consider the network shown below, where $a = 190$, $b = 180$, $c = 100$, $d = 130$, and $e = 90$.

find the maximum and minimum possible values of $x_6$.

minimum value of $x_6$ is
maximum value of $x_6$ is

Question

consider the network shown below, where $a = 190$, $b = 180$, $c = 100$, $d = 130$, and $e = 90$.

find the maximum and minimum possible values of $x_6$.

minimum value of $x_6$ is
maximum value of $x_6$ is

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Network Flow", "Kirchhoffs Junction Rule" ], "new_concepts": [], "current_concepts": [ "Network Flow", "Kirchhoffs Junction Rule", "General Solution of Linear Systems" ] } ## Set up junction equations $$ \begin{aligned} &\text{Left Node: } x_1 = x_2 + a + b \implies x_1 = x_2 + 190 + 180 \implies x_1 = x_2 + 370\ &\text{Top Node: } x_2 + x_4 = e \implies x_2 + x_4 = 90\ &\text{Center Node: } a + c = x_4 + x_5 \implies 190 + 100 = x_4 + x_5 \implies x_4 + x_5 = 290\ &\text{Bottom Node: } b + x_5 = x_3 + d \implies 180 + x_5 = x_3 + 130 \implies x_3 = x_5 + 50\ &\text{Right Node: } e + x_3 = c + x_6 \implies 90 + x_3 = 100 + x_6 \implies x_6 = x_3 - 10 \end{aligned} $$ ## Express variables in terms of a single parameter $$ \begin{aligned} &\text{Let } x_4 = t\ &x_2 = 90 - t\ &x_5 = 290 - t\ &x_3 = (290 - t) + 50 = 340 - t\ &x_6 = (340 - t) - 10 = 330 - t \end{aligned} $$ ## Apply non-negativity constraints to find bounds $$ \begin{aligned} &x_i \ge 0 \text{ for all } i\ &x_2 \ge 0 \implies 90 - t \ge 0 \implies t \le 90\ &x_5 \ge 0 \implies 290 - t \ge 0 \implies t \le 290\ &x_3 \ge 0 \implies 340 - t \ge 0 \implies t \le 340\ &x_4 \ge 0 \implies t \ge 0\ &\text{Thus, } 0 \le t \le 90\ &x_6 = 330 - t\ &\text{Minimum } x_6 = 330 - 90 = 240\ &\text{Maximum } x_6 = 330 - 0 = 330 \end{aligned} $$ Minimum value of $x_6$ is 240. Maximum value of $x_6$ is 330. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Network Flow" ] }

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QUESTION IMAGE

Question

Explanation:

Set up junction equations

Express variables in terms of a single parameter

Apply non-negativity constraints to find bounds

Answer: