Problem # $1($Planning the production of electronic components, G$.$ P$.$ S$.,8\mathrm{.}4,$ pg $113,[/30])$

To augment its competitiveness, a small business wishes to improve the production of its best selling prod$-$

ucts. One of its main activities is the production of cards with microchips and electronic badges. The

company also produces the components for these cards and badges. Good planning of the production of

these components therefore constitutes a decisive success factor for the company. The demand for compo$-$

nents is internal in this case and hence easy to anticipate.

For the next six months the production of four products with references X$43-$M$1,$ X$43-$M$2,$ Y$54-$N$1,$ Y$54-$N$2$

is to be planned. The production of these components is sensitive to variations of the level of production,

and every change leads to a non$-$negligible cost through controls and readjustments. The company therefore

wishes to minimize the cost associated with these changes whilst also taking into account the production

and storage costs.

The demand data per time period, the production and storage costs, and the initial and desired final stock

levels for every product are listed in Table $1.$ When the total production level over all products changes,

readjustments of the machines and controls have to be carried out for the current month. The cost incurred

is proportional to the increase or reduction of the production compared to the preceding month. The cost

for an increase of the production is $$1$ per unit but only $$0\mathrm{.}50$ for a decrease of the production level.

Demands Cost Stock levels

Month $123456$ Production Storage Initial Final

X$43-$M$1150030002000400020002500200\mathrm{.}41050$

X$43-$M$2130080080010001100900250\mathrm{.}5010$

Y$54-$N$1220015002900180012002100100\mathrm{.}3010$

Y$54-$N$2140016001500100011001200150\mathrm{.}3010$

Table $1$: Data for the four products

$1.$ Formulate a linear optimization model to determine a production plan that minimizes the sum of

costs incurred through changes of the production level, production costs, and storage costs over the

planning horizon. Your model should be algebraic