1 Optimization – Part II More Applications of Optimization to Operations Management In this session, we continue our discussion of “Optimization” and its application to decision problems arising in Production & Operations Management. For this session, the learning objectives are:  Using Solver to solve a Production & Inventory Planning Problem.  Using Solver to solve an Employee Scheduling Problem.

2 A Production and Inventory Planning Problem The Suny Corporation, a manufacturer of VCRs, wants to plan its production and inventory quantities for the next six months. Its demand forecasts for the next six months are given in the second column of the table below. Because of fluctuations in the costs of such things as raw materials and utilities, a VCR’s unit cost of production varies from month to month. The table’s third column specifies Suny’s forecasts of each month’s unit cost of production. Suny’s maximum production quantity also varies from month to month, because of such things as differences in each month’s required maintenance and number of working days. It is Suny’s policy not to change its workforce size from month to month. Consequently, to prevent excessive idleness, Suny has set the minimum monthly production quantity at 50% of the maximum. The table’s fourth and fifth columns contain each month’s minimum and maximum production quantities. Suny currently has 3500 VCRs in inventory. To ensure sufficient safety stock, Suny has specified 2500 VCRs as the minimum permissible inventory quantity at the end of any month. Given the available storage space, Suny’s maximum permissible inventory quantity at the end of any month is 7000 VCRs. The table’s next-to-last and last columns contain each month’s minimum and maximum inventory quantities. Suny’s accounting department has estimated that it costs $8 per month to hold a VCR in inventory, including the opportunity cost of foregone interest. Furthermore, the accounting department recommends that Suny compute each month’s inventory costs by multiplying the $8 per month figure by the average of the month’s beginning and ending inventory quantities. Given this data, Suny’s goal is to find the production and inventory quantities for each month that will minimize the total cost (total production costs plus total inventory costs) of satisfying forecasted demand with no backlogging.

3 Formulation as a Linear Program Minimize Total Costs Let P 1, P 2, P 3, P 4, P 5, & P 6 denote each month’s production quantity. Let I 1, I 2, I 3, I 4, I 5, & I 6 denote each month’s ending inventory level. Min & Max Production Min & Max Inventory Monthly Balance Equations “In” = “Out” Nonnegativity Constraints

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6 A “Rolling” Production Plan

7 An Employee Scheduling Problem Maintenance at the Haasland theme park is an ongoing process that occurs 24 hours per day. The table below summarizes how Haasland’s requirements for on-duty maintenance workers varies according to the time of day. Workers report for duty at the start of each of the above six time periods and work an 8-hour shift for 8 consecutive hours. Formulate and solve a linear program that Haasland can use to minimize the total number of workers it must hire to meet its needs throughout the day.

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9 Optimized Spreadsheet for Haasland’s Employee Scheduling Problem

10 Another Employee Scheduling Problem St. Andrew’s Hospital is currently planning the schedules for the registered nurses (RNs) who work during the so-called “swing shift” during the interval of 3:00-11:00 PM. The table below summarizes St. Andrew’s minimal daily requirement for swing-shift RNs: St. Andrew’s contract with the RNs calls for each RN to have 2 consecutive days off each week and to work the remaining 5 days. This contract further calls for an RN working the swing shift to receive a daily salary of $100 for any weekday, $150 for Saturday, and $175 for Sunday. For example, if an RN were to work Wednesday, Thursday, Friday, Saturday, & Sunday, the weekly salary would be 3($100)+$150+$175=$625. Formulate and solve an INTEGER linear program that St. Andrew’s can use to minimize the total weekly payroll costs for swing-shift RNs.