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The 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:

Tool $($hr$./$unit$)$ Total Hours

Available per

Operation Trowel Hoe Rake Shovel Month

Stamping $0\mathrm{.}040\mathrm{.}170\mathrm{.}060\mathrm{.}12500$

Drilling $0\mathrm{.}050\mathrm{.}140\mathrm{.}14400$

Assembly $0\mathrm{.}060\mathrm{.}130\mathrm{.}050\mathrm{.}10600$

Finishing $0\mathrm{.}050\mathrm{.}210\mathrm{.}020\mathrm{.}10550$

Packaging $0\mathrm{.}030\mathrm{.}150\mathrm{.}040\mathrm{.}15500$ 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:

Sheet Metal $($ft$\mathrm{.}2)$ Monthly Contracted Sales

Trowel $1\mathrm{.}21,800$

Hoe $1\mathrm{.}61,400$

Rake $2\mathrm{.}11,600$

Shovel $2\mathrm{.}41,800$

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 in order 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 overtime costs per tool for both stages are

provided in the following table:

Stage $1$ Stage $2$

Regular

Cost

Overtime

Cost

Regular

Cost

Overtime

Cost

Trowel $$6\mathrm{.}00$ $$6\mathrm{.}20$ $$3\mathrm{.}00$ $$3\mathrm{.}10$

Hoe $10\mathrm{.}0010\mathrm{.}705\mathrm{.}005\mathrm{.}40$

Rake $8\mathrm{.}008\mathrm{.}504\mathrm{.}004\mathrm{.}30$

Shovel $10\mathrm{.}0010\mathrm{.}705\mathrm{.}005\mathrm{.}40$

The cost of subcontracting in stage $1$ adds $20\%$ to the

regular production cost.

The Spring Company wants to establish a production

schedule for regular and overtime production in each stage

and for the number of tool heads subcontracted, at the minimum

cost$.$ Formulate a linear programming model for this

problem and solve the model using the computer. Which

resources appear to be most critical in the production process?