1. 
   1) What is the interpretation of the Dual Prices on the demand constraints?
2. 2) What would happen if Spring Garden was unable to subcontract in stage 1?
3. 3) Why does one variable have both a zero value and zero reduced cost? Which variable is it and what is the significance?

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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: T 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 Sheet Metal (ft.2) Monthly Contracted Sales tools is made from durable steel and has a wooden handle. Trowel 1.2 1,800 Hoe 1.6 1,400 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 at- tached to the tool heads, a finishing step, and packaging. The processing times per tool for each operation are pro- vided in the following table: Rake 2.1 1,600 Shovel 2.4 1,800 The primary reasons the company has survived and prospered are its ability always to meet customer de- mand 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 be- cause defects would be particularly hard to detect after the tool was finished and packaged. However, the com- pany does have 100 hours of overtime available cach month for each operation in both stages. The regular Total Hours Available per Tool (hr./unit) Operation Trowel Hoe Rake Shovel Month 0.04 0.05 500 Stamping Drilling Assembly Finishing Packaging 0.17 0.06 0.12 0.14 0.13 0.05 0.14 400 0.06 0.10 600 0.05 0.21 0.02 0.10 550 0.03 0.15 0.04 0.15 500 CASE PROBLEM 181 production and overtime costs per tool for both stages are provided in the following table: 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 min- Stage 1 Stage 2 Regular Cost Overtime Regular Cost Overtime imum 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? Cost Cost Trowel $6.00 $6.20 $3.00 $3.10 Ное Rake 10.70 8.50 10.00 5.00 5.40 4.30 8.00 4.00 Shovel 10.00 10.70 5.00 5.40 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: T 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 Sheet Metal (ft.2) Monthly Contracted Sales tools is made from durable steel and has a wooden handle. Trowel 1.2 1,800 Hoe 1.6 1,400 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 at- tached to the tool heads, a finishing step, and packaging. The processing times per tool for each operation are pro- vided in the following table: Rake 2.1 1,600 Shovel 2.4 1,800 The primary reasons the company has survived and prospered are its ability always to meet customer de- mand 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 be- cause defects would be particularly hard to detect after the tool was finished and packaged. However, the com- pany does have 100 hours of overtime available cach month for each operation in both stages. The regular Total Hours Available per Tool (hr./unit) Operation Trowel Hoe Rake Shovel Month 0.04 0.05 500 Stamping Drilling Assembly Finishing Packaging 0.17 0.06 0.12 0.14 0.13 0.05 0.14 400 0.06 0.10 600 0.05 0.21 0.02 0.10 550 0.03 0.15 0.04 0.15 500 CASE PROBLEM 181 production and overtime costs per tool for both stages are provided in the following table: 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 min- Stage 1 Stage 2 Regular Cost Overtime Regular Cost Overtime imum 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? Cost Cost Trowel $6.00 $6.20 $3.00 $3.10 Ное Rake 10.70 8.50 10.00 5.00 5.40 4.30 8.00 4.00 Shovel 10.00 10.70 5.00 5.40

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