(Solved): Exercise N:1 (Gauss-Jordan) (5 pts.) Use the Gauss-Jordan method to determine whether each of the f ...

Exercise N:1 (Gauss-Jordan) (5 pts.) Use the Gauss-Jordan method to determine whether each of the following linear systems \( \left(\mathrm{S}_{\mathrm{i}}\right) \) has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions (if any exist). \[ \begin{array}{l} \text { S1: }\left\{\begin{array}{c} x_{1}+x_{2}+x_{3}=2 \\ x_{2}-3 x_{3}=1 \\ 2 x_{1}+x_{2}+5 x_{3}=0 \end{array}\right. \\ \text { S2: }\left\{\begin{array}{c} -3 x_{1}+2 x_{2}-2 x_{3}=-10 \\ 2 x_{1}+x_{2}+x_{3}=4 \\ x_{1}-2 x_{2}+3 x_{3}=7 \end{array}\right. \\ \text { S3: }\left\{\begin{array}{l} 4 x_{1}+x_{2}+5 x_{3}=4 \\ x_{1}+x_{2}+2 x_{3}=1 \\ 2 x_{1}-x_{2}+x_{3}=2 \end{array}\right. \\ \end{array} \] Exercise N:2 (L.P Formulation) ( \( 5 \mathrm{pts} \).) Al-Manhal company faces a distribution problem. The company owns three selling points and a factory. The factory manufactures building material with a capacity of 350 tons per week. The storage capacities of the selling points \( \mathrm{S}_{1} \), and \( \mathrm{S}_{2} \), and \( \mathrm{S}_{3} \) are 90 tons, 50 tons, and 85 tons, respectively. Al-Manhal deals with four clients: \( \mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3} \), and \( \mathrm{C}_{4} \). The weekly demands from these clients are respectively, 60 tons, 80 tons, 55 tons, and 40 tons. Each of these four clients can purchase material either directly from the factory or from any of its selling points. The following table summarizes the distribution costs in Saudi Riyals (SR.) per ton. For instance, it is worth \( 6.5 \mathrm{SR} \) to ship one ton of material from the factory to store \( \mathrm{S}_{1} \) and 10 SR to ship from store \( \mathrm{S}_{2} \) to client \( \mathrm{C}_{3} \). Formulate the LP problem that will help the company managerial to minimize the total distribution cost. Hint: The decision variables to be used in the formulation are: - \( \mathrm{FS}_{\mathrm{i}} \) : number of tons shipped from the factory to store \( \mathrm{S}_{\mathrm{i}}(1 \leq \mathrm{i} \leq 3) \) - \( \quad \mathrm{FC}_{\mathrm{j}} \) : number of tons shipped from the factory to client \( \mathrm{C}_{j}(1 \leq \mathrm{j} \leq 4) \) - \( \quad \mathrm{SC}_{\mathrm{ij}} \) : number of tons shipped from store \( \mathrm{Si} \) to client \( \mathrm{C}_{\mathrm{j}}(1 \leq \mathrm{i} \leq 3 \) and \( 1 \leq \mathrm{j} \leq 4) \) Exercise N:3 (Solving LP graphically) (3.5 pts.) Solve the following LP graphically: \[ \begin{array}{c} \text { s.t } \quad\left\{\begin{array}{c} x \leq 4 \\ 2 y \leq 12 \\ 2 x+2 y \leq 18 \\ x, y \geq 0 \end{array}\right. \end{array} \] Exercise N:4 (Solving LP with Simplex) (5 pts.) Solve the following LP model by the simplex method (show ALL the steps in Simplex tableau): \[ \begin{array}{c} \text { s.t }\left\{\begin{array}{c} \operatorname{Max} z=2 x+4 y \\ x+3 y \leq 42 \\ x+y \leq 21 \\ x, y \geq 0 \end{array}\right. \end{array} \] Exercise N:5 (Solving LP using MS. Excel Solver) (4.5 pts.) Use the MS Excel Solver to find the optimal solution (if exists) to the following LP: \[ \text { s.t } \begin{array}{c} \text { Min } z=7 x_{1}+6 x_{2}+4 x_{3} \\ \left\{\begin{array}{c} 4 x_{2}+7 x_{2+}+9 x_{3} \geq 74 \\ x_{1}+2 x_{2}+5 x_{3} \geq 83 \\ 3 x_{1}+x_{2}+7 x_{3} \geq 66 \\ \text { all } x^{\prime} s \geq 0 \end{array}\right. \end{array} \]