Monthly Archives: July 2015

This is where the story of America’s first black president, Barack Obama, begins.

The Kenyan trade unionist turned politician Tom Mboya, who studied at Ruskin College, Oxford, wanted members of his country’s government to be adequately prepared for a post-colonial Kenya.

He recognised quite early on that there were not enough professional Africans to run an efficient civil service.

To make up the shortfall, he set up a scholarship fund that would take young bright Africans to the US and Canada.

The idea was for them to acquire the necessary skills and come back to help build a new country when the white civil servants packed up and returned to Europe.

One of those students was one Barack Obama from Kogelo, near the port city of Kisumu on the shores of Lake Victoria.

Barack Obama Senior was the first African student to study at the University of Hawaii.

There he fell in love and married a young American woman, Ann Dunham. They had a son named after his father, Barack Hussein Obama II.

Cut short

I travelled to Mr Obama’s home in Kogelo where I met his step-grandmother Sarah Obama and aunt Marsat Onyango Obama to find out what the scholarship meant for the family at the time.

We stood in the family’s small graveyard, next to the grave of Barack Obama Senior.

Ms Onyango told me that even though she had not yet been born when Mr Obama left, she knew that the family had been very proud of him.

“They said he carried their hopes and dreams.”

Tom Mboya was assassinated in central Nairobi 1969 at the age of 38.

He was minister of economic development and planning but the motive for his killing remains a mystery.

He had achieved a lot for his young age – his intellectual prowess and eloquence meant that he could articulate his vision clearly to others.

As a sign of his global significance he had appeared on the cover of Time magazine, the first Kenyan to have done so.

She replied in a quiet, confident voice: “My father could not have known that by helping one person to go to school, he was going to change the lives of so many people.

“Barack Obama has achieved a lot personally and it’s great that he is the president of the United States.

“But I think the biggest benefit that Barack Obama has brought is how he has inspired young people all over the world… and to me that is the huge thing that has come out of the scholarships.”

I asked her why she had followed in her father’s footsteps by setting up her own programme.

“I want to finish what my father started but I want to finish it in a way that brings balance to the leadership in this country.”

Thirsty man

The airlift scholarship also took the Kenyan newspaper columnist Philip Ochieng to America.

He studied a Bachelor of Arts in Literature at Chicago’s Roosevelt University.

The 76-year-old knew Mr Obama Senior very well and recalls that he was clever.

He told me in Nairobi that they used to drink whisky together.

As he put it: “America watered my thirst for knowledge.”

What if

The scholarship programme educated nearly 800 students from Kenya and elsewhere in East Africa.

Other scholars included the late Wangari Mathaai, who became the first African woman to become a Nobel Peace Prize laureate, and the late geneticist Reuben Olembo, who became a director at the United Nations environment agency, Unep.

The partner, on the other side of the Atlantic, was an American entrepreneur William X Scheinman, who was a good friend of Tom Mboya’s, and they received President John F Kennedy’s support.

Musician and activist Harry Belafonte and actor Sidney Poitier raised funds for the scholarship fund, amongst many others.

Mr Belafonte later wrote about Mr Obama Senior’s scholarship: “Imagine: perhaps, if not for support from the African American Students’ Foundation, he might not have come to America,” he said.

“Then who would be in the White House today?”

Mr Obama Senior, who also graduated from Harvard, returned to Kenya in 1968 and eventually worked for the government as an economist.

He died in a car crash in 1982.

But not before he had visited his son in Hawaii.

He gave his son his first basketball as a gift and took him to his first jazz concert, where the maestro pianist and composer Dave Brubeck was performing.

And as I left her office in Nairobi, which boasts pictures of when she met President Obama, Ms Mboya reminded me “it doesn’t take much to change a life.”

Source: http://www.bbc.com/news/world-africa-33629577

Explain which decision you would make in the light of each prediction xj , this is your policy.

EPM945 OPTIMIZATION AND DECISION MAKING – COURSEWORK

Assignment set on Thursday, 28 November 2013.

Completed assignment, to be handed in by

Thursday, 23 January 2014 at the Postgraduate Office by 4.00 pm.

Late submissions may be penalised.

Answer all questions. Show all necessary working.

 

Question 1

Use the simplex method to solve the following simplex problems.

  1. a) Maximise 2×1 + 2×2 + 3×3 subject to

x1 + 3×2 + 2×3 ≤ 6,

3×1 + x2 + 3×3 ≤ 10

x1 + x2 + x3 ≤ 4,

x1, x2, x3 ≥ 0.

 

  1. b) Maximise −3×1 − x2 − 5×3 subject to

−x1 + 4×2 + 2×3 ≤ 0,

−2×1 + x2 − 3×3 ≤ −2

7×2 + x3 ≤ 1,

x1, x2, x3 ≥ 0.

 

  1. c) Maximise 4×1 + 3×2 + 5×3 + x4 subject to

x1 + x2 + x3 + x4 ≤ 6,

3×1 + x3 = 6

4×2 − x3 + 2×4 = 1,

x1 unrestricted, x2 , x3, x4 ≥ 0.

(15 marks)

Question 2

Consider the primal linear programming problem below.

Maximise 3×1 + 4×2 + 6×3 , subject to

2×1 + 3×2 + 5×3 ≤ 15,

3×1 + 2×2 + 5×3 ≤ 12,

x1, x2, x3 ≥ 0.

  1. a) Construct the dual problem and solve it graphically.
  2. b) Find a solution to the primal problem, stating clearly any results that you use. (12 marks)

 

Question 3

Three car showrooms A, B and C are supplied by cars from three ports, 1, 2 and 3. The table below gives the cost, in pounds, of transporting a car from each port to each showroom.

 

Ports 1, 2 and 3 have stocks of 150, 100 and 200 cars respectively. A, B and C require 100, 125 and 175 cars respectively.

The aim is to minimise the total cost of transporting the cars.

 

                A             B             C

1              45           90           40

2              40           60           20

3              50           80           30

 

  1. a) Formulate, but do not solve, the problem in the form of a linear program.
  2. b) Now write the problem in the form of a transportation array, explaining how you deal with the fact that the total demand and the total supply are not equal.
  3. c) Find the optimal transportation solution.

(15 marks)

 

Question 4

Five candidates for five jobs are given an aptitude test for each job. The scores for the five candidates and the five jobs are given as follows;

 

Cand/Job             1              2              3              4              5

1                              23           19           18           18           16

2                              21           29           29           10           19

3                              12           16           17           6              21

4                              13           24           26           25           18

5                              23           17           22           8              22

 

Each of the five candidates must be assigned exactly one job. Find the optimal job assignments.

(10 marks)

 

Question 5

Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate µi with variance σ2 for i = 1, . . . , n, and the covariance of the revenues of stocks i and j is given by σij for i = j, i, j = 1, . . . , n. The proportion of stock i in the portfolio is denoted by xi .

 

  1. a) Showing your working carefully, show that the expected revenue from the port- folio is ∑n xi µi , and find an expression for the variance of the portfolio revenue, again showing your working carefully.

 

  1. b) Still for a general number of n stocks, formulate this as an optimization problem using Lagrange multipliers, and find a set of linear equations for the optimal values of the xi s. You do not need to solve the problem at this stage.
  2. c) Now consider a problem with three stocks, where the means, variances and co-variances are as follows:

µ1 = 0.12, µ2 = 0.11, µ3 = 0.08, σ2 = 0.5, σ2 = 0.3, σ2 = 0.1, σ12 = −0.2, σ13 = 0.1 and σ23 = 0.1.

Find the optimal portfolio (i.e. the one with the minimum variance) for the expected rate of return of 0.11.

  1. d) Now suppose that the target rate of return is reduced to 0.10. Find the optimal portfolio in this case.

(24 marks)

Question 6

Read the following case study based on the Applegold Cider Company. You should then apply decision analysis to the problem facing Applegold. This will involve:

  1. a) Formulating the problem as a decision tree;
  2. b) using Bayesian analysis to update prior probabilities and find the correct probabilities in the tree;
  3. c) discussing the strengths and limitations of your analysis.

 

Applegold is a major cider producer, producing draught cider for pubs and clubs, as well as bottles and cans, which has recently seen production and the number of outlets selling its products increase significantly.

 

The growth in draught cider has created some problems for the company’s managers. In particular, there is concern that when sales reach their peak in August, there might not be enough kegs (steel re-usable cider containers) available to meet demand. Applegold own about 150000 kegs, but it is felt by some managers that the stock should be increased.

 

The Operations Manager has proposed that 10000 new kegs be ordered immediately.   The Accountant was not convinced.   Kegs cost £80 each, so this would lead to an expenditure of £800000. They would be usable next year, but assuming 5% interest on capital, buying now rather than waiting until next year would cost £40000.

 

The Sales Manager proposes that the company wait until an accurate long range weather forecast is available for August,   since demand depends heavily on the weather, with hot dry months leading to high demand.   Such a forecast will be available in July. One problem is that it might be the case that other brewers had bought all of the available kegs by this time and the Operations Manager estimates a probability of 0.75 of getting the kegs if they wait until July.

 

The Sales Manager produces an interim forecast, with no knowledge of the weather, of how good sales are likely to be in August.   She estimates that they will be at least 10% higher with probability 0.5, they will increase by a lower amount with probability 0.3, and there will be no increase with probability 0.2.

The Data Processing Manager suggests three possible strategies; buying 0, 5000 or10000 kegs immediately. The associated change in profit from these three strategies were estimated, based upon the assumption that with a 10% sales increase 10000 extra kegs will be used (if available), and for a lower increase 5000 extra kegs will be used (if available), with an associated profit of £8 per keg. These are summarised in Table 1.

The Sales Manager suggests that it might still be better to wait for the weather forecast in July, and run the risk of the kegs not being available. Whether it is best to do so depends upon how accurate the forecasts are.   The data in Table 2 give some data on the recent performance of the forecasts.

Number of kegs

purchased

Increase on last year

 

nolknono

Up to 10 %

 

Over 10 %

None Up to 10% Over 10%
0 0 0 0
5000 -20000 20000 20000
10000 -40000 0 40000

 

Table 1: Predicted changes in profit (in £) for combinations of different immediate purchasing strategies and sales increases.

Actual increase

over the previous year

Predicted increase

None       Up to 10 %         Over 10 %

Total months
≥ 10%

< 10%

0

2                   5                       16

8                   21                       6

14                   7                       4

23

35

25

 

Table 2: Sales over the last 84 months.

(24 marks)

 

Question 7: Bayes Theorem

 

In considering investing in an asset you commission an advisor to make a prediction of the state of the asset. The advisors predictions are labelled x1 ≡ good, x2 ≡ fair, x3 ≡ poor, x4 ≡ bad. The true condition is labelled θ1 ≡ good, θ2 ≡ fair, θ3 ≡ poor, θ4 ≡ bad.

Your prior assessment of the true condition is given as a probability.

 

P[ѳ1] = 0:1

P[ѳ2] = 0:5

P[ѳ3] = 0:2

P[ѳ4] = 0:2

 

There are two actions, a1 ≡ bid and a2 = do not bid. The utility of the actions when the true state is known is given in Table 1.7. The advisor’s performance is not perfect and is described in Table 1.8 by the probabilities of predicting xi given a true state θj

 

U (a1 | θi ) U (a2 | θi )
θ1

θ2

θ3

θ4

60

10

−20

−60

−10

−10

−10

−10

 

 

Table 1.7: Utility Function

 

 

 

xi is the observed state; θj is the true state
P [x1 |θ1 ] = 0.80

 

P [x1 |θ2 ] = 0.10

 

P [x1 |θ3 ] = 0.00

P [x2 |θ1 ] = 0.10

 

P [x2 |θ2 ] = 0.80

 

P [x2 |θ3 ] = 0.10

P [x3 |θ1 ] = 0.10

 

P [x3 |θ2 ] = 0.10

 

P [x3 |θ3 ] = 0.70

P [x4 |θ1 ] = 0.00

 

P [x4 |θ2 ] = 0.00

 

P [x4 |θ3 ] = 0.20

               P [x1 4 ] = 0.00                       P [x2 4 ] = 0.00     P [x3 4 ] = 0.20     P [x4 4 ] = 0.80

 

Table 1.8: Observation probabilities

 

 

P [xi ]

 

P [θ1 |xi ]

 

P [θ2 |xi ]

 

P [θ3 |xi ]

 

P [θ4 |xi ]

 

x1 x2 x3

x4

 

P [x1 ] =   P [x2 ] =   P [x3 ] =   

P [x4 ] =   

 

P [x1 |θ1 ] =    P [x2 |θ1 ] =    P [x3 |θ1 ] =    

P [x4 |θ1 ] =    

 

P [x1 |θ2 ] =    P [x2 |θ2 ] =    P [x3 |θ3 ] =    

P [x4 |θ2 ] =    

 

P [x1 |θ3 ] =    P [x2 |θ3 ] =    P [x3 |θ3 ] =    

P [x4 |θ3 ] =    

 

P [x1 |θ4 ] =    P [x2 |θ4 ] =    P [x3 |θ4 ] =    

P [x4 |θ4 ] =    

 

Table 1.9: Advisor’s predictions

 

 

  1. Using the probabilities given complete the Table 1.9 of the posterior probabilities P [xi |θj ] of the true states given the predicted states.

 

  1. Using your results in Table 1.9 and the utilities calculate the expected value V (ai |xj ) of action ai given prediction

xj for each prediction and both actions.

 

  1. Explain which decision you would make in the light of each prediction xj , this is your policy.

 

  1. Lastly, calculate the expected utility of the policy.

 

Question 8 (Bayesian Treatment of Binomial Distribution).

 

A company has received several complaints about a specific type of defect on MP3 players it manufactures. The company wishes to determine the proportion p of players sold during the previous month that have the defect. Mystery shoppers buy a random sample of 100 players. The quality manager had assumed the proportion p had a Beta(1, 9) distribution.

 

  1. Determine the posterior distribution for p if there were 30 defects reported in the 100 responses.

 

  1. Estimate the proportion of players that have the defect.

 

  1. Suppose the prior distribution for p is the beta Beta(r, 9r) calculate the posterior distribution for p and sketch a plot of the posterior estimate of p (the posterior mean) against p as r varies from r = 1 to r = 10.

Ethical Issues in Employment

Mary has been working as an administrative assistant for an international company manufacturing home decorations in the USA and Europe. She is charged with the responsibility of assisting company executives in many countries. Her primary boss is based in the USA and she has many other secondary bosses in the European countries. However, after only four months in this job, Mary has observed and heard…

 

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“The buyer must give the seller sufficient notice of the vessel name, loading point and, where necessary, the selected delivery time within the agreed period.” What does the buyer’s obligation under an FOB contract to nominate the vessel entail?

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