[ewilliams]

This is a canonical type of integer programming problem I often teach in a graduate-level decision science course. A good explanation appears in Albright, S. Christian, and Wayne L. Winston. 2005. Spreadsheet Modeling and Applications: Essentials of Practical Management Science. Belmont, California: Thomson Learning, Incorporated. Briefly, the trick is this: do not concentrate on variables specifying how many workers are on-duty each day. The clever trick is: set variables to represent the number of workers STARTING their 5-day period each day of the week. Then, for example, the constraint that 19 workers must be on duty Tuesday (the busiest day) is represented as #-starting-Friday + #-starting-Saturday + #-starting-Sunday + #-starting-Monday + #-starting-Tuesday >= 19. You will have seven decision variables (all presumably positive and integer), one for each day of the week, seven constraints of the type just shown, and an objective function to minimize the sum of the seven decision variables.