What should be the shadow price corresponding to the

Question

Problem 1. You are running a blood bank. Every month, a random number of hospital patients will need blood that you will supply. Also, every month, a random number of donors will come in and give blood. You goal is to provide enough blood on hand to supply those patients who need it.

At the beginning of the current month you have no pints on hand. Your forecast for this month predicts that the number of pints needed by patients, denoted by N for needed, is normally distributed with mean 35 and standard deviation 8.

Your forecast also predicts that the number of pints given by donors who will come in this month to donate blood, denoted by D for donated, is a normally distributed with mean 20 and standard deviation 15. The amounts of pints donated and required are assumed to be independent from one another.

If the amount of blood needed exceeds the amount of donated blood, the amount of blood you 'have' at the end of the month is negative representing the deficit, (For example if this month 25 pints donated and 30 pints needed, the amount of blood you have in your bank at the end of the month is 25 - 30 = -5 pints. In general, the amount of blood you have in you bank at the end of the month is D-N)

(a) What is the likelihood that the amount of the blood donated this month exceeds 45, i.e., Pr(D>45)?

(b) What is the probability distribution describing the amount of blood you have in your bank at the end of the month? Compute its mean and variance.

(c) What is the probability that you'll run out of blood by the end of the month, i.e., Pr(D-N<0)?

(d) If you don't run out of blood at the end of the month, you pay no additional cost. If you do run out of blood before the end of the month, you will pay a fixed cost of $1000 for an emergency shipment of blood (paying this flat fee will allow you to have all the pints of blood that you need).

You have the option of spending $100 on a marketing campaign, which will definitely bring in all the donors you need (but you don't want to spend this money if you don't need it).

Draw a decision tree describing the situation you're facing, calculate the EMV, and make a recommendation.

Problem 2. Consider a system that has 4 computers. The system is down if at least one computer crashes. The system works if all 4 computers are working. During each day, the probability that a single computer crashes is 5%, independent of other computers and other days.

(a) Let us first focus on one day. What is the probability that on this given day, the system works with no downtime, i.e. what is the probability that zero computers crash?

What is the probability that the system is down on a given day? (Hint: Think complements!)

Let us denote by p the probability of the link being down on a given day. In case you have doubts about computing p in part (a), you can solve parts (b)-(d) by using the symbol p instead of its actual value. Let Y be the random variable accounting for the number of days in a month (n=30 days) that the system is down.

(b) What is the distribution of Y? What is its mean and standard deviation?
(c) What is the probability that, in a given month, the system is down for at least one day, i.e., Pr(Y>=1)?
(Hint: You can obtain an exact answer by considering the complement!)

(d) Give an approximation of the probability that, in a given month, the system is down for at least 10 days, i.e. P(Y>=10).

(Hint: Obtain an approximation via the Central Limit Theorem! Notice that random variable Y can be written as the sum of 30 random variables.)

Problem 3. In recent years, many American firms have intensified their efforts to market their products in the Pacific Rim. A consortium of U.S. firms that produce raw materials used in Singapore is interested in predicting the level of exports from the U.S. to Singapore, as well as understanding the relationship between U.S. exports to Singapore and certain variables affecting the economy of that country. The consortium hired an economist to perform an analysis.

The economist obtained monthly data on five economic variables for the period January 2006 to July 2011 (a total of 67 months) from the Monetary Authority of Singapore. These variables are as follows:

- Exports: U.S. exports to Singapore in billions of Singapore dollars
- Ml: Money supply figures in billions of Singapore dollars
- Lend: Minimum Singapore bank lending rate in percentage
- Price: Index of local prices where the base year is 2006
- Exchange: Exchange rate of Singapore dollars per U.S. dollar

Part I.

The economist performed a multiple regression analysis with Exports as the dependent variable and the four economic variables M1, Lend, Price, and Exchange as the independent variables. Part of his regression results are shown below:

Regression I

R Square Observations	0.825 67
	Coefficients	Standard Error	Lower 95%	Upper 95%
Intercept	-4.015	2.766	-9.544	1.514
MI	0.368	0.064	0.240	0.496
Lend	0.005	0.049	-0.093	0.103
Price	0.037	0.009	0.019	0.055
Exchange	0.268	1.175	-2.035	2.571

(a) Which variable(s) among the four do you think is (are) an important explanatory variable(s) for Exports? Explain your answer.

(b) The economist next computed the sample correlation between Price and Lend, which turns out to be 0.845. What problems, if any, can you identify in Regression I based on this information? How would you modify the model to avoid these problems?

Part 2.

The economist tried two other regression runs with Exports as the dependent variable. In one model, he used three independent variables: M 1, Price, and Exchange. In the other model, he used only two independent variables: M1 and Price. Part of his regression results are shown below:

Regression II
R Square Observations	0.823 67
	Coefficients	Standard Error	Lower 95%	Upper 95%
Intercept M1 Price Exchange	-3.995 0.364 0.037 0.242	2.736 0.041 0.004 1.135	-9.358 0.284 0.029 -1.983	1.368 0.444 0.045 2.467
Regression III
R Square Observations	0.821 67
	Coefficients	Standard Error	Lower 95%	Upper 95%
Intercept M1 Price	-3.423 0.361 0.037	0.541 0.039 0.004	-4.483 0.284 0.029	-2.363 0.437 0.045

(c) In your opinion, which of the three regression models (I, II, III) is the best overall?

Support your answer with any statistical reasoning that you feel is appropriate.

(d) What is your estimate of U.S. exports to Singapore in billions of Singapore dollars (using your best model) if M1=102.5, Lend=5.4, Price=126.9, and Exchange=1.26?

Problem 4. Michael has a portfolio consisting of three different stocks. The characteristics of Michael's portfolio are summarized in the following table. For example, the table shows that he bought 3,000 shares of Apollo when it was worth $20. Currently, Apollo's share is priced at $40, and is expected to have a value of $50 in one year.

Stock	Shares	Price when bought ($)	Current Price ($)	Expected Price in one year
				($)
Apollo	3,000	20	40	50
Biogen	8,000	50	70	95
Concord	1,500	10	13	20

Michael needs to immediately raise $150,000 to finance his business venture. To do this, he is planning to sell some of his shares. However, he wants to sell just enough to raise the money he needs, because the shares might be worth more in the future.

If he sells shares, then Michael pays taxes at the rate of 30% on capital gains. For example, suppose Michael sells 1,000 Apollo shares. He receives 1,000 x $40 - $40,000 for the sale. However, he owes 0.30 x (1,000 x $40 - 1,000 x $20) ---- $6,000 on capital gain taxes. So, by selling 1,000 shares of Apollo, Michael nets 40,000 - 6,000 = $34,000.

Michael asks his friend, Bill, to help him with his problem. Bill builds a linear optimization model in Excel that determines how many shares to sell in order to raise $150,000 (net of capital gains and transaction costs), while maximizing the expected value of Michael's portfolio next year.
There are three decision variables, QA, QB and QC, each corresponding to the shares of stocks Apollo, Biogen and Concord sold.

There are also four constraints: The amount of money raised should be at least $150,000. In addition, for each of the three stocks, Michael cannot sell more shares than he currently has in his portfolio.

The relevant output that Bill obtained is presented in the table below.

Cell                         Name	Final Value	Shadow Price	Constraint R.H. Side	Allowable Increase	Allowable Decrease
$J$3 Money Raised	150000	-1.48	150000	464000	48000
$J$4 QA	3000	0.47	3000	1411.76	3000
$J$5 QB	750	XXXX	8000	1E+30	7250
$J$6 QC	0	0	1500	1E+30	1500

a) Write down the objective function of the model.

b) What are the optimal quantities of shares that Michael needs to sell to meet his capital requirement of $150,000? After selling these shares, what is the expected value of his portfolio next year?

c) Write down the constraint that the amount raised from selling shares should be at least $150,000 (i.e., the constraint corresponding to J3 in the output table).

d) What should be the shadow price corresponding to the constraint of Biogen shares QB, which in the output is indicated as XXXX? Why? Why is the allowable increase to the constraint of Biogen shares QB 1E+30 ? (Recall that as we discussed in class 1E+30 represents a very large number).

e) Michael is told his business venture will cost $230,000 instead of $150,000. Consequently, he's going to have to sell even more stock. Without resolving the LP, can you determine what the new optimal expected value of his portfolio next year will be?

What is the allowable increase of the constraint corresponding to $J$3$? What does this mean intuitively?

Problem 5. John Kensey is the CEO of Showticket Productions, which manages three Broadway theaters: the Comedy Theater (C), the Drama Theater (D), and the Satire Theater (S). The following table presents information for each of the 8 shows that John is considering in the schedule. The last two columns show the anticipated weekly profit from running each show in each of the three theaters.

Show Number	Show Title	Theater Choice	Profit per Week ($ thousand)
		Comedy	635
1	Cats	Drama	423
		Satire	610
		Comedy	610
2	Chicago	Drama	540
		Satire	263
		Comedy	715
3	Les Miserables	Drama	585
		Satire	722
		Comedy	945
4	Mary Poppins	Drama	268
		Satire	492
		Comedy	515
5	My Fair Lady	Drama	175
		Satire	62
		Comedy	325
6	The Lion King	Drama	315
		Satire	56
		Comedy	385
7	Wicked	Drama	640
		Satire	600
		Comedy	1315
8	Phantom of the Opera	Drama	1276
		Satire	272

John is planning the company's upcoming season schedule and would like to determine which shows to run to which theaters during the season, in order to maximize total profit. The upcoming season is 8 weeks long. Each show can only run in one theater, and each theater can only run one show in any given week. For example, The Lion King can be assigned either to the Comedy Theater, the Drama Theater, or the Satire Theater, or to no theater, and hence not run during the season.

Any show assigned to the Drama Theater will run for 4 weeks; any show assigned to the Comedy Theater will run for 2 weeks, and similarly any show assigned to the Satire Theater will run for 2 weeks. And obviously, each theater can only accommodate a total of 8 weeks of shows in the upcoming season.

John is planning to formulate her problem as a discrete optimization problem, with decision variables C , ..., C8, D, ..., D D8, and S 1 , ..., Ss. For example D7= 1 if Wicked is assigned to the Drama Theater, and D7 = 0 otherwise. Notice that keeping track of the exact week when a play is scheduled in a theater is unnecessary, as we assume each play will generate the same expected profit on any week. This is why we only care about which play is scheduled in which theater.

a) Formulate the objective function of John's discrete optimization model. (You may use "..." notation in formulating the objective function to ease your writing burden.)

b) Formulate all the constraints in John's discrete optimization model.
Now suppose that the problem is more complicated than stated above. Model the following additional constraints:

c) Since the Comedy Theater has the largest capacity, John wants to vary the shows there as much as possible. Therefore, he does not want to bring both Cats and Phantom of the Opera (both composed by Andrew Lloyd Webber) to the Comedy Theater.

d) The lead actor in Les Miserables is dating the lead actress in Chicago and they will only agree to perform if their shows run at the same theater.

e) Suppose that after holding discussions with the VP of marketing and several executive producers, an analyst on John's staff has decided to add several more constraints. For each of the three constraints below, provide an interpretation using a sentence with no mathematical symbols.

1. C1 ±C2±C4 >1
ii. D + D > D 5 8 - 7
iii. S +S >2S
1 2 - 6

f) After running solving for the optimal solution (ignoring constraints implied by (c),(d) and (e), you discovered that the optimal solution is the following schedule:

• S1 C2 S 3 C4 C5 C6 D7 D8 1, 1, all the remaining variables set to 0

You would like to know which schedule is the "second best". How could you use discrete optimization to find it? (Hint: it is enough to add one constraint to the model)

Problem 6. Exeter Investments (EI) has decided to construct a new portfolio which will be comprised of IT (information technology) and energy companies' stocks. For this reason, EI collected the expected return, standard deviations and return correlations data for companies A, B, C, D, E, and F, which belong to those sectors. This data is displayed in Tables 1 and 2.

Company	Expected Return (°/©)	Standard Deviation	Sector
A	12	17	Energy
B	17	28	IT
C	14	19	Energy
D	4	7	IT
E	19	35	IT
F	21	41	IT

Table 1

	A		BC	D	E	F
A	1
B	-.5	1
C	.7	.4	1
	0	0	0	1
E	0	0	0	0	1
F	0	0	0	0	0	1

 

Table 2. Correlation coefficients between returns of stocks
a) EI would like the portfolio to have a maximum risk of 15% (note that we model risk of the portfolio in this problem as standard deviation). Write down a nonlinear optimization problem to determine the fraction of the total budget to invest in each stock that yield the maximum possible expected return satisfying this target risk requirement.

b) In looking at the sensitivity report with respect to the optimization problem in a), the shadow price associated with the constraint on a maximum risk of 15% was found to be 10.4163, but the sign in front of it cannot be read. That is, this shadow price is either 0.4163 or -0.4163. Which one of these possible values is correct?

c) Additionally El wants to ensure that the risk corresponding to the part of the portfolio consisting of Energy companies does not exceed 17% (note that as before we model risk as standard deviation). Incorporate this requirement into your formulation.

d) Expert revision of the data represented by Tables 1 and 2 resulted in exactly the same data as before, except that the correlation between return of stocks A and E was upgraded from 0 to 0.1. How would this affect the optimal objective value for the optimization problem you formulated for part a)? Specifically, would you expect the optimal objective value to stay the same, become larger or become smaller?

Problem 7. The primary goal of the course 15.730 Data, Models, and Decisions has been to teach you the modeling tools needed to make effective management decisions based on data. Think back to your current or last job before coming to MIT Sloan. Identify a project, activity, task, or assignment that you worked on at that job where you would now have analyzed the problem differently given your knowledge of DMD.

a) Describe the project/activity/assignment.

b) Describe either the data that you had available for the project or that you now wish you had developed in order to complete the project.

c) What modeling tool(s) from DMD would you have used on this project, and why do you think these tools would have been effective?
Try to be as concise as possible; we strongly suggest that you limit your answer to each of these questions to approximately one paragraph. Do not spend too much time on this problem.

(Note: If your job involved tasks for which the use of data and quantitative analysis was not relevant at all, then answer this question by instead discussing one of your possible job opportunities for an upcoming engagement)