Transportation Model (Powerco)

1. Transportation Model (Powerco) • Transportation between supply and demand points, with the objective of minimizing cost. • Send electric power from power plants to cities where power is needed at minimum cost • Objective: Minimize total cost of all shipments • There is a unit shipping cost on each shipping route • This is multiplied by the amount shipped and summed over all routes

2. 1 45 1 35 20 2 2 50 3 30 40 3 4 30 Powerco Contd. Constraints • Can’t ship more than is available from each power plant (supply point) • Must ship at least the amount needed to each city (demand point) Inputs • Unit shipping costs along each route • Amount of supply at each power plant • Demand at each city Decision Variables • The amount to ship along each route • There is a route from each supply point to each demand point • No other routes are allowed

4. 40 40 40 40 0 0 0 0 4 1 2 3 25 75 40 60 Producing Sailboats at Sailco(Inventory Problem Modeled as Transportation Problem) • Produce sailboats over a multiperiod horizon to meet known (forecasted) demands on time • Regular-time and overtime labor are available • Minimize total production and holding costs Supply RT OT 10 10 0 0 0 Inventory Month Demand

5. Objective • Minimize total costs, which include • Regular-time labor costs, Overtime labor costs, Inventory holding costs Inputs • Beginning inventory of sailboats • Maximum boats that can be produced per month with regular-time labor • Regular-time and overtime cost per boat • Unit holding cost per month in inventory • Monthly demands for boats Decision Variables • Number of boats to be supplied for each month from possible “supplies” • Supplies indicate the source of the boats: • Initial inventory • Regular-time labor in a particular month • Overtime labor in a particular month

7. Job Assignments at MachincoThe Assignment Problem • Assign jobs to machines so that each job is assigned and each machine does at most one job • Minimize total time to do all jobs

8. Job Assignments at MachincoModeling Approach • Model as a transportation problem, where all supplies and demands are 1 • Supplies correspond to machines (each with a supply of 1) • Demands correspond to jobs (each with a demand of 1)

9. Job Assignments at MachincoObjective • Minimize the total time to complete all jobs

10. Job Assignments at MachincoConstraints • Each job must be assigned to some machine • Each machine can do at most one job

11. Job Assignments at MachincoInputs • The time required to do each job on each machine

12. Job Assignments at MachincoDecision Variables • Which job-to-machine assignments to make

14. Critical Path ModelBasic Problem • Analyze the length of time required to complete a project composed of activities with precedence relations (some activities can’t begin until others are completed) • See which activities are critical (the total project would be delayed if they were delayed)

15. Critical Path ModelObjective • Schedule the activities in order to minimize the total project time

16. Critical Path ModelConstraints • Because of built-in precedence relations, activities can’t begin until their predecessors are completed

17. Critical Path ModelInputs • Precedence relations • Durations of activities

18. Critical Path ModelDecision Variables • The times corresponding to the nodes in the project network • These are actually the earliest times certain activities can begin (e.g., node 2 is the earliest activities C and D can begin)

19. Project Network(See “Chart1” sheet in Excel) • Precedence relations can be summarized in a graph called an “activity-on-arc” network • Each node corresponds to a point in time • Each arc corresponds to an activity • Precedence relations are obtained by joining certain nodes with certain arcs • Node 1 is a “start” node (time 0) • The last node is a “finish” node

21. Shipping Food at FoodcoBasic Problem • Ship food from production plants to customers at least cost • Food can be shipped directly to customers or from plants to warehouses and then to customers • See “Chart1” sheet in Excel

22. Shipping Food at FoodcoObjective • Minimize the total shipping cost • Each shipping cost is proportional to the amount shipped along the route

23. Shipping Food at FoodcoConstraints • Arc capacities can’t be exceeded • There must be “flow balance” at each node • There is positive net outflow at each supply point (plants) • There is zero net outflow at each transshipment point (warehouses) • There is positive net inflow (negative net outflow) at each demand point (customers)

24. Shipping Food at FoodcoInputs • Unit shipping costs • Arc capacities • Supplies at supply points • Demands at demand points

25. Shipping Food at FoodcoDecision Variables • Flows along all arcs • Includes flows into dummy node (which is excess capacity not shipped)

27. Maximum Oil Flow at SuncoBasic Problem • Ship as much oil (per unit time) from a “source” node to a “sink” (destination) node as possible along a given network of pipelines • See “Chart1” sheet in Excel

28. Maximum Oil Flow at SuncoObjective • Maximize the total flow from source to sink per unit of time

29. Maximum Oil Flow at SuncoConstraints • Don’t exceed arc (pipeline) capacities • Achieve flow balance at each node • By adding a dummy arc from the sink to the source, we can let all net outflows be zero

30. Maximum Oil Flow at SuncoInputs • Arc capacities • These indicate how much oil can go through a given pipeline per unit of time

31. Maximum Oil Flow at SuncoDecision Variables • Arc flows • These include the flow along the dummy arc (which isn’t an actual physical flow)

33. Shortest Route: Car Replacement Basic Problem • Decide on a least-cost purchasing/selling strategy for cars, given that a car is needed at all times • Economic reason for selling cars is that maintenance costs increase with age and trade-in value decreases with age

34. Shortest Route: Car ReplacementSolution Strategy • Model as a shortest route problem • Origin is year 1 • Destination is end of planning horizon • Any path from node 1 to node 6 represents a replacement strategy

35. Shortest Route: Car Replacement Objective • Minimize the total cost of owning a car during the planning horizon, including: • The cost of purchasing new cars • The maintenance cost of owning cars • The trade-in value of replaced cars

36. Shortest Route: Car Replacement Constraints • Flow balance constraints

37. Shortest Route: Car Replacement Inputs • Length of planning horizon • Cost of a new car • Maintenance cost per year, which increases with the age of the car • Trade-in value of car, which decreases with the age of the car

38. Shortest Route: Purchasing CarsDecision Variables • Flows on the arcs

40. Investing at StockcoBasic Problem • Choose the investments that stay within a budget and maximize the NPV • Each investment is an all-or-nothing decision

41. Investing at Stockco Objective • Maximize the NPV of the investments chosen

42. Investing at Stockco Constraints • Cash spent on investments can’t be greater than cash available

43. Investing at Stockco Inputs • Amount of cash required for each investment • Amount of NPV obtained from each investment

44. Investing at Stockco Decision variables • Whether to invest or not in each investment • This is indicated by a 0-1 changing cell, which is 1 for an investment that is chosen, 0 otherwise