Case Problem

Spring Garden Tools

he Spring family has owned and operated a garden tool and implements manufacturing company since $1952.$ The company sells garden tools to distributors and also directly to hardware stores and home improvement discount chains. The Spring Company's four most popular small garden tools are a trowel, a hoe, a rake, and a shovel. Each of these tools is made from durable steel and has a wooden handle. The Spring family prides itself on its high$-$quality tools.

The manufacturing process encompasses two stages. The first stage includes two operations$-$stamping out the metal tool heads and drilling screw holes in them. The completed tool heads then flow to the second stage, which includes an assembly operation where the handles are attached to the tool heads, a finishing step, and packaging. The processing times per tool for each operation are provided in the following table:

$\backslash$table$[[,,,,,\backslash$table$[[$Total Hours$],[$Available$],[$per Month$]]],[$Operation$,$Tool $($hr$./$unit$),,],[$Stamping$,0\mathrm{.}04,0\mathrm{.}17,0\mathrm{.}06,0\mathrm{.}12,500],[$Drilling$,0\mathrm{.}05,0\mathrm{.}14,-,0\mathrm{.}14,400],[$Assembly$,0\mathrm{.}06,0\mathrm{.}13,0\mathrm{.}05,0\mathrm{.}10,600],[$Finishing$,0\mathrm{.}05,0\mathrm{.}21,0\mathrm{.}02,0\mathrm{.}10,550],[$Packaging$,0\mathrm{.}03,0\mathrm{.}15,0\mathrm{.}04,0\mathrm{.}15,500]]$

The steel the company uses is ordered from an iron and steel works in Japan. The company has $10,000$ square feet of sheet steel available each month. The metal required for each tool and the monthly contracted production volume per tool are provided in the following table:

$\backslash$table$[[,$Sheet Metal $($ft$.{?}^2),$Monthly Contracted Sales$],[$Trowel$,1\mathrm{.}2,1,800],[$Hoe$,1\mathrm{.}6,1,400],[$Rake$,2\mathrm{.}1,1,600],[$Shovel$,2\mathrm{.}4,1,800]]$

The primary reasons the company has survived and prospered are its ability always to meet customer demand on time and its high quality. As a result, the Spring Company will produce on an overtime basis to meet its sales requirements, and it also has a long$-$standing arrangement with a local tool and die company to manufacture its tool heads. The Spring Company feels comfortable subcontracting the first$-$stage operations because it is easier to detect defects prior to assembly and finishing. For the same reason, the company will not subcontract for the entire tool because defects would be particularly hard to detect after the tool was finished and packaged. However, the company does have $100$ hours of overtime available each month for each operation in both stages. The regular production and

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CHAPTER $4$ LINEAR PROGRAMMING: MODELING EXAMPLES

vertime costs per tool for both stages are provided in the ollowing table: $$

$\backslash$table$[[,$Stage $1,,$Stage $2],[,\backslash$table$[[$Regular$],[$Cost$]],\backslash$table$[[$Overtime$],[$Cost$]],,\backslash$table$[[$Regular$],[$Cost$]],\backslash$table$[[$Overtime$],[$Cost$]]],[$Trowel$,$$$6\mathrm{.}00,$$$6\mathrm{.}20,,$$$3\mathrm{.}00,$$$3\mathrm{.}10$