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[Point 25] A diner is open from 11am to 7pm. Each employee i…

[Point 25] A diner is open from 11am to 7pm. Each employee it hires works for four consecutive hours. The diner can decide the shift for each employee. Employees can start at either 11am, 12pm, 1pm, 2pm or 3pm. The diner has recorded the ideal number of employees it would need during each hour in the following table: Hours # of employees 11am-12pm 5 12pm-1pm 8 1pm-2pm 11 2pm-3pm 7 3pm-4pm 13 4pm-5pm 10 5pm- 6pm 15 6pm-7pm 16   The diner can have more or less than the ideal number of employees at each hour. However, if it is short of staff then it will pay a penalty of $20 per time period, per employee under the desired number. Each employee makes $30 per four-hour shift, except for those starting at 11am and at 3pm, who make $40 per shift because they have to open (or close) the diner. In this model variable xi represents the number of employees that start working on shift i, variable sj represents the number of employees below the ideal number during hour j. Moreover, the constraints describe that the number of employees that work during a given hour plus the number of employees we are short of during this hour should reach at least the ideal number of employees needed during this hour. To obtain a shift schedule that minimizes the cost of operating the diner, we write and solve the following GAMS model: The solution and sensitivity reports produced by GAMS are presented in the following figure.   (a) The manager realizes that the employees who open the diner have much more work than the others. Therefore, he would like to pay those employees $45 per four-hour shift. By how much does the minimum operating cost change?   (b) The manager realizes that, to ensure that the lunch service runs smoothly, it is critical to staff the 1pm-2pm period carefully. Therefore, he decides to increase the penalty for being short of staff during this hour to $25. By how much does the minimum cost change after this decision?   (c) Following some complaints from patrons who arrive between 6pm and 7pm, the manager decides that the ideal number of employees during this hour should be 18. By how much does the minimum cost change after this decision?   (d) Identify one constraint that could be removed from the problem without altering the optimal primal solution. Justify your answer.