Mathematical modeling

2.7.2 Workforce PlanningProblem DefinitionConsider a restaurant that is open seven days a week. Based on past experience, thenumber of workers needed on a particular day is given as follows:Day Mon Tue Wed Thu Fri Sat SunNumber 14 13 15 16 19 18 11Every worker works five consecutive days, and then takes two days off, repeating thispattern. How can we minimize the number of workers that staff the restaurant?ModelA first attempt at this problem is to let x i be the number of people working on day i. Notethat such a variable definition does not match up with what we need. Some workers willwork both Monday and Tuesday, some only one day, some neither of those days. Instead,let the days be numbered 1 through 7 and let x i be the number of workers who begin theirfive day shift on day i. Our objective is clearly minimize: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7Consider the constraint for Monday's staffing is 14. Who works on Mondays? Clearlythose who start their shift on Monday (x 1 ). Those who start on Tuesday (x 2 ) do not workon Monday, nor do those who start on Wednesday (x 3 ). Those who start on Thursday (x 4 )do work on Monday, as do those who start on Friday, Saturday, and Sunday. This givesthe constraint: x 1 + x 4 + x 5 + x 6 + x 7 ≥ 14. Similar arguments give a total formulation of:Minimize x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7Subject to x 1 + x 4 + x 5 + x 6 + x 7 ≥ 14x 1 + x 2 + x 5 + x 6 + x 7 ≥ 13x 1 + x 2 + x 3 + x 6 + x 7 ≥ 15x 1 + x 2 + x 3 + x 4 + x 7 ≥ 16x 1 + x 2 + x 3 + x 4 + x 5 ≥ 19x 2 + x 3 + x 4 + x 5 + x 6 ≥ 18x 3 + x 4 + x 5 + x 6 + x 7 ≥ 11x i ≥ 0 (for all i)Systems Analysis 35 Dr. Emad Elbeltagi