Linear Programming Examples For A Calculator Company

Linear Programming Examples A Calculator Company Produces A Scientifi

Given a scenario involving a calculator company producing scientific and graphing calculators, the goal is to determine the optimal production quantities to maximize profit based on specific demand, capacity, and contractual constraints. The problem specifies minimum and maximum production capacities, profit/loss per calculator type, and shipping requirements. The mathematical model involves defining variables for the number of calculators produced, establishing constraints to reflect production limits, demand, and shipping obligations, and formulating an objective function to maximize profit. Using linear programming techniques, such as identifying boundary points and evaluating the objective function at these points, the optimal production quantities are determined. The solution indicates producing 100 scientific calculators and 170 graphing calculators daily to achieve maximum net profit.

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Linear programming is a powerful mathematical technique used to determine the best possible outcome in a given problem, subject to a set of constraints. It is widely employed in various industries for optimizing resource allocation, production scheduling, transportation, and more. The example involving a calculator company provides a clear illustration of how linear programming can be used to maximize profit under practical constraints.

In this specific scenario, the company produces two types of calculators: scientific and graphing. The key objectives include meeting demand, adhering to capacity limitations, fulfilling shipping commitments, and maximizing profit. The problem states that the company expects to produce at least 100 scientific calculators and 80 graphing calculators daily. The maximum production capacities are 200 and 170 calculators respectively. Additionally, there's a shipping requirement to send a combined total of at least 200 calculators each day. The profit per scientific calculator results in a loss of $2, while each graphing calculator yields a profit of $5.

To formulate the problem mathematically, define variables: let x represent the number of scientific calculators produced per day, and y represent the number of graphing calculators produced per day. The constraints can be written as inequalities: x ≥ 100 (minimum scientific calculators), y ≥ 80 (minimum graphing calculators), x ≤ 200 (capacity for scientific calculators), y ≤ 170 (capacity for graphing calculators), and x + y ≥ 200 (shipping requirement). Additionally, since negative production isn't feasible, x and y are non-negative.

The profit function to maximize is:

P = –2x + 5y

subject to the constraints:

  • x ≥ 100
  • y ≥ 80
  • x ≤ 200
  • y ≤ 170
  • x + y ≥ 200

Plotting these constraints in the xy-plane defines a feasible region—a convex polygon where all the inequalities are satisfied. The vertices (corner points) of this feasible region are critical since, according to the fundamental theorem of linear programming, the maximum or minimum of the objective function occurs at one of these vertices.

Testing these vertices involves calculating the profit function at each point and selecting the one with the highest value. The corner points identified are (100, 170), (200, 170), (200, 80), (120, 80), and (100, 100). Evaluating P at these points yields:

  • (100, 170): P = –2(100) + 5(170) = –200 + 850 = 650
  • (200, 170): P = –2(200) + 5(170) = –400 + 850 = 450
  • (200, 80): P = –2(200) + 5(80) = –400 + 400 = 0
  • (120, 80): P = –2(120) + 5(80) = –240 + 400 = 160
  • (100, 100): P = –2(100) + 5(100) = –200 + 500 = 300

Among these, the maximum profit of $650 occurs at (100, 170). Therefore, the company should produce 100 scientific calculators and 170 graphing calculators daily to maximize profits under the given constraints.

This example demonstrates the practical application of linear programming in manufacturing and supply chain management. By modeling production constraints and profit functions, companies can make informed decisions that optimize resource utilization and maximize profitability. Modern software tools, such as linear programming solvers and optimization packages, facilitate solving more complex real-world problems with higher numbers of variables and constraints.

References

  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson/Brooks/Cole.
  • Ozdemir, G., & Ozdemir, A. (2019). Mathematical Modeling and Optimization Techniques. Springer.
  • Hillier, F. S., & Lieberman, G. J. (1995). Introduction to Operations Research (6th ed.). McGraw-Hill.
  • Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear Programming and Network Flows. Wiley.
  • Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and Combinatorial Optimization. Wiley.
  • Rardin, R. (1998). Optimization in Operation Research. Prentice Hall.
  • Chvatal, V. (1983). Linear Programming. W. H. Freeman & Co.
  • Murty, K. G. (1983). Optimization Techniques. Wiley-Interscience.
  • Hillier, F. S., & Lieberman, G. J. (2005). Introduction to Operations Research. McGraw-Hill Education.

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