session 6a. decision models -- prof. juran2 overview multiple objective optimization two dimensions...

Session 6a

Decision Models -- Prof. Juran

2

OverviewMultiple Objective Optimization• Two Dimensions• More than Two Dimensions• Finance and HR Examples• Efficient Frontier• Pre-emptive Goal Programming

Intro to Decision Analysis


3

Scenario Approach RevisitedYear Ford Lilly Kellogg Merck HP 1983 14.13 14.47 8.09 5.02 42.38 1984 15.21 16.50 10.00 5.22 33.88 1985 19.33 27.88 17.38 7.61 36.75 1986 28.13 37.13 25.88 13.76 41.88 1987 37.69 39.00 26.19 17.61 58.25 1988 50.50 42.75 32.13 19.25 47.25 1989 43.63 68.50 33.81 25.83 31.88 1990 26.63 73.25 37.94 29.96 57.00 1991 28.13 83.50 65.38 55.50 69.88

Use the scenario approach to determine the minimum-risk portfolio of these stocks that yields an expected return of at least 22%, without shorting.


4

Using the same notation as in the GMS case, the percent return on the portfolio is represented by the random variable R.

In this model, xi is the proportion of the portfolio (i.e. a number between zero and one) allocated to investment i. (In the GMS case, we used thousands of dollars as the units.)

Each investment i has a percent return under each scenario j, which we represent with the symbol rij.

5

1iiixrR


5

W e c a l c u l a t e t h e p e r c e n t r e t u r n o n e a c h o f t h e s t o c k s i n e a c h y e a r : Y e a r F o r d L i l l y K e l l o g g M e r c k H P 1 9 8 4 0 . 0 7 6 0 . 1 4 0 0 . 2 3 6 0 . 0 4 0 - 0 . 2 0 1 1 9 8 5 0 . 2 7 1 0 . 6 9 0 0 . 7 3 8 0 . 4 5 8 0 . 0 8 5 1 9 8 6 0 . 4 5 5 0 . 3 3 2 0 . 4 8 9 0 . 8 0 8 0 . 1 4 0 1 9 8 7 0 . 3 4 0 0 . 0 5 0 0 . 0 1 2 0 . 2 8 0 0 . 3 9 1 1 9 8 8 0 . 3 4 0 0 . 0 9 6 0 . 2 2 7 0 . 0 9 3 - 0 . 1 8 9 1 9 8 9 - 0 . 1 3 6 0 . 6 0 2 0 . 0 5 2 0 . 3 4 2 - 0 . 3 2 5 1 9 9 0 - 0 . 3 9 0 0 . 0 6 9 0 . 1 2 2 0 . 1 6 0 0 . 7 8 8 1 9 9 1 0 . 0 5 6 0 . 1 4 0 0 . 7 2 3 0 . 8 5 2 0 . 2 2 6

F o r e x a m p l e , F o r d w e n t f r o m $ 1 4 . 3 1 t o $ 1 5 . 2 1 i n 1 9 8 4 , s o t h e r e t u r n o n F o r d s t o c k i n 1 9 8 4 w a s :

076.013.14

13.1421.15

0

011

S

SSr j

j


6

The portfolio return under any scenario j is given by:

5

1iiijj xrR


7

Let Pj represent the probability of scenario j occurring.

The expected value of R is given by:

8

1jjjR PR

8

1

2

jjRjR PR

The standard deviation of R is given by:


8

In this model, each scenario is considered to have an equal probability of occurring, so we can simplify the two expressions:

8

8

1 j

j

R

R

8

8

1

2

j

Rj

R

R


9

Decision VariablesWe need to determine the proportion of our portfolio to invest in each of the five stocks.

ObjectiveMinimize risk.

ConstraintsAll of the money must be invested. (1)The expected return must be at least 22%.(2)No shorting. (3)

Managerial Formulation


10

Mathematical FormulationDecision Variablesx1, x2, x3, x4, and x5 (corresponding to Ford, Lilly, Kellogg, Merck, and HP). ObjectiveMinimize Z = Constraints

(1)

(2)

For all i, xi ≥ 0 (3)

8

8

1

2

j

Rj

R

R

0.15

1

i

ix

22.08

8

1 j

j

R

R


11

123

45678910111213141516171819202122232425262728

A B C D E F G H I JTotal Ford Lilly Kellogg Merck HP

Exp Return = 0.127 1 1.000 0.000 0.000 0.000 0.000StDev = 0.265

req. return = 0.220 Historical dataYear Ford Lilly Kellogg Merck HP1983 14.13 14.47 8.09 5.02 42.381984 15.21 16.50 10.00 5.22 33.881985 19.33 27.88 17.38 7.61 36.751986 28.13 37.13 25.88 13.76 41.881987 37.69 39.00 26.19 17.61 58.251988 50.50 42.75 32.13 19.25 47.251989 43.63 68.50 33.81 25.83 31.881990 26.63 73.25 37.94 29.96 57.001991 28.13 83.50 65.38 55.50 69.88

Historical data on returnsreturn deviation^2 Year Ford Lilly Kellogg Merck HP0.076 0.003 1984 0.076 0.140 0.236 0.040 -0.2010.271 0.021 1985 0.271 0.690 0.738 0.458 0.0850.455 0.108 1986 0.455 0.332 0.489 0.808 0.1400.340 0.045 1987 0.340 0.050 0.012 0.280 0.3910.340 0.045 1988 0.340 0.096 0.227 0.093 -0.189-0.136 0.069 1989 -0.136 0.602 0.052 0.342 -0.325-0.390 0.267 1990 -0.390 0.069 0.122 0.160 0.7880.056 0.005 1991 0.056 0.140 0.723 0.852 0.226

mean 0.127 0.265 0.325 0.379 0.114stdevp 0.265 0.235 0.271 0.290 0.341

=AVERAGE(B19:B26)

=SQRT(AVERAGE(C19:C26))

=SUM(G2:K2)

=SUMPRODUCT($F$2:$J$2,F19:J19)

=(B19-$C$2)^2


12

The decision variables are in F2:J2.

The objective function is in C3.

Cell E2 keeps track of constraint (1).

Cells C2 and C5 keep track of constraint (2).

Constraint (3) can be handled by checking the “assume non-negative” box in the Solver Options.


13


14

123

45678910111213141516171819202122232425262728

A B C D E F G H I JTotal Ford Lilly Kellogg Merck HP

Exp Return = 0.220 1 0.173 0.426 0.054 0.105 0.241StDev = 0.128

req. return = 0.220 Historical dataYear Ford Lilly Kellogg Merck HP1983 14.13 14.47 8.09 5.02 42.381984 15.21 16.50 10.00 5.22 33.881985 19.33 27.88 17.38 7.61 36.751986 28.13 37.13 25.88 13.76 41.881987 37.69 39.00 26.19 17.61 58.251988 50.50 42.75 32.13 19.25 47.251989 43.63 68.50 33.81 25.83 31.881990 26.63 73.25 37.94 29.96 57.001991 28.13 83.50 65.38 55.50 69.88

Historical data on returnsreturn deviation^2 Year Ford Lilly Kellogg Merck HP0.042 0.032 1984 0.076 0.140 0.236 0.040 -0.2010.450 0.053 1985 0.271 0.690 0.738 0.458 0.0850.366 0.021 1986 0.455 0.332 0.489 0.808 0.1400.205 0.000 1987 0.340 0.050 0.012 0.280 0.3910.076 0.021 1988 0.340 0.096 0.227 0.093 -0.1890.194 0.001 1989 -0.136 0.602 0.052 0.342 -0.3250.175 0.002 1990 -0.390 0.069 0.122 0.160 0.7880.253 0.001 1991 0.056 0.140 0.723 0.852 0.226

mean 0.127 0.265 0.325 0.379 0.114stdevp 0.265 0.235 0.271 0.290 0.341


15

Invest 17.3% in Ford, 42.6% in Lilly, 5.4% in Kellogg, 10.5% in Merck, and 24.1% in HP.

The expected return will be 22%, and the standard deviation will be 12.8%.

Conclusions


16

2. Show how the optimal portfolio changes as the required return varies.


17

3031323334353637383940414243444546474849505152535455

A B C D E F G HRequired Return Risk Return Ford Lilly Kellogg Merck HP

0.000 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.010 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.020 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.030 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.040 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.050 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.060 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.070 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.080 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.090 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.100 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.110 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.120 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.130 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.140 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.150 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.160 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.170 0.115 0.179 0.289 0.407 0.000 0.000 0.3040.180 0.115 0.180 0.285 0.413 0.000 0.000 0.3020.190 0.116 0.190 0.249 0.430 0.029 0.007 0.2860.200 0.119 0.200 0.224 0.429 0.038 0.039 0.2710.210 0.123 0.210 0.198 0.428 0.046 0.072 0.2560.220 0.128 0.220 0.173 0.426 0.054 0.105 0.2410.230 0.133 0.230 0.148 0.425 0.063 0.138 0.2260.240 0.139 0.240 0.122 0.424 0.071 0.171 0.211


18

Optimal Portfolio

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

15% 20% 25% 30% 35%

Required Return

Pro

po

rtio

n o

f P

ort

folio

Lilly

HP

Merck

Kellogg

Ford


19

3. Draw the efficient frontier for portfolios composed of these five stocks.


20

Efficient Frontier

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Risk (Standard Deviation)

Ex

pe

cte

d R

etu

rn

Lilly

FordHP

Kellogg

Merck


21

Repeat Part 2 with shorting allowed.


22

75767778798081828384858687888990919293949596979899100

A B C D E F G HRequired Return Risk Return Ford Lilly Kellogg Merck HP

0.000 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.010 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.020 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.030 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.040 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.050 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.060 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.070 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.080 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.090 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.100 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.110 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.120 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.130 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.140 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.150 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.160 0.114 0.166 0.311 0.432 0.009 -0.074 0.3220.170 0.114 0.170 0.300 0.432 0.013 -0.059 0.3150.180 0.115 0.180 0.274 0.431 0.021 -0.026 0.3000.190 0.116 0.190 0.249 0.430 0.029 0.007 0.2860.200 0.119 0.200 0.224 0.429 0.038 0.039 0.2710.210 0.123 0.210 0.198 0.428 0.046 0.072 0.2560.220 0.128 0.220 0.173 0.426 0.054 0.105 0.2410.230 0.133 0.230 0.148 0.425 0.063 0.138 0.2260.240 0.139 0.240 0.122 0.424 0.071 0.171 0.211


23

Efficient Frontier

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Risk (Standard Deviation)

Ex

pe

cte

d R

etu

rn

Lilly

FordHP

Kellogg

Merck


24

GMS Case Revisited

Assuming that Torelli's goal is to minimize the standard deviation of the portfolio return, what is the optimal portfolio that invests all $10 million?


25

FormulationDecision Variables

The decision variables are four amounts: x1, x2, x3, and x4, representing GMS stock, Put Option A, Put Option B, and Put Option C, respectively.

Objective

Minimize Z =

7

1

2

jjRjR PR

Constraints

000,104

1

i

ix

0ix for all investments i.


26

Optimal Solution

1234567891011121314151617181920

21222324

A B C D E F G H I J K LGMS price 100

Scenarios for GMS stock in one monthScenario GMS price Probability GMS Put A Put B Put C Return Sqdev

1 150 0.05 50% -100% -100% -100% 2446 52016742 130 0.10 30% -100% -100% -100% 786 3859783 110 0.20 10% -100% -100% -100% -873 10778104 100 0.30 0% -100% -100% -20% 198 10685 90 0.20 -10% -100% 56% 60% 230 42376 80 0.10 -20% 355% 213% 140% 225 35437 70 0.05 -30% 809% 369% 220% 219 2912

Put options on GMS stock that expire in one monthOption A B CStrike price 90 100 110Option price $2.20 $6.40 $12.50

Investment decision (thousands of dollars spent on each investment) Return from portfolio ($1000)GMS Put A Put B Put C Total Budget Mean 1658297 -8 -665 2376 10000 = 10000 Stdev 718

Units of investments purchased (shares for GMS, number of puts for options)GMS Put A Put B Put C

82972 -3798 -103843 190058

Returns from one unit of each investment Portfolio

Now the nonnegativity conditions for the changing cells is removed, and the investor sells short on the put A and B options. This lowers the standard deviation of the portfolio (and also increases its mean).


27

Efficient Frontier for GMSGMS Risk vs. Return

$-

$100

$200

$300

$400

$500

$600

$- $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 $2,000

Std Dev of Return (x 1000)

Ex

pe

cte

d R

etu

rn (

x 1

00

0)

Minimum Risk - No ShortingGMS Stock Only

Minimum Risk with Shorting

"One-for-One"

Efficient Frontier with Shorting

Efficient Frontier - No Shorting


28

Parametric Approach Revisited

(a) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with no shorting of stocks allowed.

(b) Draw the efficient frontier for portfolios composed of these three stocks.

(c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with no shorting of stocks allowed.

From Session 5a:


29

Formulation

Objective

Minimize Z = 3,1313,2322,12123

23

22

22

21

21 222 COVxxCOVxxCOVxxxxx

Constraints

12.0332211 xxx (1)

0.13

1

i

ix (2)

For all i, 0ix (3)


30

Optimal Solution

123456789

10111213

14

15161718192021

A B C D E F G H I JStock 1 Stock 2 Stock 3

Mean return 0.140 0.110 0.100Variance of return 0.200 0.080 0.180

StDev of return 0.447 0.283 0.424

Correlations CovariancesStock 1 Stock 2 Stock 3 Stock 1 Stock 2 Stock 3

Stock 1 1.00 0.80 0.70 Stock 1 0.2000 0.1012 0.1328Stock 2 0.80 1.00 0.90 Stock 2 0.1012 0.0800 0.1080Stock 3 0.70 0.90 1.00 Stock 3 0.1328 0.1080 0.1800

Investment decisionStock 1 Stock 2 Stock 3 Total Required

Fractions to invest 0.333 0.667 0.000 1 = 1

Expected portfolio returnActual Required0.120 >= 0.120

Portfolio variance 0.103Portfolio stdev 0.321


31

SolverTable


32


33

SolverTable Output3

45678910111213141516171819202122

A B C D E FInput (cell $D$18) values along side, output cell(s) along top

Stock 1 Stock 2 Stock 3 Ret

urn

Ris

k

0.100 -0.244 1.861 -0.616 0.109 0.2400.101 -0.244 1.861 -0.616 0.109 0.2400.102 -0.244 1.861 -0.616 0.109 0.2400.103 -0.244 1.861 -0.616 0.109 0.2400.104 -0.244 1.861 -0.616 0.109 0.2400.105 -0.244 1.861 -0.616 0.109 0.2400.106 -0.244 1.861 -0.616 0.109 0.2400.107 -0.244 1.861 -0.616 0.109 0.2400.108 -0.244 1.861 -0.616 0.109 0.2400.109 -0.240 1.859 -0.619 0.109 0.2400.110 -0.212 1.849 -0.637 0.110 0.2400.111 -0.185 1.840 -0.655 0.111 0.2410.112 -0.158 1.830 -0.673 0.112 0.2410.113 -0.130 1.821 -0.691 0.113 0.2420.114 -0.103 1.811 -0.709 0.114 0.2440.115 -0.075 1.802 -0.726 0.115 0.2450.116 -0.048 1.792 -0.744 0.116 0.2470.117 -0.021 1.783 -0.762 0.117 0.249


34

Efficient Frontier

8%

9%

10%

11%

12%

13%

14%

15%

16%

20% 25% 30% 35% 40% 45% 50%

Risk

Ex

pe

cte

d R

etu

rn

Stock 2

Stock 1

Stock 3


35

Parametric Approach, cont.

(c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with shorting of stocks allowed.

All we need to do here is remove the non-negativity constraint and re-run SolverTable.


36

Efficient Frontier

8%

9%

10%

11%

12%

13%

14%

15%

16%

20% 25% 30% 35% 40% 45% 50%

Risk

Ex

pe

cte

d R

etu

rn

Stock 2

Stock 1

Stock 3


37

Preemptive Goal Programming:Consulting Example

The Touche Young accounting firm must complete three jobs during the next month. Job 1 will require 500 hours of work, job 2 will require 300 hours, and job 3 will require 100 hours. At present the firm consists of five partners, five senior employees, and five junior employees, each of whom can work up to 40 hours per month.


38

The dollar amount (per hour) that the company can bill depends on the type of accountant assigned to each job, as shown in the table below. (The "X" indicates that a junior employee does not have enough experience to work on job 1.)

Job 1 Job 2 Job 3 Partner 160 120 110 Senior employee 120 90 70 Junior employee X 50 40


39

All jobs must be completed.

Touche Young has also set the following goals, listed in order of priority:• Goal 1: Monthly billings should exceed $74,000.

• Goal 2: At most one partner should be hired.

• Goal 3: At most three senior employees should be hired.

• Goal 4: At most one junior employee should be hired.


40

Decision Variables

There are three types of decisions here.

First, we need to decide how many people to hire in each of the three employment categories.

Second, we need to assign the available human resources (which depend on the first set of decisions) to the three jobs.

Finally, since it is not apparent that we will be able to satisfy all of Touche Young’s goals, we need to decide which goals not to meet and by how much.

Managerial Formulation


41

Objective

In the long run we want to minimize any negative difference between actual results and each of the four goals. Of course, our optimization methods require that we only have one objective at a time, so we will use a variation of goal programming to solve the problem four times.

The approach here will be to treat each of the goals as an objective until it is shown to be attainable, after which we will treat it as a constraint. For example, we will solve the model with the goal of minimizing any shortfall in the $74,000 revenue target. Once we find a solution that has no shortfall, we will solve the problem again, with an added constraint that the shortfall be zero.


42

ConstraintsThe numbers of people hired must be integers.

(1)

Each project must receive its required number of man-hours. (2)

Our model must take into account any difference between the actual performance of the plan and the four targets.(3)

We can’t assign people to jobs unless we hire them.(4)


43

Define tk to be the target amount for goal k. For example, the target for goal 1 (the billing goal) is $74,000; therefore t1 = 74,000.

Define δk to be the “negative difference” between what we have achieved and the target for “current” goal k. (δ is the Greek letter delta.) In the case of the billing goal, the negative difference would be any amount less than 74,000. In the case of goal 2, the negative difference would be any amount of new partners hired greater than 1. If our current solution yields $60,000 in billings, then δ1 = 74,000 – 60,000 = 14,000.

Define vk to be the best value for goal k that has previously been achieved in our model. For example, if the best solution we can find yields only $60,000 in billings, then v1 = 14,000.

Define xi to be the number of new hires of type i.

Define Aij to be the number of man-hours of type i assigned to job j.

Define Rij to be the number of man-hours of type i required for job j.

Mathematical Formulation


44

Decision Variables

xi (three decisions), Aij (nine decisions), δk (up to three decisions)

Objective

Minimize Z =

Constraints

All xi are integers. (1)

Aij = Rij for all i, j. (2)

δ Goals ≠ k = vk for all goals < k (3)

for all i. (4)

k

ij

ij xA 403

1


45

1234567891011121314151617181920

A B C D E F G H I J K LAssigned Hours Job 1 Job 2 Job 3 Assigned Available Billling Rate Job 1 Job 2 Job 3

Partners 0 0 0 0 <= 200 Partners 160 120 110Seniors 0 0 0 0 <= 200 Seniors 120 90 70Juniors 0 0 0 0 <= 200 Juniors NA 50 40

Assigned 0 0 0= = = Present staff Added Total

Needed 500 300 100 Partners 5 0 5Seniors 5 0 5

Goals Actual Under Over Net Goal Juniors 5 0 5Billings 0 0 0 0 = 74000

Partners Hired 0 0 0 0 = 1 Hours/month 40Seniors Hired 0 0 0 0 = 3Juniors Hired 0 0 0 0 = 1

Deviation Already attained PriorityBillings 0 <= 0 1

Partners Hired 0 <= 0 2Seniors Hired 0 <= 0 3Juniors Hired 0 <= 0 4

=SUM(B4:D4)

=$J$11*L9

=C10

=D11

=SUMPRODUCT(J2:L4,B2:D4) =B10+C10-D10

=J7+K7

=SUM(D2:D4)


46

The tk targets are in cells G10:G13.

The δk “negative difference” variables will be in B16:B19. We use the range C10:D13 to track all deviations (both positive and negative), and then refer to the “undesirable” one in B16:B19. For example, it is undesirable to have billings under 74,000, so B16 refers to C10. It is undesirable for the number of new partners to be over 1, so B17 refers to D11.

The vk “best achieved” variables will be in D16:D19.

The xi are in K7:K9.

The Aij assignments are in B2:D4.

The Rij requirements are in B7:D7.

Cells G2:G4 keep track of constraint (4).

We constrain B4 to be zero.


47

First Iteration At first we’ll ignore all of the goals except the billing target of $74,000.

Decision Variables

xi (three decisions, cells K7:K9), Aij (nine decisions, cells B2:D4)

Objective

Minimize Z = 1d (the shortfall, if any, between planned billings and $74,000)

Constraints



A11 = 0. (4)

ij

ij xA 403

1

£å=

for all i. (5)

Note that constraint (3) doesn’t matter in this iteration.

Also note the balance equation constraint, forcing E10 = G10.


48


49

We have verified that it is feasible to have billings of $74,000.

1

234

56

789

10

111213141516

171819










50

Second Iteration M eeting the first goal w ill now be a constraint, and w e’ll focus on the second goal.

D ecision V ariables

xi (as before), A i j (as before), δk (for the billings goal)

Objective

M inimize Z = 2 (the number of partners hired more than the goal of 1)

Constraints

A ll xi are integers. (1)

A i j = R i j for all i, j. (2)

δ 1 = v1 (forcing the 74,000 bill ing goal to be met) (3)

A 11 = 0. (4)

ij

ij xA 403

1

for all i. (5)

N ote that v1 = 0 from the previous iteration.


51


52

Now we know that it is feasible to achieve both of the first two goals.

1

234

56

789

1011

121314151617

1819





Goals Actual Under Over Net Goal Juniors 5 5 10

Billings 79000 0 5000.000171 74000 = 74000Partners Hired 1 0 0 1 = 1 Hours/month 40

Seniors Hired 13 0 0 13 = 3Juniors Hired 5 0 0 5 = 1


Partners Hired 0 <= 0 2

Seniors Hired 0 <= 0 3Juniors Hired 0 <= 0 4


53

Third Iteration

Meeting both the first and second goals will now be constraints, and we’ll focus on the third goal. Decision Variables

xi (three decisions), Aij (nine decisions), δk (two decisions)

Objective

Minimize Z = 3d (the number of senior employees hired above the goal of 3)

Constraints



δ Goals k = vk (for goals 1 and 2) (3)

A11 = 0. (4)

ij

ij xA 403

1

£å=

for all i. (5)

Note that v1 = v2 = 0 from the previous iteration.


54


55

All of the first three goals are feasible.

1

234

56

789

101112

131415161718

19


Partners 195.25 6E-07 6.2152E-07 195.25 <= 200 Partners 160 120 110Seniors 304.75 0 0 304.75 <= 320 Seniors 120 90 70Juniors 0 300 99.99999938 400 <= 400 Juniors NA 50 40




Billings 86810 0 12809.99833 74000 = 74000Partners Hired 0 1 0 1 = 1 Hours/month 40Seniors Hired 3 0 0 3 = 3

Juniors Hired 5 0 0 5 = 1


Partners Hired 0 <= 0 2Seniors Hired 0 <= 0 3

Juniors Hired 0 <= 0 4


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Fourth Iteration Meeting all of the first three goals will now be constraints, and we’ll focus on the

fourth goal.

Decision Variables

xi (three decisions), Aij (nine decisions), δk (for the first three goals)

Objective

Minimize Z = 4d (the number of new juniors hired above the goal of 1)

Constraints



δ Goals k = vk for goals 1, 2, 3 (3)

A11 = 0. (4)

ij

ij xA 403

1

£å=

for all i. (5)

Note that v1 = v2 = v3 = 0 from the previous iteration.


57


58

1

234

56

789

10111213

141516171819






Billings 89990 0 15990 74000 = 74000Partners Hired 1 0 0 1 = 1 Hours/month 40Seniors Hired 3 0 0 3 = 3Juniors Hired 4 0 3 1 = 1




59

F ifth I teration W e can do one more round here. G iven that w e are going to hire 1 partner, 3 seniors, and 4 juniors, w hy not maximize the revenue from that combination of w orkers?

D ecision V ariables

xi (three decisions), A i j (nine decisions), δk (for the 2nd, 3rd, and 4th goals)

O bjective

M aximize Z = R evenue

C onstraints

A ll xi are integers. (1)

A i j = R i j for all i, j. (2)

δk = vk for goals 2, 3, 4 (3)

A 11 = 0. (4)

ij

ij xA 403

1

for all i. (5)

N ote that v2 = v3 = 0, and v4 = 3, from the prev ious iteration.


60


61

1

234

56

789

10

111213

141516171819









There are two constraints that don’t show in the window: $E$2:$E$4<=$G$2:$G$4, and $K$7:$K$9 = integer.


62

It would appear that it is infeasible to reach all four of the goals.

However, the first three can be met while delivering all of the jobs as required.

The only goal that can’t be attained is the limit of hiring only one new junior-level employee. The best solution we found requires us to hire four new employees at this level.

It is possible to have revenues of $91,000 under this hiring plan.

Conclusions


63

Decision Analysis is a tool for studying situations in which there are several decision alternatives and a set of uncertain future events.

Decision Alternatives are elements of a set of possible choices, represented by

d1, d2, d3, … di

States of Nature are elements of a set of N random future events, represented by

s1, s2, s3, … sN

A Payoff Table lists outcomes associated with some combination of decision alternative and state of nature. The payoff for decision alternative i under state of nature j is symbolized by vij.

Decision analysis is a natural extension of our previous work with conditional probability and systems of probabilities and payoffs; the only new element here is the opportunity for a decision maker to make choices at certain discrete points in time.


64

A buyer for a large department store chain must place orders with an athletic shoe manufacturer 6 months prior to the time the shoes will be sold in the department stores. In particular, the buyer must decide on November 1 how many pairs of the manufacturer's newest model of tennis shoes to order for sale during the upcoming summer season.

Assume that each pair of this new brand of tennis shoes costs the department store chain $45 per pair. Furthermore, assume that each pair of these shoes can then be sold to the chain's customers for $70 per pair. Any pairs of these shoes remaining unsold at the end of the summer season will be sold in a closeout sale next fall for $35 each. Finally, assume that the department store chain must purchase these tennis shoes from the manufacturer in lots of 100 pairs.


65

The probability distribution of consumer demand for these tennis shoes (in hundreds of pairs) during the upcoming summer season has been assessed by market research specialists and is provided in the table below.

Consumer Demand Probability 1 0.05 2 0.15 3 0.25 4 0.30 5 0.15 6 0.10


66

B a s i c a l l y , t h e c o n t r i b u t i o n i s g i v e n b y t h e f o l l o w i n g f o r m u l a , i n w h i c h p r o fi t i s s y m b o l i z e d b y t h e G r e e k l e t t e r π , p u r c h a s e q u a n t i t i e s ( i n h u n d r e d s ) a r e r e p r e s e n t e d b y p a n d d e m a n d s t a t e s ( i n h u n d r e d s ) a r e r e p r e s e n t e d b y d .

i j = R e v e n u e f r o m R e g u l a r P r i c e + R e v e n u e f r o m C l o s e o u t P r i c e – C o s t t o P u r c h a s e

45*100*,0max*100*35$,min*100*70$ ijiji pdpdp

F o r e x a m p l e , i f w e p u r c h a s e 4 0 0 s h o e s a n d d e m a n d i s 2 0 0 , t h e n p = 4 a n d d = 2 .

i j 45*100*,0max*100*35$,min*100*70$ ijiji pdpdp

45*100*42*100*35$2*100*70$

000,18$000,7$000,14$

000,3$


67

1234

5

678910111213141516

A B C D E F G H$45 Cost of each pair of new tennis shoes$70 Selling price of each pair of shoes$35 Closeout sale price of each leftover pair

Purchased 1 2 3 4 5 61 $2,500 $2,500 $2,500 $2,500 $2,500 $2,5002 $1,500 $5,000 $5,000 $5,000 $5,000 $5,0003 $500 $4,000 $7,500 $7,500 $7,500 $7,5004 ($500) $3,000 $6,500 $10,000 $10,000 $10,0005 ($1,500) $2,000 $5,500 $9,000 $12,500 $12,5006 ($2,500) $1,000 $4,500 $8,000 $11,500 $15,000

Probability 0.05 0.15 0.25 0.3 0.15 0.1

Payoff TableConsumer Demand

=($A$2*100*MIN(G$7,$A13))+($A$3*100*MAX(0,($A13-G$7)))-($A13*$A$1*100)


68

F o r each p o ssib le str ateg y , w e can calcu late th e exp ected r ev en u e an d th e stan d ar d d ev iatio n o f r ev en u e.

5

6789

1 01 11 21 31 41 51 61 71 8

A B C D E F G H I J K

P u r c h a s e d 1 2 3 4 5 6 E x p e c te d R e v e n u e S td D e v o f R e v e n u e V a r ia n c e1 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 0 02 $ 1 ,5 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 4 ,8 2 5 $ 1 8 4 3 3 9 9 3 .7 53 $ 5 0 0 $ 4 ,0 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 6 ,6 2 5 $ 6 2 5 3 9 0 4 6 8 .84 ( $ 5 0 0 ) $ 3 ,0 0 0 $ 6 ,5 0 0 $ 1 0 ,0 0 0 $ 1 0 ,0 0 0 $ 1 0 ,0 0 0 $ 7 ,5 5 0 $ 1 ,1 9 7 1 4 3 2 0 2 55 ( $ 1 ,5 0 0 ) $ 2 ,0 0 0 $ 5 ,5 0 0 $ 9 ,0 0 0 $ 1 2 ,5 0 0 $ 1 2 ,5 0 0 $ 7 ,4 2 5 $ 1 ,4 6 7 2 1 5 3 2 4 46 ( $ 2 ,5 0 0 ) $ 1 ,0 0 0 $ 4 ,5 0 0 $ 8 ,0 0 0 $ 1 1 ,5 0 0 $ 1 5 ,0 0 0 $ 6 ,7 7 5 $ 1 ,6 1 3 2 6 0 2 8 1 9

P r o b a b il i t y 0 .0 5 0 .1 5 0 .2 5 0 .3 0 .1 5 0 .1

P a y o f f T a b leC o n s u m e r D e m a n d

= S U M P R O D U C T ( B 1 3 :G 1 3 ,$ B $ 1 4 :$ G $ 1 4 )

= ( ( $ B $ 1 4 * ( B 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ C $ 1 4 * ( C 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ D $ 1 4 * ( D 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ E $ 1 4 * ( E 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ F$ 1 4 * ( F1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ G $ 1 4 * ( G 1 3 - H 1 3 ) ) ^ 2 )

= S Q R T ( J 1 3 )

T h e v ar ian ce fo r m u la lo o k s u g ly , bu t i t w o r k s.


69

Risk Profile

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800

Std. Deviation of Revenue

Ex

pe

cte

d R

ev

en

ue

Buy 100 Pairs

Buy 200 Pairs

Buy 300 Pairs

Buy 400 Pairs Buy 500 Pairs

Buy 600 Pairs


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SummaryMultiple Objective Optimization• Two Dimensions• More than Two Dimensions• Finance and HR Examples• Efficient Frontier• Pre-emptive Goal Programming

Intro to Decision Analysis

session 6a. decision models -- prof. juran2 overview multiple objective optimization two dimensions...

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