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**ENGM 732 Network Flow Programming**

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Solutions 1-1 Eight tankers are scheduled to arrive and depart from a port at the times given in the table below. Set up a minimum cost flow model that will find the smallest number of docks required to service the tankers within the arrival and departure times and the schedule of the tankers at the docks. Tanker Arrival Date Departure Date 1 3 2 5 4 7 9 6 10 8

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**Tanker – try an assignment**

[1] T1 [Fixed] (flow, cost) T2 D1 [1] [-1] How can we adjust for minimal number of docks and how can we adjust for days in dock? T3 [1] D2 [-1] D3 [-1] T8 [1]

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**Tanker – try a rethink on days**

[1] D0 D1 [0,-M,1] [Fixed, Slack, Cost] (flow, cost) D2 [1] D2 [0,-M,1] Assumptions Load / unload takes 1 day Tanker 1 arrives day 0 and must leave by day 3. D3 [1] D3 [0,-M,1] D4 D4 [2] [0,-M,1] D6 [1] D10 [0,-M,1] D7 [2]

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**Tanker – try a rethink on days**

[1] D0 D1 [0,-M,1] [Fixed, Slack, Cost] (flow, cost) D2 [1] D2 [0,-M,1] Assumptions Load / unload takes 1 day Tanker 1 arrives day 0 and must leave by day 3. Tanker 2 arrives day 2 and must leave by day 5. D3 [1] D3 [0,-M,1] D4 D4 [2] [0,-M,1] D6 [1] D10 [0,-M,1] D7 [2]

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**Tanker – try a rethink on days**

[1] D0 D1 [0,-M,1] [Fixed, Slack, Cost] (flow) D2 [1] D2 [0,-M,1] Assumptions Load / unload takes 1 day Tanker 1 arrives day 0 and must leave by day 3. Tanker 2 arrives day 2 and must leave by day 5. 2 tankers arrive day 4 and 2 tankers arrive day 7 D3 [1] D3 [0,-M,1] D4 D4 [2] [0,-M,1] D6 [1] D10 [0,-M,1] D7 [2]

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Solutions 1-2 Over the next three months, 20,000 boxcars of Wyoming coal must be transported from Butte to distribution points at St. Louis, Houston and New Orleans. The three cities have contracted for 4000, 10,000, and 6000 boxcars, respectively. Because of different rail carriers’ practices, three different routes are possible. Direct transport to St. Louis, Houston and New Orleans at a cost of $1200, $1400 and $1500 per boxcar, respectively, Transport first to Wichita at a cost of $600 per boxcar; then a decoupling-coupling charge of $30 per car, finally a charge of $600, $400 and $400 per car for final transport to St. Louis, Houston and New Orleans. Transport first to St. Louis at a cost of $1200 per car; then by river barge to New Orleans at $250 per car (equivalent); finally by boat across the Gulf of Mexico to the port of Houston at a cost of $150 per car (equivalent). Various government regulations stipulate that no more than 20% of the 20,000 cars may be sent to Wichita. Because of other commitments, the direct rail lines from Butte to Houston and New Orleans are limited to 1000 and 3000 cars per month, respectively. Formulate a network flow model to minimize the total costs of transporting the 20,000 cars of coal.

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