Elizabeth Burke wants to develop a model to more effectively plan production for the next year. Currently, PLE has a planned capacity of producing 9100 mowers each month, which is approximately the average monthly demand over the previous year. However, looking at the unit sales figure for the previous year, she observed that the demand for mowers has a seasonal fluctuation, so with this level production strategy there is over-production 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 and under time (producing more or less than the average monthly demand), but this incurs additional cost, 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 by optimizing the production plan. Ms. Burke saw a presentation at a conference about a similar model that another company used but didn’t fully understand the approach. The power point notes didn’t have all the details but they did explain the variables and types of constraints used in the model. she thought they would be helpful to you in implementing an optimization model. Here are the highlights of the presentation.

X^t=planned production in period t

I^t=inventory held at the end of period t

l^t=number of lost sales incurred in period t

O^t=amount of overtime scheduled in period t

U^t=amount of under time scheduled in period t

R^t=increase in production rate from period t-1 to period t

D^t=decrease in production rate from period t-1 to period t

material in balance constraint: (X^t) + (I^t) – (I^t) + (L^t) =demand of the month

Overtime /undertime constraint: (O^t)-(U^t) = (X^t)- normal production capacity

Production rate-change constraint: (X^t) - (X^(t-1)) = (R^t) – (D^t)

Ms. burke also provided the following data and estimates for the next year: unit cost production = $70: inventory holding cost= $1.40 per unit per month; lost sales cost=$200 per unit; overtime cost =$6.50 per unit; undertime cost is $3.00 per unit; and production rate change cost = $5.00 per 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 9100 units. she believed that the 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 forecast.

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 the solution with the level of production strategy of producing 9100 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 cost.

Data:

Month	NA	SA	Europe	Pacific	China	World
Jan-18	6210	270	400	200	0	7080
Feb-18	8030	280	750	190	0	9250
Mar-18	8540	300	970	210	5	10025
Apr-18	9120	340	1310	220	16	11006
May-18	9570	390	1260	200	22	11442
Jun-18	10230	380	1240	210	26	12086
Jul-18	9580	350	1300	230	14	11474
Aug-18	7680	340	1250	220	15	9505
Sep-18	6870	320	1210	220	11	8631
Oct-18	5930	310	970	230	3	7443
Nov-18	5260	300	650	240	1	6451
Dec-18	4830	290	300	230	0	5650

If possible could you show the excel step by step calculation for i^t, x^t etc.