กำหนดปัญหาโปรแกรมเชิงเส้นต่อไปนี้ Maximize \( P=5 x_{1}-3 x_{2}+x_{3} \) Subject to \( \begin{array}{r}2 x_{1}-4 x_{2}+x_{3} \leq-3 \\ x_{1}+3 x_{2} \geq 5 \\ 4 x_{1}-x_{3}=7 \\ x_{2}+x_{3} \leq 12 \\ x_{1}, x_{2}, x_{3} \geq 0\end{array} \) ข้อใดต่อไปนี้ ไม่ใช่ เงื่อนไขของ modified problem \( \begin{array}{l}\text { (a) } 2 x_{1}-4 x_{2}+x_{3}+S_{1}=-3 \\ \text { (c) } 4 x_{1}-x_{3}+a_{2}=7\end{array} \)

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)

Decision variables in an OR model are : (a) Controllable (b) Uncontrollable (c) Parameters (d) Constants

Consider the Solow growth model with government spending but without technological progress. Output \( Y \) is produced according to the production function \[ Y=F(K, L)=\sqrt[3]{K} \cdot \sqrt[3]{L^{2}} \] where \( K \) is the capital stock and \( L \) is the labour force. Labour force is equal to employment and total population and grows at the rate \( n \). Capital depreciates at the rate \( \delta \). The government spends ('consumes') resources without contributing to production or capital accumulation. Government purchases, \( G \), are financed with proportional income taxation. The government budget in per capita terms is balanced, \[ g=\tau y \] where \( y \equiv Y / L \) and \( g \equiv G / L \) are output and government purchases per capita, and \( \tau \in[0,1) \) denotes the proportional tax rate. Investment \( I \) and consumption \( C \) absorb, correspondingly, constant fractions \( s \) and \( (1-s) \) of disposable income. (a) [5 marks] Show that the function \( F(K, L) \) has constant returns to scale and diminishing marginal returns to capital.

15. (a) (i) Let \( \quad(L, \leq) \quad \) be a lattice. For any \( a, b \in L \), \( a \leq b \Leftrightarrow a \wedge b=a \Leftrightarrow a \vee b=b \).

Calcular la proyección del vector \( \vec{u}=(-3,2,1) \) sobre el vector \( \vec{v}=(-1,2,4) \) a. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) b. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21}, \frac{44}{21}\right) \) c. \( \vec{w}=\left(-\frac{11}{21},-\frac{22}{21}, \frac{44}{21}\right) \) d. \( \vec{w}=\left(\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \)