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Give step-by-step solution with explanation and final answer:The Spring family has owned and operated a garden tool and implements manufacturing company since 1952. The company sells garden tools to distributors and 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 ————————— Avallable per Operation Trowel Hoe Rake Shovel Month Samping 004 017 006 0.12 500 Drilling ~~ 005 014 — 014 00 Assembly 006 013 005 0.10 60 Finishing 005 021 002 0.10 550 Packiging 003 015 004 015 500 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: Sheet Metal (ft.) Monthly Contracted Sales Trowel 12 1.800 Hoe 16 1400 Rake 21 1.600 Shovel 24 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 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.

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Give step-by-step solution with explanation and final answer:Uploaded ImageThe Spring family has owned and operated a garden tool and implements manufacturing company since 1952. The company sells garden tools to distributors and 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 ————————— Avallable per Operation Trowel Hoe Rake Shovel Month Samping 004 017 006 0.12 500 Drilling ~~ 005 014 — 014 00 Assembly 006 013 005 0.10 60 Finishing 005 021 002 0.10 550 Packiging 003 015 004 015 500 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: Sheet Metal (ft.) Monthly Contracted Sales Trowel 12 1.800 Hoe 16 1400 Rake 21 1.600 Shovel 24 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 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.

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Asked by: gvr
Created at: 2025-07-26 20:26:44

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Let's break down the problem and solve it step by step: ## **Step 1: Identify Constraints** ### **1. Processing Hours Available per Month** Each operation (Stamping, Drilling, Assembly, Finishing, Packaging) has a maximum number of regular hours available per month: - Stamping: 500 hours - Drilling: 400 hours - Assembly: 550 hours - Finishing: 550 hours - Packaging: 500 hours ### **2. Overtime** - 100 hours of overtime for both stages (stamping & drilling) **combined** per month. - 100 hours of overtime for both stages (assembly, finishing, packaging) **combined** per month. ### **3. Sheet Metal Availability** - 10,000 sq. ft. per month. ### **4. Monthly Contracted Sales (Minimum Production)** - Trowel: 1,800 units - Hoe: 1,400 units - Rake: 1,600 units - Shovel: 1,800 units ## **Step 2: List Resource Requirements** ### **A. Tool Operations Table** | Operation | Trowel | Hoe | Rake | Shovel | Hours Available (Regular) | |-----------|--------|-----|------|--------|---------------------------| | Stamping | 0.04 | 0.17| 0.06 | 0.12 | 500 | | Drilling | 0.05 | 0.14| 0.09 | 0.14 | 400 | | Assembly | 0.05 | 0.13| 0.05 | 0.10 | 550 | | Finishing | 0.06 | 0.15| 0.02 | 0.10 | 550 | | Packaging | 0.03 | 0.15| 0.04 | 0.15 | 500 | ### **B. Sheet Metal Requirements** | Tool | Sheet Metal (ft²/unit) | Contracted Sales (units) | |--------|------------------------|--------------------------| | Trowel | 1.2 | 1,800 | | Hoe | 1.6 | 1,400 | | Rake | 2.1 | 1,600 | | Shovel | 2.4 | 1,800 | ## **Step 3: Calculate Total Resources Needed to Meet Contracted Sales** ### **A. Sheet Metal** Calculate total sheet metal needed for contracted sales: - Trowel: \(1,800 \times 1.2 = 2,160\) sq. ft. - Hoe: \(1,400 \times 1.6 = 2,240\) sq. ft. - Rake: \(1,600 \times 2.1 = 3,360\) sq. ft. - Shovel: \(1,800 \times 2.4 = 4,320\) sq. ft. **Total Needed:** \(2,160 + 2,240 + 3,360 + 4,320 = 12,080\) sq. ft. **Constraint:** Only 10,000 sq. ft. available – **not enough** to meet all contracted sales. ### **B. Processing Hours** #### 1. Stamping (per tool) - Trowel: \(1,800 \times 0.04 = 72\) hours - Hoe: \(1,400 \times 0.17 = 238\) hours - Rake: \(1,600 \times 0.06 = 96\) hours - Shovel: \(1,800 \times 0.12 = 216\) hours **Total Stamping:** \(72 + 238 + 96 + 216 = 622\) hours **Available:** 500 regular + 100 overtime = 600 hours **Still short by 22 hours.** #### 2. Drilling - Trowel: \(1,800 \times 0.05 = 90\) hours - Hoe: \(1,400 \times 0.14 = 196\) hours - Rake: \(1,600 \times 0.09 = 144\) hours - Shovel: \(1,800 \times 0.14 = 252\) hours **Total Drilling:** \(90 + 196 + 144 + 252 = 682\) hours **Available:** 400 regular + 100 overtime = 500 hours **Short by 182 hours.** #### 3. Assembly - Trowel: \(1,800 \times 0.05 = 90\) hours - Hoe: \(1,400 \times 0.13 = 182\) hours - Rake: \(1,600 \times 0.05 = 80\) hours - Shovel: \(1,800 \times 0.10 = 180\) hours **Total Assembly:** \(90 + 182 + 80 + 180 = 532\) hours **Available:** 550 regular + 100 overtime = 650 hours **Enough capacity.** #### 4. Finishing - Trowel: \(1,800 \times 0.06 = 108\) hours - Hoe: \(1,400 \times 0.15 = 210\) hours - Rake: \(1,600 \times 0.02 = 32\) hours - Shovel: \(1,800 \times 0.10 = 180\) hours **Total Finishing:** \(108 + 210 + 32 + 180 = 530\) hours **Available:** 550 regular + 100 overtime = 650 hours **Enough capacity.** #### 5. Packaging - Trowel: \(1,800 \times 0.03 = 54\) hours - Hoe: \(1,400 \times 0.15 = 210\) hours - Rake: \(1,600 \times 0.04 = 64\) hours - Shovel: \(1,800 \times 0.15 = 270\) hours **Total Packaging:** \(54 + 210 + 64 + 270 = 598\) hours **Available:** 500 regular + 100 overtime = 600 hours **Just enough capacity.** --- ## **Step 4: Identify the Bottleneck** - **Sheet Metal:** Only 10,000 sq. ft. available (need 12,080 sq. ft. for all contracts). - **Drilling and Stamping:** Both over capacity even with overtime. - Drilling is the most restrictive (short by 182 hours). --- ## **Step 5: Final Answers and Recommendations** ### **A. Is it possible to meet all contracted sales?** **No,** because both the sheet metal and labor hours for drilling and stamping are insufficient to meet all contracted sales. ### **B. What is the bottleneck resource?** - **Drilling** is the most restrictive bottleneck, followed by sheet metal and stamping. ### **C. What should the company do?** - **Prioritize production** based on the most constrained resource (drilling hours), then by sheet metal. - **Consider subcontracting** for stamping and drilling, as suggested in the problem statement, to meet contracted sales. - **Negotiate with suppliers** for more sheet metal or attempt to optimize tool mix to maximize profit within the given constraints. --- ## **Summary Table of Constraints and Shortfalls** | Resource | Needed | Available | Shortfall | |-------------|--------|-----------|-----------| | Sheet Metal | 12,080 sq.ft. | 10,000 sq.ft. | 2,080 sq.ft. | | Stamping | 622 hrs| 600 hrs | 22 hrs | | Drilling | 682 hrs| 500 hrs | 182 hrs | | Assembly | 532 hrs| 650 hrs | 0 | | Finishing | 530 hrs| 650 hrs | 0 | | Packaging | 598 hrs| 600 hrs | 0 | --- ## **Final Answer** **The company cannot meet all monthly contracted sales using only its regular and allowed overtime resources, primarily due to insufficient sheet metal and a shortage of drilling and stamping hours. The most severe bottleneck is drilling hours. To meet contracted sales, the company must either subcontract the bottleneck operations (especially drilling and stamping) or renegotiate contracted quantities.**

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