INPUTS City Supply Demand Metric (1) Newark 200 units (2) Boston 101 units (3) Columbus 60 units (4) Richmond 80 units (5) Atlanta 170 units (6) Mobile 70 units (7) Jacksonville 300 units Per-Unit Shipment Costs Origin Destination Cost (1) Newark (2) Boston 30.00 (1) Newark (4) Richmond 40.00 (2) Boston (3) Columbus 50.00 (3) Columbus (5) Atlanta 35.00 (5) Atlanta (3) Columbus 40.00 (5) Atlanta (4) Richmond 30.00 (5) Atlanta (6) Mobile 35.00 (6) Mobile (5) Atlanta 25.00 (7) Jacksonville (4) Richmond 50.00 (7) Jacksonville (5) Atlanta 45.00 (7) Jacksonville (6) Mobile 50.00 RESULTS Amount Shipped (Decision Decision Variable Origin Destination Variable) (Notation) Metric (1) Newark (2) Boston 120 X12 units (1) Newark (4) Richmond 80 X14 units (2) Boston (3) Columbus 19 X23 units (3) Columbus (5) Atlanta 0 X35 units (5) Atlanta (3) Columbus 41 X53 units (5) Atlanta (4) Richmond 0 X54 units (5) Atlanta (6) Mobile 0 X56 units (6) Mobile (5) Atlanta 0 X65 units (7) Jacksonville (4) Richmond X74 units (7) Jacksonville (5) Atlanta 211 X75 units (7) Jacksonville (6) Mobile 70 X76 unitsTable 3: Constraints City Constraint (1) Newark X12 + X14 s 200 (2) Boston X12 - X23 2 100 (3) Columbus X23 + X53 - X35 2 60 (4) Richmond X14 + X54 + X74 2 80 (5) Atlanta X35 + X65 + X75 - X53 - X54 - X56 2 170 (6) Mobile X56 + X76 - X65 2 70 (7) Jacksonville X74 + X75 + X76 Supply, which ensures that all of the cars leave the port cities. Since it costs money to ship extra cars, these constraints would be tight. The demand node constraints should have this form: Inflow - Outflow s Demand. These constraints ensure that no distributor receives more cars than requested. Since there are not enough cars to go around, some of these constraints must be loose. If the total supply equals the total demand, then all constraints are equality constraints: Outflow - Inflow = Supply (for supply-nodes); Inflow - Outflow = Demand (for demand-nodes. These constraints are, by definition, tight. Developing the BMC Model Open the file titled "Shell for BMC.xIs," which is an incomplete version of the model shown in Figures 2 and 3. Figure 2 presents the Inputs section. B C D E INPUTS City Supply Demand Metric W N (1) Newark 200 units (2) Boston 101 units (3) Columbus 60 units (4) Richmond 80 units (5) Atlanta 170 units (6) Mobile 70 units (7) Jacksonville 300 units Per-Unit Shipment Costs Origin Destination Cost (1) Newark (2) Boston 30.00 (1) Newark (4) Richmond 40.00 (2) Boston (3) Columbus 50.00 (3) Columbus (5) Atlanta 35.00 (5) Atlanta (3) Columbus 40.00 (5) Atlanta (4) Richmond 30.00 (5) Atlanta 6) Mobile 35.00 (6) Mobile (5) Atlanta 25.00 (7) Jacksonville (4) Richmond 50.00 (7) Jacksonville 5) Atlanta 45.00 (7) Jacksonville (6) Mobile 50.00 Figure 2: Inputs Section of Excel ModelTable 2: Decision Variables Decision Variable Origin City Destination City Default Plan X12 {1] Newark (2} Boston 115 X14 (1) Newark (4) Richmand 85 X23 (2) Boston (3) Columbus 10 X35 (3) Columbus (5) Atlanta 2 X53 (5} Atlanta (3) Columbus 55 X54 {5} Atlanta (4} Richmond 0 X56 {5) Atlanta (6) Mobile 0 x65 (6) Mobile (5] Atlanta 228 X74 (7) Jacksonville [4) Richmond X75 (7) Jacksonville (5) Atlanta X76 (7) Jacksonville (6) Mobile The objective function is to minimize total transportation cost, which is the sum of the transportation costs along each arc in the network: MIN 30X\" +40qu +50X3 +35%35 +10.133 +30X$4 +35X56 +2537\" +50XH +45XL' +50X75 The constraints in Kevin's problem are owbalance constraints, which ensure that distributors demands are met and no vehicle; are lost in transit. Each of the seven nodes (cities) in Figure 1 has a constraint. In this scenario. the supply of cars {200+300=500) exceeds the overall demand (too+6o+80+170+70 = 480), so every distributor will receive all the cars it needs, and there will be some cars left over at the ports. Therefore the constraints at the supplynodes (port cities Newark and Jacksonville) have the following form: Outow Inow 5 Supply. (Here \"Outflow\" refers to the total number of cars shipped out of the city, i.e. the sum of the decision variables corresponding to arcs pointing away from the city. likewise, \"Inflow\" refers to the total number of cars shipped into the city, which is the sum of the decision variables corresponding to arcs pointing towards the city.) These constraints ensure that Kevin doesn't send more cars than are available. We call a constraint light when the value on the left side of the constraint equals the value on the right side. and the constraint is loose (or slack) when the two values are different. Since there will be some cars left over at the ports, one or both of the supplynode constraints will be loose. The mnstraints at the demand-nodes (cities containing distributors) have this form: Inflow Outflow z Demand. These constraints ensure that every distributor will receive at least the number of cars it requested. Since it costs money to ship extra cars, no extra cars will be sent, and every demand-node constraint will be tight. The exact constraints are listed in Table 3. Pagezofr