A company manufactures two products, A and B. Product A requires 3 hours of labor and yields a profit of $40, while product B requires 5 hours of labor and yields a profit of $60. If the company has 120 hours of labor available, what is the maximum profit they can achieve by optimally allocating labor between the two products?

["Maximizing Profit: How Much Profit Can a Company Earn with Limited Labor?", "In manufacturing, efficient labor allocation is key to maximizing profit. Consider a company that produces two products—Product A and Product B—with different labor requirements and profit margins. Product A takes 3 hours and earns $40 profit, while Product B requires 5 hours and brings in $60 profit. With only 120 hours of labor available, how can the company optimize production to maximize earnings?", "This article explores the mathematical and strategic approach to solving this labor allocation problem, demonstrating how to achieve the highest possible profit using operations research principles.", "---", "### Understanding the Problem", "Let’s define the variables:\n- Let ( x ) be the number of units of Product A produced\n- Let ( y ) be the number of units of Product B produced", "Each product has specific labor and profit characteristics:\n- Product A: 3 hours per unit → $40 profit\n- Product B: 5 hours per unit → $60 profit\n- Total labor available: 120 hours", "The constraint is:\n[ 3x + 5y \leq 120 ]\nand ( x, y \geq 0 ) (non-negative quantities only)", "The objective function to maximize is:\n[ \ ext{Profit} = 40x + 60y ]", "---", "### Optimizing Labor Use", "To find the best labor allocation, we analyze the profit per hour for each product:", "- Product A: ( \frac{40}{3} \approx 13.33 ) dollars per labor hour\n- Product B: ( \frac{60}{5} = 12 ) dollars per labor hour", "Even though Product A delivers slightly higher profit per labor hour, Product B offers better total profit per unit. However, maximizing labor efficiency doesn’t always mean favoring the higher per-hour profit—because of spatial limits (120 labor hours), the true optimization depends on how many of each product can be produced within the constraint.", "---", "### Testing Integer Solutions at Constraints", "Because ( x ) and ( y ) must be whole units, we need to test feasible combinations that fully or nearly exhaust 120 labor hours.", "We rewrite the constraint:\n[ 3x + 5y = 120 \quad \ ext{(assume full utilization)} ]", "Solve for integer solutions:\nTry values of ( y ) from 0 up and check whether ( x = \frac{120 - 5y}{3} ) is an integer.", "Start with ( y = 0 ):\n( 3x = 120 \Rightarrow x = 40 ) → Profit: ( 40(40) + 60(0) = 1600 )", "( y = 3 ):\n( 5(3) = 15 ), ( 120 - 15 = 105 ), ( x = 35 )\nProfit: ( 40(35) + 60(3) = 1400 + 180 = 1580 )", "( y = 6 ):\n( 5(6) = 30 ), ( 3x = 90 \Rightarrow x = 30 )\nProfit: ( 40(30) + 60(6) = 1200 + 360 = 1560 )", "( y = 9 ):\n( 5(9) = 45 ), ( 3x = 75 \Rightarrow x = 25 )\nProfit: ( 40(25) + 60(9) = 1000 + 540 = 1540 )", "Continue:", "| ( y ) | ( x = \frac{120 - 5y}{3} ) | Profit |\n|--------|-------------------------------|------------|\n| 12 | ( (120 - 60)/3 = 20 ) | ( 40(20) + 60(12) = 800 + 720 = 1520 ) |\n| 15 | ( (120 - 75)/3 = 15 ) | ( 40(15) + 60(15) = 600 + 900 = 1500 ) |\n| 18 | ( (120 - 90)/3 = 10 ) | ( 40(10) + 60(18) = 400 + 1080 = 1480 ) |", "We observe that profit decreases as ( y ) increases beyond 6, and the highest profit occurs when ( y = 0 ), ( x = 40 ), yielding $1600.", "But is this truly optimal?", "Wait—what if we reduce ( y ) and increase ( x ) further? Our earlier values already show declining profit.", "Let’s verify the continuous relaxation (ignoring integrality for insight):", "Use linear programming.\nMaximize: ( P = 40x + 60y )\nSubject to: ( 3x + 5y = 120 ), ( x, y \geq 0 )", "Express ( x = \frac{120 - 5y}{3} )\nSubstitute:\n( P = 40\left(\frac{120 - 5y}{3}\right) + 60y = \frac{4800 - 200y}{3} + 60y )\n( = 1600 - \frac{200}{3}y + 60y = 1600 + \left(60 - \frac{200}{3}\right)y = 1600 - \frac{20}{3}y )", "Since the coefficient of ( y ) is negative, profit decreases as ( y ) increases — meaning maximum occurs at ( y = 0 ), ( x = 40 )", "---", "### Real-World Implication", "Even though Product B is more profitable per unit, the higher labor intensity restricts output when 120 hours are limited. Product A’s lower labor demand allows more units within the same time, resulting in higher total profit.", "### Conclusion", "The company achieves maximum profit of $1,600 by producing 40 units of Product A and 0 units of Product B, fully utilizing the 120-hour labor constraint.", "This case illustrates a critical lesson: profit optimization depends on both margin and resource efficiency. Choosing the high-margin product exclusively may not be optimal if it exhausts labor too quickly—using lower-margin but faster-turning products can yield greater total returns.", "For manufacturers, the key is balancing product profitability, resource consumption, and labor availability—this problem serves as a practical model for strategic capacity planning.", "---", "Key Takeaways:\n- Optimize not just per-unit profit, but profit per labor hour and total output potential.\n- Integer programming may be needed, but continuous relaxation provides strong guidance.\n- Labor constraint can favor one product over another—never assume higher margin always means higher profit.", "Maximize your labor hours. Choose wisely. Increase your profit."]