Transportation Model (Powerco) Send electric power from power plants to cities where power is needed at minimum cost Transportation between supply and demand points, with the objective of minimizing 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

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) Constraints Unit shipping costs along each route Amount of supply at each power plant Demand at each city Inputs The amount to ship along each route –There is a route from each supply point to each demand point –No other routes are allowed Decision Variables Powerco Contd

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 Demand Supply Month RTOT Inventory

Minimize total costs, which include –Regular-time labor costs, Overtime labor costs, Inventory holding costs 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 Objective Inputs 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 Decision Variables

Job Assignments at Machinco The 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

Job Assignments at Machinco Modeling 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)

Job Assignments at Machinco Objective Minimize the total time to complete all jobs

Job Assignments at Machinco Constraints Each job must be assigned to some machine Each machine can do at most one job

Job Assignments at Machinco Inputs The time required to do each job on each machine

Job Assignments at Machinco Decision Variables Which job-to-machine assignments to make

Critical Path Model Basic 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)

Critical Path Model Objective Schedule the activities in order to minimize the total project time

Critical Path Model Constraints Because of built-in precedence relations, activities can’t begin until their predecessors are completed

Critical Path Model Inputs Precedence relations Durations of activities

Critical Path Model Decision 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)

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

Shipping Food at Foodco Basic 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

Shipping Food at Foodco Objective Minimize the total shipping cost –Each shipping cost is proportional to the amount shipped along the route

Shipping Food at Foodco Constraints 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)

Shipping Food at Foodco Inputs Unit shipping costs Arc capacities Supplies at supply points Demands at demand points

Shipping Food at Foodco Decision Variables Flows along all arcs –Includes flows into dummy node (which is excess capacity not shipped)

Maximum Oil Flow at Sunco Basic 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

Maximum Oil Flow at Sunco Objective Maximize the total flow from source to sink per unit of time

Maximum Oil Flow at Sunco Constraints 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

Maximum Oil Flow at Sunco Inputs Arc capacities –These indicate how much oil can go through a given pipeline per unit of time

Maximum Oil Flow at Sunco Decision Variables Arc flows –These include the flow along the dummy arc (which isn’t an actual physical flow)

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

Shortest Route: Car Replacement Solution 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

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

Shortest Route: Car Replacement Constraints Flow balance constraints

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

Shortest Route: Purchasing Cars Decision Variables Flows on the arcs

Investing at Stockco Basic Problem Choose the investments that stay within a budget and maximize the NPV Each investment is an all-or-nothing decision

Investing at Stockco Objective Maximize the NPV of the investments chosen

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

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

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