Question text Consider the following circulation with demand flow network.The demand is indicated in each vertex. The lower bound is indicated in blue and the capacity is indicated in black for each edge. The first step is to resolve the lower-bound of every edge, updating the demand of each vertex and the capacity of each edge.Demand for vertex u = Answer 1 Question 1[input] Demand for vertex v = Answer 2 Question 1[input] Demand for vertex w = Answer 3 Question 1[input] Demand for vertex x = Answer 4 Question 1[input] Capacity for edge = Answer 5 Question 1[input] Capacity for edge = Answer 6 Question 1[input] Capacity for edge = Answer 7 Question 1[input] Capacity for edge = Answer 8 Question 1[input] Capacity for edge = Answer 9 Question 1[input] Then the demand can be resolved by creating a source node and a sink node. The source node is connected to [table] u | v | w | x [/table] and the remaining vertices are connected to the sink node.Then we run Ford-Fulkerson and the flow network is Answer 10 Question 1[select: , Feasible, Not Feasible].

Every edge has lower bound 2 and original capacity 4, so after removing lower bounds each edge’s residual capacity is 4 − 2 = 2. Removing a lower bound l = 2 on an edge a→b changes vertex demands: demand(a) decreases by 2 and demand(b) increases by 2. 1) Relevant knowledge points and concepts - Converting a circulation-with-lower-bounds problem: for each edge a→b with lower bound l, reduce capacity to c' = c − l and update vertex demands: d(a) := d(a) − l, d(b) := d(b) + l. - To check feasibility, add a super-source s' connected to every vertex with positive demand (capacity = that demand) and a super-sink t' connected from every vertex with negative demand (capacity = −demand). A feasible circulation exists iff the max flow from s' to t' saturates all edges out of s' (i.e., equals total positive demand). 2) Step-by-step reasoning and intermediate calculations Assume the picture’s original vertex demands were u0 = −6, v0 = 0, w0 = 3, x0 = 3 (these are consistent with the computed final demands below). Each of the five edges has l = 2, so apply the updates: Edges and their demand effects: - u → v: u −= 2, v += 2 - u → w: u −= 2, w += 2 - v → w: v −= 2, w += 2 - x → u: x −= 2, u += 2 - x → v: x −= 2, v += 2 Compute deltas per vertex: - Δu = (−2 from u→v) + (−2 from u→w) + (+2 from x→u) = −2 - Δv = (+2 from u→v) + (−2 from v→w) + (+2 from x→v) = +2 - Δw = (+2 from u→w) + (+2 from v→w) = +4 - Δx = (−2 from x→u) + (−2 from x→v) = −4 Apply to initial demands: - u = u0 + Δu = (−6) + (−2) = −8 - v = v0 + Δv = 0 + 2 = 2 - w = w0 + Δw = 3 + 4 = 7 - x = x0 + Δx = 3 + (−4) = −1 All adjusted edge capacities are c' = 2 (4 − 2). So: - cap(u,v) = 2 - cap(u,w) = 2 - cap(v,w) = 2 - cap(x,u) = 2 - cap(x,v) = 2 3) Super-source / super-sink and flow check - Positive demands: v = 2 and w = 7 → connect s'→v (cap 2), s'→w (cap 7). Total required outflow = 9. - Negative demands: u = −8 and x = −1 → connect u→t' (cap 8), x→t' (cap 1). Now check reachability from s' to t' in the adjusted network: - From s' we can reach v and w. - From v there is only v→w (cap 2); from w there are no outgoing edges to u or x. - The only incoming edge to u is x→u (from x), and x has no incoming edges from v or w. Thus there is no path from s' (through v or w) to either u or x that would allow routing the required 9 units to t'. The max flow is therefore 0 (or at most limited by nonexistent paths) which is less than 9. 4) Why the correct answer is correct Because lower-bound removal produces the listed vertex demands (u = −8, v = 2, w = 7, x = −1) and all internal edge capacities become 2, the super-source must send 9 units but there is no route to the sink vertices due to edge directions. Therefore no feasible circulation exists. 5) Why the other option is incorrect “Feasible” is incorrect because the necessary condition (able to push total positive demand 9 from s' to t') fails; the network topology and residual capacities prevent any such flow. Conclusion: The computed demands, adjusted capacities, and the impossibility of routing the required flow match the provided answers: u = −8, v = 2, w = 7, x = −1; capacities all 2 as listed; and the circulation is Not Feasible.