSIGNMENT PROBLEM


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Assignment problem

Assignment model is generally made on one-to-onebasis & if there are more jobs to do than can be done,one can decide which job to leave undone and whatresource to add.

In assignment model number of source is equal tonumber of destination ,supply at each source is unity(1)& demand at each destination is also unity(1), it isdifferent from transportation problem.

Unit allocated to cell is either 1 or 0 ,it is also differentfrom transportation model

However transportation algorithm is not very useful tosolve this model because of degeneracy.

ASSIGNMENT PROBLEM

Difference between transportation & assignment model


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Assignment problem

Assignment problem is completely degenerate form oftransportation model.

In n*n assignment problem, there will be n assignment

instead of 2n-1 assignment in transportation model.

Special structure of assignment model allows a moreconvenient and simple method of solution than

transportation model.

ASSIGNMENT PROBLEM



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Assignment problem

Represent the assignment of n facilities ton jobs.Cij is cost of assigning ith facility to

jth job and Xij represents the assignment

of ith facility to jth job. If ith facility can be assigned to jth job,

xij=1,otherwise 0.

The objective is to make assignment thatminimize the total assignment cost ormaximize the total associated gain.

ASSIGNMENT PROBLEM

Structure & formulation of assignment model


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Assignment problem

Xij=0,if ith facility is not assigned to j th job,

Xij=1, if the ith facility is assigned to jth job.

Then the model is given by

Minimize z= CijXij

Subject to constraints Xij=1,i=1,2,3n(one

job is assigned to ith facility)

Xij=1,j=1,2,3,..n(one facilityassign to Jth job.

ASSIGNMENT PROBLEM

Structure & formulation of assignment model

J=1

n

i=1

n

J=1

i=1

n

n


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Assignment problem

& Xij=0 or 1.

ASSIGNMENT PROBLEM


C11 C12 . C1n

C21 C22 C2n

C31 C32 C3n

Cn1 Cn2 Cnn

1

1

:

1

1 1 1

Jobs

1 2 n

1

2

:

n

facilities


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Assignment problem

Step 1: prepare the assignment table.

Step 2: convert assignment table into balanced problemby add dummy row or column if it is not balancedproblem.

Step 3:convert problem into minimization problem bysubtracting all the elements from the largest element if itis maximization problem.

Obtain reduced cost table:

(a) Subtract the minimum element in each row from all theelements of that row and then

(b) Subtract the minimum element in each column from allthe elements of that column.

ASSIGNMENT PROBLEM

Hungarian method


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Assignment problem

Step 5:Draw minimum number of lines to cover

all the 0s in the table.

Step 6:check the lines drawn equal to the orders

of the matrix or not. if not then, subtract the smallest uncovered elements

from all the uncovered elements and add it to

intersection of line. keep the remaining elements

unchanged & go to step 5 & then 6.

Step 7:get the optimum solution.

ASSIGNMENT PROBLEM

Hungarian method


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Assignment problem

A machine tool company decides to make

four subassemblies through four

contractors. each contractor is to receive

only one sub assembly. the cost of eachsubassembly is determined by the bids

submitted by each contractor and is show

in table in hundreds of rupees.

ASSIGNMENT PROBLEM

EXAMPLE-TRANSPORTATION PROBLEM


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Assignment problemASSIGNMENT PROBLEM


15 13 14 17

11 12 15 13

13 12 10 11

15 17 14 16

Contractors

1 2 3 4

S

u

ba

s

s

e

m

bli

e

s

1

2

3

4


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Assignment problem

Step III: (a) constraints on subassemblies are

X11+x12+x13+x14=1

X21+x22+x23+x24=1

X31+x32+x33+x34=1 X41+x42+x43+x44=1

(b) constraints on contractors are:

X11+x21+x31+x41=1

X12+x22+x32+x42=1

X13+x23+x33+x43=1

X14+x24+x34+x44=1

ASSIGNMENT PROBLEM



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Assignment problem

A school bus has to start from the school located

at A and return after picking up children from

B,C,D and E .the table below shows distances

between various pick-up points on differentroutes.the distance from A to B is 8 KM but from

B to A is 6 KM Because of one way streets and

other restrictions on movement between points.

as no of movement is possible from adestination to itself, these cells are marked with

dash.

ASSIGNMENT PROBLEM


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Assignment problem

A school bus has to start from the school located

at A and return after picking up children from

B,C,D and E .the table below shows distances

between various pick-up points on differentroutes.the distance from A to B is 8 KM but from

B to A is 6 KM Because of one way streets and

other restrictions on movement between points.

as no of movement is possible from adestination to itself, these cells are marked with

dash.

ASSIGNMENT PROBLEM


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TO

A B C D E

A - 8 7 9 5

FROM B 6 - 11 5 5

C 15 11 - 6 14

D 9 5 6 - 8

E 7 3 6 4 -


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Assignment problem

Five workers are available to work with fivemachines. the respective cost associated witheach worker and machine assignment is given inthe table below. a sixth machine is available toreplace one of the existing machines and itsassociated costs are also given.

(a) determine, whether the new machine can beaccepted.

(b) determine the optimal assignment andsaving in cost.

ASSIGNMENT PROBLEM

Problem with dummy row or column


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Assignment problem

A company has a team of four salesmen

and there are four districts where the

company wants to start its business. after

taking into account the capabilities ofsalesman and the nature of district, the

company estimates that the profit per day

in rupees for each salesman in eachdistrict is as below.

ASSIGNMENT PROBLEM

Maximization problem


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Maximization problem

1 2 3 4

A 16 10 14 11

B 14 11 15 15C 15 15 13 12

D 13 12 14 15

DISTRICT

SALESMAN

Find the assignment of salesmen to various district which will yield maximumprofit?


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Maximization problem-2

A small garment making unit has five tailors switching five different types of

garments. the output per day per tailor and the profit for each type of

garment are given below

1 2 3 4 5A 7 9 4 8 6

B 4 9 5 7 8

C 8 5 2 9 8

D 6 5 8 10 10

E 7 8 10 9 9

Profit per garment 2 3 2 3 4

garments

tailors


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Maximization problem-2

1. Which type of garment should be assigned to which tailor in order to maximize

profit, assuming that there are no other constraints?

2. If tailor D is absent for a specified period and no other substitute tailor is

available, what should be the optimal assignment?


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Assignment problem

Consider a manufacturing company that is thinking ofseveral alternative methods to increase its production tomeet the increase market demand.

step 1:list all the viable alternatives

1.expand the present plant

2.construct the new plant

3.subcontract production for extra demand.- these alternatives term as course of actions or simply

actions or strategies.

Decision theory approach

1DECISION THEORY


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Assignment problem

step 2:identify the expected future event

List all the future event they may occur. but it is very tough to

identifying that particularly which event will occur.

These future events are termed as state of nature

In our example state of nature are

- high demand

- moderate demand

- Low demand

- no demand


DECISION THEORY2


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Assignment problem

step 3:construct pay off table or conditional gain or profittable

- Make a pay off table for each possible combination ofalternative course of action and state of nature.

3


ALTERNATIVES STATE OF NATURE

HIGH MODERATE LOW NIL

EXPAND Rs.50000 Rs.25000 -25000 -45000CONSTRUCT 70000 30000 -40000 -80000

SUBCONTRACT 30000 15000 -1000 -10000

DECISION THEORY


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Assignment problem

Step 4 : select optimum decision criterion

Finally the decision-maker will choose criterionwhich results in largest payoff. the criterion may be

economic, quantitative or qualitativeetc.

DECISION THEORY 4



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Assignment problem

decision-making under conditions of certainty.

decision-making under conditions of uncertainty.

decision-making under conditions of risk.

5

Decision making environment

DECISION THEORY


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Assignment problem

- in this environment, only one state of nature exists foreach alternative i.e., there is complete certainty aboutthe future. it is easy to analyze the situation and makegood decisions.

- Decision-maker has perfect knowledge about the futureoutcomes, he simply choose the alternative havingoptimum pay-off.

6

Decision making under condition of certainty

DECISION THEORY


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Assignment problem

EXAMPLE:-

A company can manufacture two different products k1 andk2.the two products have to be processed through twomachines M1 & M2 whose available capacities in terms ofhours per week are 480 and 410 respectively. product k1 need

3 hours of M1 and 1 hour of M2 and product K2 requires 1hour of M1 and 2 hours of M2. find the number of units ofproducts K1 & K2 that can be manufactured every week byutilizing fully the available capacities of the machine.

7

MACHINES

PRODUCT TIMES

AVAILABILITY

M1 3 1 480

M2 1 2 410


DECISION THEORY


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Assignment problem

Let x be the number of units of k1 and y be the number

of units k2 that can be manufactured. then, as per the

given data we have the following simultaneous equation:

3x+y=480

X+2y=410

Solving we get x=110,y=150

Therefore 110 units of product k1 and 150 units of

product k2 can be manufactured.

8


DECISION THEORY


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Assignment problem

Example: Linear programming when resources required,

resources available, cost or profit per unit of the product

are know with certainty.

9


DECISION THEORY


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Assignment problem

Here, more than one states of nature exist but the

decision-maker lacks sufficient knowledge to allow him

assign probabilities to the various states of nature.

Under conditions of uncertainty, a few decision criteria

are available which could be of help to the decision-maker and a choice among them is determined by the

companys policy and attitude of the decision maker.

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Decision making under condition of uncertainty

DECISION THEORY

Maximax criterion or criterion of optimism


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Assignment problem



11DECISION THEORY



EXPAND Rs.50000 Rs.25000 -Rs.25000 -Rs.45000

CONSTRUCT 70000 30000 -40000 -80000

SUBCONTRACT 30000 15000 -1000 -10000


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Assignment problem


12


DECISION THEORY

Maximum of rows

50000

70000

30000


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Assignment problem

When dealing with costs ,the minimum of each

alternative is considered and the alternative which

minimizes the above minimum costs is selected. this is

called minimin criterion.

13


DECISION THEORY


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Assignment problem

To use this criterion, the decision-maker maximizes his

minimum possible payoffs.

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Maximin criterion or criterion of pessimism or wald criterion

DECISION THEORY



EXPAND Rs.50000 Rs.25000 -Rs.25000 -Rs.45000

CONSTRUCT 70000 30000 -40000 -80000

SUBCONTRACT 30000 15000 -1000 -10000


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Assignment problem 15DECISION THEORY

Maximin criterion or criterion of pessimism or wald criterion

Minimum of rows

-45000

-80000

-10000


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Assignment problem

This criterion developed by L.J.Savage.

Minimize our maximum regret.

16

Minimax regret criterion or savage criterion

DECISION THEORY



EXPAND Rs.20000 Rs.5000 Rs.24000 35000

CONSTRUCT 0 0 39000 70000

SUBCONTRACT 40000 15000 0 0


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Assignment problem


17


DECISION THEORY

Maximum of row

35000

70000

40000


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Assignment problem

Hurwicz criterion (criterion of Realism)

This is also called as a weighted average criterion

This concept allow decision maker to take into

account both the maximum and minimum for each

alternative and assign them weights according to

his degree of optimism.

=degree of optimism

1-=degree of pessimismhere is called coefficient or index of

optimism

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DECISION THEORY


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Assignment problem

Step 1:choose appropriate degree of optimism

Step 2:Determine the maximum as well as

minimum of each alternative and obtain

P=.maximum+(1-).minimum Step 3:choose the alternative that yields

the maximum value of p.

19


DECISION THEORY

H i it i ( it i f R li )


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Assignment problem 20


DECISION THEORY

maximum of row minimum of row p

50000 -45000 31000

70000 -80000 40000

30000 -10000 22000

L l C it i C it i f R ti lit


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Assignment problem

This criterion assigns equal probabilities to

all the events of each alternative decision

and selects the alternative associated with

the maximum expected payoff.

If n denotes the number of events and Ps

denote the payoffs, then expected value

for strategy, say S1 is 1/n[p1+p2++Pn].

21

Laplace Criterion or Criterion of Rationality

DECISION THEORY

R li h i it i


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Realism or hurwicz criterion

DECISION THEORY

Expected payoff

1000/4[50+25-25-45]=-1250

-5000

85000

D i i ki d diti f i k


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Decision making under conditions of risk

DECISION THEORY

Most business decisions may have to be made underconditions of risk. here more than one states of natureexist and the decision-maker has sufficient information toassign probabilities to each of these states.

These probabilities could be obtained from the pastrecords or from simply the subjective judgment

E t d t l t d l it i


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Expected monetary value or expected value criterion

DECISION THEORY

This criterion requires the calculation of the expected

value of each decision alternative which is the sum of the

weighted payoffs for that alternative ,where the weights

are the probabilities assigned to the states of nature that

can happen.

Step 1: construct a conditional payoff table listing thealternative decisions and the various states of nature.

Enter the conditional profit for each decision-event

combination along with the associated probabilities.

Step 2: calculate the EMV for each decision alternativeby multiplying the conditional profits by assigned

probabilities and adding the resulting conditional values.

Step 3: select alternatives that yields the highest EMV.


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A news paper boy has the following probabilities ofselling a magazine:

No. of copies sold probabilities

10 .10

11 .15

12 .20

13 .25

14 .30

Cost of a copy is 30 paise and sale price is 50 paise.hecannot return unsold copies.how many copies should heorder?



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possible

demand probabilities

possible stock action

(no. of

copies)

10

copies

11

copies

12

copies

13

copies

14

copies

10 0.1 200 170 140 110 80

11 0.15 200 220 190 160 130

12 0.2 200 220 240 210 180

13 0.25 200 220 240 260 230

14 0.3 200 220 240 260 280

Conditional profit table



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Expected profit table

possibledemand probabilities possible stock action

(no. of

copies)

10

copies

11

copies

12

copies

13

copies

14

copies

10 0.1 20 17 14 11 8

11 0.15 30 33 28.5 24 19.5

12 0.2 40 44 48 42 36

13 0.25 50 55 60 65 57.5

14 0.3 60 66 72 78 84

total

expected

profit 200 215 222.5 220 205



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The newsboy must, therefore, order 12

copies to earn the highest possible

average daily profit of 222.5 paise



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An alternative approach is to minimize expected

opportunity loss. Step 1:prepare the conditional profit table for each

decision event combination and write the associatedprobabilities.

Step 2:for each event, determine the conditional

opportunity loss by subtracting the pay off from themaximum payoff for that event.

Step 3:clculate EOL for each decision alternative bymultiplying the COLs by the associated probabilities and

then adding the values. Step 4;select the alternative that yields the lowest EOL.

Expected opportunity loss criterion


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possible

demand probabilities possible stock action

(no. of

copies)

10

copies

11

copies

12

copies

13

copies

14

copies

10 0.1 200 170 140 110 80

11 0.15 200 220 190 160 130

12 0.2 200 220 240 210 180

13 0.25 200 220 240 260 230

14 0.3 200 220 240 260 280

conditional profit table


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conditional LOSS TABLE

possible


(no. of

copies)

10

copies

11

copies

12

copies

13

copies

14

copies

10 0.1 0 30 60 90 120

11 0.15 20 0 30 60 90

12 0.2 40 20 0 30 6013 0.25 60 40 20 0 30

14 0.3 80 60 40 20 0


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EXPECTED LOSS TABLE

possible


(no. of

copies)

10

copies

11

copies

12

copies

13

copies

14

copies

10 0.1 0 3 6 9 12

11 0.15 3 0 4.5 9 13.5

12 0.2 8 4 0 6 1213 0.25 15 10 5 0 7.5

14 0.3 24 18 12 6 0

EOL 50 35 27.5 30 45


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THE OPTIMUM STOCK ACTION IS THE ONE WHICH WILL MINIMIZE

EXPECTED OPPORTUNITY LOSSES;THIS ACTION CALLS FOR THESTOCKING OF 12 COPIES EACH DAY,AT WHICH POINT THERE IS

MINIMUM EXPECTED LOSS OF 27.5 PAISE.


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EXPECTED VALUE OF PERFECT INFORMETION

possibledemand possible stock action

(no. of copies) 10 copies 11 copies 12 copies 13 copies 14 copies

10 200

11 220

12 240

13 26014 280

CONDITIONAL PROFIT TABLE UNDER CERTAINTY


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EXPECTED PROFIT TABLE WITH PERFECT INFORMATION

market size

conditional

profit

probability of market

size expected profit

(no. of

copies

under

certainty with perfect

information

10 200 0.1 20

11 220 0.15 33

12 240 0.2 48

13 260 0.25 65

14 280 0.3 84

EPPI=250


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Assignment problemDECISION THEORY


EVPI=250-222.5=27.5So, EVPI is no more than 27.5

36

f


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EMV for item that have selvage value

36

An ice-cream retailer buys ice cream at a cost of Rs.5per cup and sells it for Rs. 8 per cup; any remainingunsold at the end of the day can be disposed of at asalvage price of Rs.2 per cup. past sales have rangedbetween 15 and 18 cups per day; there is no reason tobelieve that sales volume will take on any othermagnitude in future. find the EMV if the sale history hasthe following probabilities:

Market size: 15 16 17 18

Probability: 0.1 0.2 0.4 0.3

EMV f it th t h l l


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37

If CP denotes conditional profit. S the quantity in stockand D the market demand, then

CP=(8-5)S,WHEN D=>SAND

CP=8D-5S+2(S-D),WHEN D


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38

possible stock action (alternative)

15 16 18

15 0.1 45 42 3616 0.2 45 48 42

17 0.4 45 48 48

18 0.3 45 48 54

Possible

demand

(event)

probabilities

Conditional profit table

EMV f it th t h l l


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39

possible demand

(event) probabilities

possible stock action

(alternative)

15 16 17 18

15 0.1 4.5 4.2 3.9 3.6

16 0.2 9 9.6 9 8.4

17 0.4 18 19.2 20.4 19.2

18 0.3 13.5 14.4 15.3 16.2

EMV 45 47.4 48.6 47.4

EXPECTED PROFIT TABLE

EMV=Rs.48.6 for stock action of 17 ice cream cups each day

E l


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Example

40

GM of calcutta motor is planning staffing of thedealershipps garage facilities in Ahmedabad. frominformation provided by the manufacturer and from othernearby dealership, he has estimated the number ofannual mechanic hours that the garage will be likely to

needed Hours 10000 12000 14000 16000

Probability 0.15 0 .25 0.4 0.2

The manager plans to pay each mechanic Rs.9 per hourand to charge customer Rs.16. mechanics will work a 40

hours per week and get an annual 2 week vacation.Determine how many mechanics Calcutta motor shouldhire?

Calculate EOL and EVPI also?

DECISION TREES


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DECISION TREES

41

The decision problems discussed so far

with only single stage decision making. we

now consider situations that involve

multiple stages. they are characterized bya sequence of decision with each decision

influencing the next. such problems, called

sequential decision problems, areanalyzed best with the help of decision

trees.

DECISION TREES


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DECISION TREES

42

A client asks an estate agent to sell three properties A,Band c for him and agrees to pay him 5% commission oneach sale. he specifies certain conditions. the estateagent must sell property A first, and this he must dowithin 60 days. if and when A is sold, the agent receives

his 5% commission on that sale. he can then either backout at this stage or nominate and try to sell one of theremaining two properties within 60 days. if he does notsucceed in selling the nominated property in that period,he is not given the opportunity to sell the other. if hedoes sell it in the period, he is given the opportunity to

sell the third property on the same conditions. Thefollowing table summarizes the prices, selling cost andthe estate agents estimated probability of making a sale.

DECISION TREES


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DECISION TREES

43

Property Prices of

property

Selling

costs

Probability

of sales

A 12000 400 0.7

B 25000 225 0.6

C 50000 450 0.5

1)DRAW UP AN APPROPRIATE DECISION TREE FOR THE

ESTATE AGENT.2)WHAT IS THE ESTATE AGENTS BEST STRATEGY UNDER

EMV APPROACH.

DECISION TREES


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DECISION TREES

44

Mr Sinha had to decide whether or not to drill a well on his farm. in his

village, only 40% of the wells drilled were successful at 200 feet of depth.some of the farmers who did not get water at 200 feet, drilled further upto

250 feet but only 20% struck water at 250 feet. cost of drilling is Rs.50

per foot. Mr. Sinha estimated that he would pay Rs.18,000 during a 5-

year period in the present value terms, if he continues to buy water from

the neighbor rather than go for the well which would have a life of 5

years. Mr. Sinha has three decision to make:(a)should he drill up to 200feet and (b) if no water is found at 200 feet, should he drill up to 250

feet?(c) should he continue to buy water from his neighbor?

Posterior probabilities and bayesian analysis


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Posterior probabilities and bayesian analysis

45

In a preceding analysis of decision making under

probabilities used in the expected value criterion are usually obtained

from the past historical data. these probabilities are called priorprobabilities and the analysis that uses them to find expected payoffs is

known as prior analysis.

If the prior analysis results in high expected opportunity loss (EOL) or

EVPI,it may be desirable to obtain additional information through

sampling or experimentation or test research. prior probabilities may thenbe revised in the light of this additional information by using bayes

theorem to yield posterior probabilities or bayes probabilities.

Revised analysis of the problem using these probabilities is called

posterior analysis or bayesian analysis. this analysis gives a revised

value of EMV which is expected to be better than the EMV based on prior

analysis. the difference between the two should at least be equal to the

UTILITY THEORY


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UTILITY THEORY

46

Think about the situation in which you have two option:-

(1)Get 10000 rupees fix

(2)5% chance of getting Rs.250000 and 95% chance of getting nothing

What is your answer?

The worth of money is different for different people and is also different in

different situations.

Each rupee is not equally valuable to an individual.

The determination of utility is subjective.it depends upon our attitude of

accepting risk.The utility theory approach attempts to determine a utility function or a

utility curve for a decision maker.

UTILITY THEORY


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UTILITY THEORY

47

Here function converts money into an arbitary utility measure.

Risk avoider: slope decrease as money values increase. firm with

unsound finansial position.

Risk neutral:expected utility give the same results as expected

return(EMV) analysis.

EXAMPLE:LARGE COMPANIES & GOVERNMENT.

Risk taker:curve increase faster than the money value.

In practice utility curve is a combinetion of risk taker and risk avoider

UNBALANCED ASSIGNMENT


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Assignment problem

UNBALANCED ASSIGNMENT

PROBLEM

You are given the information about the

cost of performing different jobs by

different persons. the job-person marking

(@) indicates that the individual involvedcannot perform the particular job. using

this information, state (i) the optimal

assignment of jobs, and (ii) the cost ofsuch assignment


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Traveling salesman problems

Mr Patel is a salesman with Delite

Manufecturing company. he wants to visit

six cities, say, 1,2,3,4,5 and 6,starting with

city 1where he is stationed. the distancesbetween various cities are given in the

table.Patel wants to develop a tour

through the five other cities and return tohis home city in such a way that he was to

travel a minimum distance.

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Ni Rav Assignment