Plant-Location Problem  A new company has won contracts to supply a product to customers in Central America, United States, Europe, and South America. The company has determined three potential locations for plants.  Relevant costs are:

Plant-Location Problem: Customer Demands and Shipping Costs Numbers on arcs represent shipping costs (in $100 per unit). Which plant location/shipping plan combination minimizes production and distribution costs? Plant Customer Brazil Philippines Mexico Central America United States Europe South America

Plant-Location Model  Indices: Let B represent the Brazil plant, and similarly use P (Philippines), M (Mexico), C (Central America), U (United States), E (Europe), and S (South America)  Decision Variables:  Plant opening decisions: and define y P and y M similarly.  Production decisions: p B = # of units to produce in Brazil and similarly define p P and p M.  Distribution decisions: x BC = # of units to ship from Brazil to Central America, and define x BU, x BE, …, x MS similarly.

Plant-Location Model  Objective Function: Total cost = fixed + variable + shipping costs Total variable cost is: VAR = 1,000 p B + 1,200 p P + 1,600 p M. Total shipping cost is: SHIP = 900 x BC x BU x BE x BS x PC +700 x PU x PE x PS +300 x MC x MU x ME x MS. Total fixed cost: FIX = 50,000 y B + 40,000 y P + 60,000 y M

Plant-Location Model (continued)  Constraints: Plant-production definitions: There are constraints to define total production at each plant. For example, the total production at the Mexico plant is: p M = x MC + x MU + x ME + x MS This can be thought of as a “flow in = flow out” constraint for the Mexico node.  Plant-Capacity Constraints: Production cannot exceed plant capacity, e.g., for Brazil p B  30  Demand constraints: There are constraints to ensure demand is met for each customer. For example, the constraint for Europe is: x BE + x PE + x ME = 20. This is a “flow in = flow out” constraint for the Europe node.

Additional Constraints (continued)  If y B = 0 we want to rule out production at the Brazil plant  If y B = 1, the plant is open and its “available” capacity is 30 units per month  Let’s try this: p B  30 y B If y B = 0 then the constraint becomes p B  0. If y B = 1 then the constraint becomes p B  30.  Alternatively, if p B  0 (and y B can only take on the values 0 or 1) then y B = 1

Plant Location Integer Programming Model min VAR + SHIP + FIX  Cost definitions: (VAR Def.) VAR = 1,000 p B + 1,200 p P + 1,600 p M. (SHIP Def.) SHIP = 900 x BC x BU x BE +500 x BS x PC +700 x PU, x PE x PS x MC x MU x ME x MS (FIX Def.) FIX = 50,000 y B + 40,000 y P + 60,000 y M  Plant production definitions: (Brazil) p B = x BC + x BU + x BE + x BS (Philippines) p P = x PC + x PU + x PE + x PS (Mexico) p M = x MC + x MU + x ME + x MS  Demand constraints: (Central America) x BC + x PC + x MC = 18 (United States) x BU + x PU + x MU = 15 (Europe) x BE + x PE + x ME = 20 (South America) x BS + x PS + x MS = 12  Modified plant capacity constraints: (Brazil) p B  30 y B (Philippines) p P  25 y P (Mexico) p M  35 y M  Binary variables: y B, y P, y M = 0 or 1  Nonnegativity: All variables  0

The Petromor Bidding Example  Petromor, the national oil company of a small African state, is selling unexploited land with good oil-extraction potential to private oil companies  For each piece of land (or zone, in the oil lingo), Petromor has projected the number of barrels that can potentially be extracted in this zone.  Petromor has organized a public bid in which private oil companies present sealed offers ($ per barrel) for the zones they are interested in buying

The Petromor Bidding Problem  The companies present different bids for different zones.  If a company wins a certain zone, the final price it pays is determined by multiplying the offer in the bid by the zone's potential oil volume, as estimated by the government.  No oil company can be awarded more than one zone as a result of the public offering

The Petromor Bidding Problem  Petromor would like to maximize the revenue resulting from these sales: Table 1. Bids (in $ per Barrel) A B C D E F Zone 1 $8.75 $8.70 $8.80 $8.65 $8.60$8.50 Zone 2 $6.80 $7.15 $7.25 $7.00 $7.20 $6.85 Zone 3 $8.30 $8.20 $8.70 $7.90 $8.50 $8.40 Zone 4 $7.60 $8.00 $8.10 $8.00 $8.05 $7.85 Table 2. Zone potential (in # of Barrels) Potential Zone 1205,000 Zone 2 240,000 Zone 3 215,000 Zone 4 225,000 What is the most profitable assignment of zones to the companies in this case?