Elizabeth Burke wants to develop a model to more effectively plan production for the next year. Currently, PLE has a planned capacity of producing 9,100 mowers each month, which is approximately the average monthly demand over the previous year. However, looking at the unit sales figures for the previous year, she observed that the demand for mowers has a seasonal fluctuation, so with this level production strategy, there is overproduction in some months, resulting in excess inventory buildup and underproduction in others, which may result in lost sales during peak demand periods.

In discussing this with her, she explained that she could change the production rate by using planned overtime or undertime (producing more or less than the average monthly demand), but this incurs additional costs, although it may offset the cost of lost sales or of maintaining excess inventory. Consequently, she believes that the company can save a significant amount of money by optimizing the production plan.

Ms. Burke saw a presentation at a conference about a similar model that another company used but didnt fully understand the approach. The PowerPoint notes didnt have all the details, but they did explain the variables and the types of constraints used in the model. She thought they would be helpful to you in implementing an optimization model. Here are the highlights from the presentation:

Variables:

Xt= planned production in period t

It= inventory held at the end of period t

Lt= number of lost sales incurred in period t

Ot= amount of overtime scheduled in period t

Ut= amount of undertime scheduled in period t

Rt= increase in production rate from period t1 to period t

Dt= decrease in production rate from period t1 to period t

Material balance constraint:

Xt+It1It+Lt=demand in montht

Overtime/undertime constraint:

OtUt=Xtnormal production capacity

Production rate-change constraint:

XtXt1=RtDt

Ms. Burke also provided the following data and estimates for the next year: unit production cost=$70.00; inventory-holding cost=$1.40per unit per month; lost sales cost=$200per unit; overtime cost=$6.50per unit; undertime cost=$3.00per unit; and production-rate-change cost=$5.00per unit , which applies to any increase or decrease in the production rate from the previous month. Initially, 900 units are expected to be in inventory at the beginning of January, and the production rate for December 2012 was 9,100 units. She believes that monthly demand will not change substantially from last year, so the sales figures for last year in the PLE database should be used for the monthly demand forecasts.

Your task is to design a spreadsheet that provides detailed information on monthly production, inventory, lost sales, and the different cost categories and solve a linear optimization model for minimizing the total cost of meeting demand over the next year. Compare your solution with the level production strategy of producing 9,100 units each month. Interpret the Sensitivity report, and conduct an appropriate study of how the solution will be affected by changing the assumption of the lost sales costs. Summarize all your results in a report to Ms. Burke.