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Questions and Answers of
Small Business Management
A telecommunication network is a set of nodes and directed arcs on which data packets flow. We assume that the flow between each pair of nodes is known and constant over time; please note that the
In the portfolio optimization models that we considered in this chapter, risk is represented by variance or standard deviation of portfolio return.An alternative is using MAD (mean absolute
In the minimum cost lot-sizing problem, we assumed that demand must be satisfied immediately; by a similar token, in the maximum profit lotsizing model, we assumed that any demand which is not
In Section 12.4.2 we have illustrated a few ways to represent logical constraints. Suppose that activity i must be started if and only if both activities j and k are started. By introducing customary
Extend the knapsack problem to cope with logical precedence between activities. For instance, say that activity 1 can be selected only if activities 2, 3, and 4 are selected. Consider alternative
Extend the production planning model (12.27)in order to take maintenance activities into account. More precisely, we have M resource centers, and each one must be shut down for exactly one time
In Example 12.12 we considered a single-period blending problem with limited availability of raw materials. In practice, we should account for the possibility of purchasing raw materials at a
Consider the constrained problem:min a:3— 3xy s.t. 2x — y = — 5 5x + 2y> 37 x,y > 0• Is the objective function convex?• Apply the KKT conditions; do we find the true minimizer?
Solve the optimization problem max xyz s.t. x + y + z < 1 x,y,z>0 How can you justify intuitively the solution you find?
Consider the domain defined by the intersection of planes:3x + y + z = 5 x + y + z — 1 Find the point on this domain which is closest to the origin.
Is the function f(x) = xe 2x convex? Does the function feature local minima? What can you conclude?
Assume that functions /¿(x), i = 1,..., m, are convex. Prove that the function mi=l where Q¿ > 0, is convex.
In Example 5.9 we assumed that the presenter opens box C knowing where the prize is. Now, let us assume that he has no information on where the prize is. Does this change our conclusions?
Assume that P(^4) = P(-B) for two events A and B. Then prove that, given another event E V{A\E) _ Ρ(Ε|Λ)P{B\E) ~ P(E\B)Find an interpretation of the result as a probability inversion formula.
Consider two events E and G, such that E Ç G. Then prove that P(£) < P(G).
Consider a moving-average algorithm with time window n. Assume that the observed values are i.i.d. variables. Show that the autocorrelation function for two forecasts that are k time buckets apart is
Prove Eqs. (11.31) and (11.32).
Prove that the weights in Eq. (11.18) add up to one. (Hint: Use the geometric series.)
We want to apply the Holt-Winter method, assuming a cycle of one year and a quarterly time bucket, corresponding to ordinary seasons. We are at the beginning of summer and the current parameter
In the table below, "-" indicates missing information and "??" is a placeholder for a future and unknown demand:Quarter Year I II III IV 2008 - - 40 28 2009 21 37 46 30 2010 29 43 ?? ??Initialize a
The following table shows quarterly demand data for 3 consecutive years:Quarter Year I II III IV 2008 21 27 41 13 2009 19 32 42 12 2010 22 33 38 10 Choose smoothing coefficients and apply exponential
Consider the demand data in the table below:t 1 2 3 4 5 6 Yt 35 50 60 72 83 90 We want to apply exponential smoothing with trend:• Using a fit sample of size 3, initialize the smoother using linear
Prove that, for a symmetric matrix A, we haveΣ Σ4 = Σ*fc = l where λ&, k = 1,..., n, are the eigenvalues of A.
Show that, if the eigenvalues of A are positive, those of A + A - 1 are not less than 2.
Show that if λ is an eigenvalue of A, then 1/(1 + λ) is an eigenvalue of (I + A)- 1.
Prove that two orthogonal vectors are linearly independent.
Prove that hh T— h T h I is singular.
For a square matrix A, suppose that there is a vector x φ 0 such that Ax = 0. Prove that A is singular.
Find the inverse of each of the following matrices Ai 6 0 0 2 0 0 00- 5, A2 =0 0 5 0 2 0 3 0 0, A3 =1 1 0 0 1 1 1 0 1
Check that the determinant of diagonal and triangular matrices is the product of elements on the diagonal.
Consider the matrix C = I„ — -J n , where I„ € M.n'n is the identity matrix and Jn G Rn , n is a matrix consisting of 1. This matrix is called a centering matrix, since x TC = {xi — x},
Consider the matrix H = I —hhT, where h is a column vector in M.n and I is the properly sized identity matrix. Prove that H is orthogonal, provided that h T h = 1. This matrix is known as the
Unlike usual algebra, in matrix algebra we may have AX though A^BandX/0 . Check with BX, even 1 0 2 0 1 1 2 0 2 B =1 3 0 4 2 3 0- 1 0x =6 5 7 2 2 4 3 3 6
Let A e Rm'n, and let D be a diagonal matrix in W""n. Prove that the product AD is obtained by multiplying each element in a row of A by the corresponding element in the diagonal of D. Check with A =
Prove that the representation of a vector using a basis is unique.
Express the derivative of polynomials as a linear mapping using a matrix.
Solve the system of linear equations:xi + 2x2 - X3 — - 3 ari + 4x3 = 9 2ar2 + ar3 = 0 using both Gaussian elimination and Cramer's rule.
Prove the result of Eq. (9.40). You should use the result established in Problem 7.11 to find the expected value of X(n), given a sample of n independent observations from the uniform distribution on
Apply the method of maximum likelihood to estimate the parameters of a uniform distribution on the interval [a, b].
Consider an exponential distribution with rate λ. On the basis of a random sample of size n, apply the method of moments to estimate λ.
Consider a sequence of random variables 0 with probability 1 Xn n with probability —~n Does this sequence converge in probability to a number? What about convergence in quadratic mean?
Consider the simulation of a continuous review (Q, R) inventory control policy. Define the relevant events for the system, and outline a procedure for the management of each event. To deal with a
Define an algorithm to generate pseudorandom variables characterized by the following density function:/(*) = 4 x - 1, if 1 < x < 2 3-x, if 2 < a; < 3 0, otherwise
A m-Erlang distribution with rate λ is obtained when summing m independent exponential random variables with rate λ. This distribution may be used to model more realistic random service times in
Apply one-way ANOVA to check equality of means for the following sample:i = 1 82.31 160.98 230.84 522.06 449.25 t = 2 240.80 228.27 278.73 278.16 172.16 t = 3 181.55 188.83 334.07 326.81 327.55
In one-way ANOVA we define the sum of squares SS& and SS™. Prove the identity s s - = Σ Σ xij -nm*2- -SSb i j
In order to estimate the fraction of defective parts, you take a sample of size 1000 and find that 63 are not acceptable. Find a 99% confidence interval for the fraction of defective parts.
The following dataset is a random sample from a normal distribution:103.23, 111.00, 86.45, 105.17, 101.91 92.15, 97.40, 102.06, 121.47, 116.62 Find a 95% confidence interval for variance.
A study was done to measure the impact of fatigue on human performance when carrying out a certain task. The performance is measured by an appropriate index, the larger the better, which is measured
You want to compare the reliability of two machines that insert chips onto electronic cards. The main problem is the occurrence of jams in the feeding mechanism, as this requires stopping production
Air quality is measured by the concentration of a dangerous pollutant.The mayor of a city has engaged in a program to improve traffic conditions in order to decrease the concentration ofthat
TakeltEasy produces special shoes for runners, whose average life is 1250 km. In order to improve the product, they experiment with a new design, and test prototypes with a sample of 30 runners. The
In standard confidence intervals, you use the sample mean as an estimator of expected value. Now suppose that a friend of yours suggests the following alternative estimator:-^"= Τ7 ρ-^1 "t" TT-^2 +
Find the 97% confidence interval, given a sample mean of 128.37, sample standard deviation of 37.3, and sample size of 50. What is the width of the confidence interval? Suppose that you want to cut
You have to compute a confidence interval for the expected value of a random variable. Using a standard procedure, you take a random sample of size N = 20, and the sample statistics are X = 13.38 and
The director of a Masters' program wants to assess the average IQ of her students. A sample of 18 students yields the following results:130, 122, 119, 142, 136, 127, 120, 152, 141 132, 127, 118, 150,
Consider two random variables X and Y, not necessarily independent.Prove that Cov(X -Y,X + Y)=0.
You have invested $10,000 in IFM stock shares and $20,000 in Peculiar Motors stock shares. Compute the one-day value at risk, at 95% level, assuming normally distributed daily returns. Daily
You are in charge of component inventory control. Your firm produces end items P\ and P2, which share a common component C. You need two components C for each piece of type P\ and three components
You have to decide how much ice cream to buy in order to meet demand at two retail stores. Demand is modeled as follows:£>i = 100X + eu D2= 120X + e2 where X, e\, and ei are independent normal
Batteries produced by a company are known to be defective with a probability of 0.02. The company sells batteries in packages of eight and offers a money-back guarantee that at most one of them is
According to an accurate survey, 40% of people checked at the exit of a well-known pub have made excessive use of alcoholic drinks. If we take a random sample of 25 persons, what is the probability
You are about to launch a new product on the market. If it is a success you will make $16 million; otherwise, you lose $5 million. The probability of success is 65%. You could increase chances of
Using the binomial expansion formula (2.3), prove that the PMF of the binomial distribution [see Eq. (6.16)] adds up to one.
Using Eq. (6.15), prove that the expected value of a geometric random variable X with parameter p is E[X] = 1/p. (Hint: Use the result in Example 2.40.)
Consider a generalization of the Bernoulli random variable, i.e., a variable taking values X\ with probability p and X2 with probability I—p. Which values of p maximize and minimize variance?
Consider the following extension of the EOQ model, often labeled economic manufacturing quantity (EMQ). Most of the assumptions of the EOQ and EMQ models are the same, but in the latter, rather than
Assume that you work from age 25 to age 65. At the end of each year, you contribute C to your pension fund, which is invested at a yearly rate r = 5%. At age 65 you retire and plan to consume $20,000
Consider a stream of constant cash flows Ft= C, for t = 1,..., T.From Example 2.39, we know how to find its present value at time t = 0.Now imagine that these cash flows are the amount you invest to
Prove that the intersection of two convex sets Si and S2 is a convex set.
Consider the following functions and tell if they are convex, concave, or neither:fi{x) = e'a+\ / 2 (x) = ln(x + l), / 3 (x) = x 3- x 2+ 2
In Example 2.33 we assume discrete-time compounding of interest. This results in the need for introducing modified duration, in Eq. (2.19), to get rid of an annoying factor 1 + y, where y is yield to
Consider function and find linear (first-order) and quadratic (second-order) approximations around points xo — 0 and xo = 10. Check the quality of approximations around these points.
Consider the following functions defined on piecewise domains:j-x, x0'h{x) = Ì3X, *>0 'h{x)~-3 2 In other cases, one should define an integral by considering the value of the function at the
Consider functions /(x) = x3— x and g(x) — x3+ x. Use derivatives to sketch the function graphs, and look for maxima and minima.
Find the first-order derivative of the following functions:
Find the equation of a line• With slope -3 and intercept 10• With slope 5 and passing through point (-2,4)• Passing through points (1,3) and (3,-5)
Find the domain of functions 1 1 f(x) = ,, —7> 9(x) y/l - x2- 1 ' Vx2+ 1 - x
Use the concepts that we have introduced in Chapter 2 to check the qualitative properties of the logistic function of Eq. (16.11).
Apply the formulas of multiple regression to the case of a single regressor, and verify that the familiar formulas for simple regression are obtained.
Prove the identity in Eq. (14.26)
Consider the fraction È of defective items in a batch of manufactured parts. Say that the prior distribution of È is a beta distribution with parameters «i = 5 and ct2 = 10 (see Section 7.6.2 for
Consider again the Bayesian coin flipping experiment of Example 14.14, where the prior is uniform. If we use Eq. (14.21) to find the Bayesian estimator, what is the estimate of È after the first
Consider the data of the Braess' paradox example in Section 14.5, but imagine that a central planner can assign routes to drivers, in order to minimize total travel cost. Check that adding the new
Two firms have a production technology involving a fixed cost and constant marginal cost, as represented by the cost function:TCi(f t) = {f +C| * *« > 0 . i =1, 2 1^0 if
Two firms have the same production technology, represented by the cost function:T Cite ) = ^ + 2«·>» , «_i, S[0 if ft = 0 3 5 There is also an alternative way of interpreting Black-Litterman
Consider the Cournot competition outcome of Eqs. (14.12) and (14.13).Analyze the sensitivity of the solution with respect to innovation in production technology, i.e., how a reduction in production
Find the Nash equilibria in the games in Tables 14.2 and 14.4. Are they unique?
Consider a point-to-point transportation network consisting of M nodes. By "point-to-point" we mean that, given transportation requirements between all pair of nodes, there is a direct transportation
Consider the plant location model of Section 12.4.5 [see Eqs. (12.54-12.55)]. Adapt the model to cope with uncertain demand scenarios, building a two-stage stochastic linear programming model with
We know that VaR, in general, is not a subadditive risk measure.Consider a portfolio of two assets, with jointly normal returns.• Show that, in this specific case, VaR is a subadditive risk
You have invested $150,000 in stock shares of Doom and $200,000 in stock shares of Mishap. Assume that daily returns follow a multivariate normal distribution; daily volatilities for the two stock
An investor has an initial wealth WQ that must be allocated between a risk-free asset, with certain return r{, and a risky asset. We assume a simple binomial model of uncertainty, like we did in
You own a plant whose value is $100,000. In case of a fire, the value of your property might be significantly reduced or even destroyed, depending on how severe the accident is. Let us represent risk
A decision maker with a quadratic utility function of the form (13.12)is offered the following lottery:Probability Payoff 0.20 $10,000 0.50 $50,000 0.30 $100,000 If the risk aversion coefficient is
You are the manager of a pension fund, and your fee depends on the return attained. You can play it safe and allocate wealth to a risk-free portfolio earning 4% per year. Alternatively, you can
The Research and Development (R&D) division of your firm has developed a new product that could be immediately launched on the market. If so, the probability of success is 60%, in which case profit
Consider again the "fancy coin flipping" example of Section 1.2.3, i.e., the decision of producing a movie or not. Formalize the problem with a proper decision tree.
A firm sells a perishable product, with a time window for sales limited to 1 month. The product is ordered once per month, and the delivery lead time is very small, so that the useful shelf life is
Given the observed data x 45 50 55 60 65 70 75 y 24.2 25.0 23.3 22.0 21.5 20.6 19.8 build a 95% confidence interval for the slope.
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