Question

Suppose that the daily demand (𝐷) for a certain item in the capital market is random and behaves according to a uniform probability (quantity) law with 𝑑 = 0.1, … ,19. Every day, 𝑥 units of the item are brought for sale, which are sold at 5 mu per unit and, if not sold, 2 mu are lost (for storage, return or other). Determine the number of these items, which should be brought at the beginning of the day, so that the expected utility is maximum.

128

likes

641 views

Answer to a math question Suppose that the daily demand (𝐷) for a certain item in the capital market is random and behaves according to a uniform probability (quantity) law with 𝑑 = 0.1, … ,19. Every day, 𝑥 units of the item are brought for sale, which are sold at 5 mu per unit and, if not sold, 2 mu are lost (for storage, return or other). Determine the number of these items, which should be brought at the beginning of the day, so that the expected utility is maximum.

$Expert avatar$

Murray

4.5

92 Answers

\text{Expected Profit} = \frac{\int_{0.1}^{19} \left[ \min(5x, 7D - 2x) \right] \, dD}{19 - 0.1}

Since there's a uniform distribution, we consider the two cases where the profit equations exist:

1. **Demand

$ D \geq x $

:**

\text{Profit} = 5x

2. **Demand

$ D < x $

:**

\text{Profit} = 7D - 2x

Integrate over both segments accordingly:

E[\text{Profit}] = \int_{0.1}^{x} \left( 7D - 2x \right) \frac{dD}{18.9} + \int_{x}^{19} 5x \frac{dD}{18.9}

Solve these integrals:

E[\text{Profit}] = \left( \frac{7}{18.9} \int_{0.1}^{x} D \, dD - \frac{2x}{18.9} \int_{0.1}^{x} dD \right) + \frac{5x}{18.9} \int_{x}^{19} dD

= \frac{7}{18.9} \cdot \left[ \frac{D^2}{2} \right]_{0.1}^{x} - \frac{2x}{18.9} \left[ D \right]_{0.1}^{x} + \frac{5x}{18.9} \left[ D \right]_{x}^{19}

Evaluating these:

1. First part:

= \frac{7}{18.9} \left( \frac{x^2}{2} - \frac{0.1^2}{2} \right) - \frac{2x}{18.9} (x - 0.1)

2. Second part:

= \frac{5x}{18.9} (19 - x)

Combine and simplify:

E[\text{Profit}] = \frac{7x^2 - 7 \cdot 0.01}{2 \times 18.9} - \frac{2x^2 - 0.2x}{18.9} + \frac{5x (19 - x)}{18.9}

After simplifying, the expected profit equation in terms of

$ x $

E[\text{Profit}] = \frac{(7x^2 - 0.07) - (2x^2 - 0.2x) + 5x(19 - x)}{18.9}

Set the derivative of accumulated profit with respect to

$ x $

to 0 and solve for

$ x $

\frac{d}{dx} \left[ \frac{5x (19 - x) + 7 \frac{x^2}{2} - 0.07 - 2x^2 + 0.2x}{18.9} \right] = 0

\frac{d}{dx} \left[ \frac{13.6x - \frac{7x^2}{2} - 0.07 - 2x^2 + 0.2x}{18.9} \right] =0

13.6 \quad \text{results in maximum expected profit}

Therefore:

x = 13.6

Frequently asked questions (FAQs)

Solve the inequality: 5x - 3 < 12.

What is the variance of the dataset {2, 5, 9, 12, 17}?

Question: What is the value of x in a triangle with two sides measuring 5 units each and an included angle of 60 degrees?

New questions in Mathematics

Y=-x^2-8x-15 X=-7

58+861-87

We have spent 1/4 of the inheritance on taxes and 3/5 of the rest on buying a house. If the inheritance was a total of €150,000 How much money do we have left?

find all matrices that commute with the matrix A=[0 1]

What is the total tolerance for a dimension from 1.996" to 2.026*?

20% of 3500

find f(x) for f'(x)=3x+7

3. A rock is dropped from a height of 16 ft. It is determined that its height (in feet) above ground t seconds later (for 0≤t≤3) is given by s(t)=-2t2 + 16. Find the average velocity of the rock over [0.2,0.21] time interval.

The market for economics textbooks is represented by the following supply and demand equations: P = 5 + 2Qs P = 20 - Qd Where P is the price in £s and Qs and Qd are the quantities supplied and demanded in thousands. What is the equilibrium price?

A researcher is interested in voting preferences on change of the governing constitution in a certain country controlled by two main parties A and B. A questionnaire was developed and sent to a random sample of voters. The cross tabs are as follows Favour Neutral Oppose Membership: Party A 70 90 85 Party B 50 50 155 Test at α = 0.05 whether party membership and voting preference are associated and state the conditions required for chi-square test results to be valid.

3/9*4/8=

TEST 123123+123123

User One of the applications of the derivative of a function is its use in Physics, where a function that at every instant t associates the number s(t), this function s is called the clockwise function of the movement. By deriving the time function we obtain the velocity function at time t, denoted by v(t). A body has a time function that determines its position in meters at time t as S(t)=t.³√t+2.t . Present the speed of this body at time t = 8 s.

find missing measure for triangle area = 48 m square base = 10m heaighy = ? m

94 divided by 8.75

2x-5-x+2=5x-11

Given two lines 𝐿1: 𝑥 + 4𝑦 = −10 and 𝐿2: 2𝑥 − 𝑦 = 7. i. Find the intersection point of 𝐿1 and 𝐿2.

a) 6x − 5 > x + 20

the product of a 2-digit number and a 3-digit number is about 50000, what are these numbers

A plant found at the bottom of a lake doubles in size every 10 days. Yeah It is known that in 300 days it has covered the entire lake, indicate how many days it will take to cover the entire lake four similar plants.

You might be interested in