CASE: PERFORMANCE LAWN EQUIPMENT

Elizabeth Burke wants to develop a model to more effec$-$tively 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.

Ms$.$ Burke explained that she could change the pro$-$duction rate by using planned overtime or undertime $($pro$-$ducing 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. Con$-$sequently, she believes that the company can save a signif$-$icant 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 didn$$t fully

understand the approach. The PowerPoint notes didn$$t

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 optimiza$-$tion model. Here are the highlights from the presentation:

Variables:

Xt $=$planned production in period t$.$ Month NA SA Eur Pac China World

Jan$-1460000571130911045074662$

Feb$-1477184611176791111096585$

Mar$-14778856582275910680102369$

Apr$-14861907782796612370116171$

May$-14961178862789513130126210$

Jun$-14971438823056611760129768$

Jul$-14847578482944413590116409$

Aug$-14798047352836412380110141$

Sep$-1464800657283931215095065$

Oct$-1459307595244441154085500$

Nov$-1452157553180001262071972$

Dec$-1445049462124531386059349$

Jan$-1558627553127781443073401$

Feb$-1576200615182141515096545$

Mar$-15828716582388913730108791$

Apr$-15849047842945514420116584$

May$-15931008462946412150124625$

Jun$-15930008382741413330122585$

Jul$-15830487632736814150112594$

Aug$-15748546942732112960104164$

Sep$-1560769625294441402092241$

Oct$-1555619610237741468081470$

Nov$-1548155571173081351067386$

Dec$-1542647512129411389057489$

Jan$-1657885537109621509070892$

Feb$-1677647595152731402094917$

Mar$-16818456592055615240104583$

Apr$-16860957562678615740115211$

May$-16917768782482814680118949$

Jun$-161006808252473715600127801$

Jul$-16861907562482814410113216$

Aug$-1671887714251791545099325$

Sep$-1660000651245451667086863$

Oct$-1655566643192861698077193$

Nov$-1650857619152731810068558$

Dec$-164259654891071731053982$

Jan$-175809558185711887069135$

Feb$-1775566614131581845091182$

Mar$-17802866221965519230102486$

Apr$-1785140727251791981125113027$

May$-1790093826231031810142115832$

Jun$-1795472783242861942226122482$

Jul$-1787308681247371961230114686$

Aug$-1774476646266072000122103729$

Sep$-17616986252298220755687381$

Oct