1 A Logistics Problem The Dispatch Manager for ABC Logistics needs to send a fleet of 8 small trucks and 4 large trucks from a depot to pick up items at five plants located near Kyoto and deliver those items to a central warehouse (each truck can go to only one plant). The number of (identical) items to be picked up at each plant as well as the distances in kilometers from the depot to each plant and from each plant to the warehouse are given in the following table: Plant 1Plant 2Plant 3Plant 4Plant 5 Depot (kms) Warehouse (kms) Number of Items Each small truck can hold up to 50 items and each large truck can hold up to 100 items.

2 A Logistics Problem Draw a graph to represent this problem. Indicate the meaning of the vertices and edges. Include all of the data in your diagram. Use the following variables to formulate a model with objective function and constraints to determine how to use the fleet of trucks to pick up the items at the plants and deliver those items to the warehouse so that total distance traveled by all the trucks from the depot to the warehouse is as small as possible. Is your model a linear program? Why or why not? S j = the number of small trucks to send from the Depot to Plant j (j = 1, …, 5) L j = the number of large trucks to send from the Depot to Plant j (j = 1, …, 5)

3 A Scheduling Problem An ambulance service must schedule its drivers on a daily basis. The driver’s schedules vary and may be 4 hours (part-time) or 8 hours (full-time). Starting times of these drivers may be anytime but, for practical purposes, are limited to midnight, 4 A.M., 8A.M., Noon, or 4 P.M. Part-time drivers can also start at 8 P.M. but full-time drivers cannot. The demand for these vehicles depends on the day of the week and is given in the following table for Monday.

4 A Scheduling Problem The company estimates that each vehicle staffed with a part-time driver costs $75/hour, whereas one with full- time driver cost $60/hour. Formulate a mathematical model with variables, objective function and constraint to decide the number of full-time and part-time drivers to schedule at each starting time so as to satisfy the demand with least cost. Clearly define all the decision variables used.