Joe thinks that there must be a middle ground, and that is where he wants to position his company. Parket Sisters makes high$-$quality items, with limited trim and diversity, but also offers lifetime warranties. Furthermore, its pens and pencils are ergonomically efficient. Joe knows that some people want the status of the Mont Blanc Meister stuck pen, for example, but he has never met a person who said that writing with such a pen is enjoyable. The pen is too large and clumsy for smooth writing. Parket Sisters$$ products, on the other hand, have a reputation for working well, are easy to hold and use, and cause limited $$writer$$s fatigue.$$Parket Sisters makes only three items$$a ballpoint pen, a mechanical pencil, and a fountain pen. All are available in just one color, black, and are sold mostly in specialty stores and from better catalog companies. The per$-$unit profit of the items is $$3\mathrm{.}00$ for the ballpoint pen, $$3\mathrm{.}00$ for the mechanical pencil, and $$5\mathrm{.}00$ for the fountain pen. These values take into account labor, the cost of materials, packing, quality control, and so on$.$ The company is trying to plan its production mix for each week. Joe believes that the company can sell any number of pens and pencils it produces, but production is currently limited by the available resources. Because of a recent strike and certain cash$-$flow problems, the suppliers of these resources are selling them to Parket Sisters in limited amounts. In particular, Joe can count on getting at most $1,000$ ounces of plastic, $1,200$ ounces of chrome, and $2,000$ ounces of stainless steel each week from his suppliers, and these figures are not likely to change in the near future. Because of Joe$$s excellent reputation, the suppliers will sell Joe any amount $($up to his limit$)$ of the resources he needs when he requires them. That is$,$ the suppliers do not require Joe to buy some fixed quantities of resources in advance of his production of pens and pencils; therefore, these resources can be considered variable costs rather than fixed costs for the pens and pencils. Each ballpoint pen requires $1\mathrm{.}2$ ounces of plastic, $0\mathrm{.}8$ ounces of chrome, and $2$ ounces of stainless steel. Each mechanical pencil requires $1\mathrm{.}7$ ounces of plastic, no chrome, and $3$ ounces of stainless steel. Each fountain pen requires $1\mathrm{.}2$ ounces of plastic, $2\mathrm{.}3$ ounces of chrome, and $4\mathrm{.}5$ ounces of stainless steel. Joe believes LP could help him decide what his weekly product mix should consist of$.$ Getting his notes and notebooks, Joe grapples with the LP formulation. In addition to the constraints of the available resources, he recognizes that the model should include many other constraints $($such as labor time availability and materials for packing$).$ However, Joe wants to keep his model simple. He knows that eventually, he$$ll have to take other constraints into account, but as a first$-$pass model, he$$ll restrict the constraints to just the three resources: plastic, chrome, and stainless steel.