# Problem 4. Maximum 14 points

A high-tech Japanese company has presented a unique robot capable of producing construction blocks, which can be sold for 90 monetary units each. Due to a shortage of specialized chips, it is impossible to replicate this robot or even repair the existing one if it becomes inoperative. If the robot works for \( \mathrm{L} \) hours a day on block production, it will become completely inoperative after \( 8 - \sqrt{L} \) days. It is known that the average hourly productivity of the robot during a shift is determined by the function \( \frac{1}{\sqrt{L}} \), and the value of \( \mathrm{L} \) can be set once and cannot be changed.

(a) What value of \( \mathrm{L} \) should be chosen if the main goal is to maximize the number of construction blocks produced by the robot per day?

(b) What value of \( \mathrm{L} \) should be chosen if the main goal is to maximize the number of construction blocks produced by the robot over its entire operational life?

(c) The Japanese company owning the robot decided to earn as much money as possible from it. Therefore, it was decided to use the robot not only for production but also to send it to a round-the-clock exhibition in its free time from block production, where it will stand as an exhibit and bring in 4 monetary units per hour. If the robot becomes inoperative, it cannot be used as an exhibit. What value of \( \mathrm{L} \) should be chosen to ensure the maximum income from owning the robot? What will the company's income be?

(d) What is the minimum hourly income from using the robot at the exhibition that the company would accept to forgo block production?

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# Solution

(a) Let's write down the company's objective function. Let $Q_{d}$ be the number of blocks the robot produces in a day. $Q_{d}=L \cdot \frac{1}{\sqrt{L}}=\sqrt{L} \rightarrow \max$. Notice that the function $Q_{d}(L)$ is monotonically increasing, which means $Q_{d}$ reaches its maximum value when $L=L_{\max }$. There are 24 hours in a day $\rightarrow L^{*}=24$.

Answer: 24.

(b) Let's write down the company's objective function. Let $Q_{l}$ be the number of blocks the robot produces over its entire operational period. $Q_{l}=Q_{d} \cdot(8-\sqrt{L})=8 \sqrt{L}-L$. Let $a=\sqrt{L}$, then $Q_{l}(a)=8 a-a^{2}$. The graph of the function is a parabola opening downwards, the maximum is achieved at the vertex $-a^{*}=4, L^{*}=16$.

Answer: 16.

(c) Let $N$ be the number of hours in a day the robot works as an exhibit, then $N+L=24$. Let $Q$ be the number of blocks the robot produces in its lifetime, then the revenue from selling blocks is $90 Q$. The revenue from working as an exhibit is 4 monetary units per hour, so the total revenue from this activity is $4 N \cdot(8-\sqrt{L})$. The total revenue is $T R=90 Q+4 N \cdot(8-\sqrt{L})$. Considering the previous part, it is clear that $Q=8 \sqrt{L}-L$ and $N=24-L$:

$T R=90 \cdot(8 \sqrt{L}-L)+4(24-L)(8-\sqrt{L})$

$T R=720 \sqrt{L}-90 L+768-96 \sqrt{L}-32 L+4 L^{1.5}$

$T R=4 L^{1.5}-122 L+625 \sqrt{L}+768$

Let's find the maximum of the function. $T R^{\prime}=6 L^{0.5}-122+\frac{312}{L^{0.5}}=0$

$T R^{\prime}=\frac{6 L-122 L^{0.5}+312}{L^{0.5}}=0$

$T R^{\prime}=\frac{\left(L^{0.5}-3\right)\left(6 L^{0.5}-104\right)}{L^{0.5}}$

$L_{1}=9, L_{2}=\frac{2704}{9}$. Notice that $L \in[0 ; 24]$.

$T R^{\prime}(L) \geq 0, L \in(0 ; 9]$ and $T R^{\prime}(L) \leq 0, L \in[9 ; 24]$.

The maximum total revenue is achieved at $L^{*}=9$.

$T R_{\max }=90(8 \cdot 3-9)+4(24-9)(8-3)=1650$

Answer: $L^{*}=9, T R_{\max }=1650$.

(d) Let $x$ be the hourly revenue of the company from using the robot at the exhibition, then:

$T R=90(8 \sqrt{L}-L)+a(24-L)(8-\sqrt{L})$

We need to find such $a$ that $L^{*}=0$.

$T R=720 \sqrt{L}-90 L+192 a-8 a \cdot L-24 a \cdot \sqrt{L}+a \cdot L^{1.5}$

$T R_{L}^{\prime}=\frac{360}{\sqrt{L}}-90-8 a-\frac{12 a}{\sqrt{L}}+1.5 a \cdot L^{0.5}$

$T R_{L}^{\prime}=0$ when $a0$.

When $a=30, T R_{L}^{\prime}<0, L \in(0 ; 24]$. Therefore, $T R_{\max }=T R(0)$. Clearly, the optimum at $L=0$ will be preserved with increasing $a$. The minimum hourly revenue from using the robot at the exhibition, at which the company will decide to stop producing blocks, is 30.

Answer: 30.

## Grading Criteria

(a) The objective function is formulated - (1p), the optimal value of $\mathrm{L}$ is determined - (1p).

(b) The objective function is formulated - (1p), the optimal value of $\mathrm{L}$ is determined - (2p).

(c) The objective function is formulated - (1p), the optimal value of $\mathrm{L}$ is determined - (2p), the maximum revenue of the firm is found - (1p).

(d) The firm's objective function is determined - (2p), the minimum hourly revenue from using the robot at the exhibition is determined - (3p).

A penalty of 1 point for arithmetic errors that did not lead to significant distortion of the results. If the optimal value of the function is found correctly but lacks justification for optimization - 0 points. Answers without solutions are scored 0 points. Alternative solutions may be scored full points if they contain a correct and justified sequence of actions.