Maximizing Income For A Bakery: Formulating And Analyzing Co

Maximizing Income for a Bakery: Formulating and Analyzing Constraints

The bakery makes Jumbo biscuits and Regular biscuits, with several limitations including oven capacity, flour availability, and cooking time constraints for each biscuit type. Revenue, cost, and production constraints must be formulated into inequalities. Additionally, the feasibility of producing specific quantities of biscuits must be assessed, followed by graphing the inequalities, identifying feasible solutions, and determining the combination that maximizes profit.

Paper For Above instruction

The task involves modeling the production constraints of a bakery producing two types of biscuits—Jumbo and Regular—using inequalities, analyzing production feasibility, and optimizing profit within these constraints.

Formulation of Constraints as Inequalities

Let x represent the number of Regular biscuits and y denote the number of Jumbo biscuits.

The constraints are as follows:

  • Oven capacity: Since the oven can bake at most 150 biscuits per day, the total number of biscuits produced cannot exceed this limit:

    x + y ≤ 150

  • Flour availability: Each Jumbo biscuit requires 2 ounces of flour, and each Regular requires 1 ounce. With 200 ounces of flour available per day:

    1x + 2y ≤ 200

  • Maximum Regular biscuits due to cooking time: The total Regular biscuits should not exceed 130:

    x ≤ 130

  • Maximum Jumbo biscuits due to cooking time: The total Jumbo biscuits should not exceed 70:

    y ≤ 70

These constraints ensure production remains within equipment and resource limits.

Additionally, both types of biscuits cannot be produced in negative quantities, leading to non-negativity inequalities:

  • x ≥ 0
  • y ≥ 0

Feasibility of Producing Specific Quantities

To determine whether specific production quantities are feasible, each quantity pair is tested against the inequalities:

  • Case 1: 20 regular and 80 jumbo biscuits
  • Check constraints:
  • x + y = 20 + 80 = 100 ≤ 150 → OK
  • 120 + 280 = 20 + 160 = 180 ≤ 200 → OK
  • x = 20 ≤ 130 → OK
  • y = 80 ≤ 70 → VIOLATES the jumbo limit
  • Conclusion: The production of 20 regular and 80 jumbo biscuits exceeds the jumbo limit; thus, infeasible.
  • Case 2: 124 regular and 32 jumbo biscuits
  • Check constraints:

  • li>124 + 32 = 156 ≤ 150 → VIOLATES the oven capacity

  • 1124 + 232 = 124 + 64 = 188 ≤ 200 → OK
  • 124 ≤ 130 → OK
  • 32 ≤ 70 → OK
  • Conclusion: The total biscuit count exceeds oven capacity; infeasible.
  • Case 3: 80 regular and 50 jumbo biscuits
  • Check constraints:
  • 80 + 50 = 130 ≤ 150 → OK
  • 180 + 250 = 80 + 100 = 180 ≤ 200 → OK
  • 80 ≤ 130 → OK
  • 50 ≤ 70 → OK
  • Conclusion: All constraints satisfied; production feasible.
  • Case 4: 115 regular and 61 jumbo biscuits
  • Check constraints:
  • 115 + 61 = 176 ≤ 150 → VIOLATES oven capacity
  • 1115 + 261 = 115 + 122 = 237 ≤ 200 → VIOLATES flour constraint
  • 115 ≤ 130 → OK
  • 61 ≤ 70 → OK
  • Conclusion: Both total biscuit count and flour constraints are violated; infeasible.
  • Graphing Constraints and Identifying Feasible Solutions
  • Graphing each inequality on a coordinate plane with x (Regular biscuits) on the x-axis and y (Jumbo biscuits) on the y-axis allows visual identification of the feasible region. The area where all inequalities overlap—bounded by the constraints—represents feasible solutions. Coordinates corresponding to the intersections of these constraints are potential optimal points for maximizing profit.
  • Finding Intersection Points
  • The intersections are determined by solving equations derived from the boundary lines of the inequalities. For example:
  • From x + y = 150 and x = 0, y = 150
  • From x + y = 150 and y = 0, x = 150
  • From 1x + 2y = 200 and x = 0, y = 100
  • From 1x + 2y = 200 and y = 0, x = 200
  • Other intersections are found by solving pairs of equations simultaneously.
  • Calculating all such points helps identify feasible candidates for maximum profit.
  • Profit Equation and Optimization
  • The revenue for each biscuit is given: $1 per Jumbo, $0.65 per Regular. Costs are $0.20 and $0.15, respectively. The profit per unit is revenue minus cost, resulting in:
  • Profit per Jumbo = 1 - 0.20 = $0.80
  • Profit per Regular = 0.65 - 0.15 = $0.50
  • The total profit function is then:
  • Profit = 0.50x + 0.80y
  • To maximize profit, we evaluate this function at each intersection point within the feasible region. The point yielding the highest profit indicates the optimal production mix.
  • Conclusion
  • In conclusion, by translating production constraints into inequalities, analyzing specific production options, graphing the feasible region, and calculating profits at intersection points, the bakery can determine the most profitable production plan within its resource limitations. This process highlights the utility of linear programming and geometric analysis in operational decision-making.
  • References
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. 4th ed. Belmont, CA: Thomson Brooks/Cole.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. 10th Edition. McGraw-Hill Education.
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  • Kerzner, H. (2017). Project Management: A Systems Approach to Planning, Scheduling, and Controlling. Wiley.
  • Gallo, P., & Grande, S. (2014). Linear Programming and Optimization. Springer International Publishing.
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  • Beasley, J. E. (2014). Operations Management: An Integrated Approach. Pearson.
  • Maravelias, C., & Sung, S. (2020). Optimization in Manufacturing and Logistics. Elsevier.

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