The purpose of the ratio test is to 1. Determine the slack ✓ Solved

1. The purpose of the ratio test is to: Determine the slack, surplus and artificial variables; Determine the variable to enter the next tableau; Determine the variable to exit the next tableau; All of the above; None of the above.

2. Which of the following statement is true? There are three constraints associated with this problem; There are three slack variables that are associated with this problem; The initial tableau will have X1=0 and X2=0; All of the above; None of the above.

3. The objective function in linear programming usually: Specifies the rate of use of a resource by each activity; Specifies the mathematical expression to be maximized or minimized; Is a dimensionless quantity and is expressed without measurable units; Is constrained to some level of attainment of the specified objective; Is represented by a less than or equal to inequality.

4. The first step in the simplex algorithm is to: Convert the constraints of the LP to the standard form by removing the inequalities and adding slack, surplus and artificial variables; Determine the variable to enter the next tableau by conducting a test on the Z row; Determine the exiting variable by conducting the ratio test; Setting up the tableau with the objective function values and constraints.

5. The simplex algorithm process uses ____________ principles to form the basis of solving for an optimal solution: Bernoulli's equation; Quadratic equations; The method of least squares; Gauss-Jordan Method.

6. A nonbasic variable: Is determined by conducting the ratio test; Is a nonnegative number; Has a value of zero; All of the above; None of the above.

7. Match the simplex algorithm term to the appropriate definition: Ratio Test; Slack Variable; Surplus Variable; Real Decision Variables; Basic Variables: A. Determines variable to exit next tableau; B. Solution variables of the problem; C. Associated with less than or equal to constraints; D. Associated with greater than or equal to constraints; E. Have values other than zero.

8. Refer to the LP model and solution shown in figure 6: The following statement is true: The optimum profit is . The value for L is . One of the resources uses exactly 125 units; All of the above; None of the above.

9. For the tableau to be optimal, what should the values be for c1, c2 and b: c1=-2, c2=1 and b=. c1=4, c2=3, and b=. c1=3, c2=6, b=-; All of the above; None of the above.

10. Which of the following is not required for all linear programming problems? There must be an objective function to be maximized or minimized; The variables may only assume continuous values that are greater than or equal to zero; The constraints and objective function must be linear; The number of constraints must be less than or equal to the number of variables; All of the above are required.

11. Which of the following is true in the standard form of a linear programming problem? Each constraint is expressed as an equality; There will be the same number of variables as constraints; Redundant constraints will have to be removed; All of the above is true; None of the above is true.

12. Suppose you are a manager of a watch making firm operating in a competitive market. Your cost of production is given by C = 100 + Q2, where Q is the level of output and C is the total cost. The marginal cost is 2Q. If the price of watch is $60, determine how many watches should you produce to maximize profit. Additionally, calculate your profit level and assess whether there will be entry or exit. At what minimum price will you produce a positive output? Finally, establish the short run supply curve of the firm.

Paper For Above Instructions

The Ratio Test in linear programming and optimization is a strategic technique used to analyze whether a given sequence converges or diverges, particularly in the context of the simplex algorithm. It primarily serves to determine the slack, surplus, and artificial variables, which are pivotal in constructing feasible solutions to optimization problems (Winston, 2004). The indication of implementing ratio testing corroborates the selection of variables that exit or enter the tableau, thereby facilitating the search for an optimal solution.

Investigating the fundamentals of linear programming reveals that an objective function must typically be a mathematical expression that is maximized or minimized according to set constraints (Taha, 2017). Each step in the simplex algorithm is meticulously crafted, beginning with converting linear program constraints into standard form. This initial setup often involves the removal of inequalities through the addition of slack, surplus, and artificial variables, permitting a structured approach towards optimality (Dantzig, 1963).

Nonbasic variables assume a critical role within this framework as they are precisely determined through conducting the ratio test; these variables remain at zero while basic variables carry the load of the solution (Chvátal, 1983). In matching simplex terminology to definitions, the identification of slack variables, surplus variables, and basic variables illuminates their respective functionalities: slack variables actively address less-than-or-equal-to constraints, while surplus variables correspond with greater-than-or-equal-to constraints (Korte & Vygen, 2018).

Quantifying the dimensions of constraints and objectives, it becomes clear that for any linear programming problem, having linear constraints is an intrinsic necessity. Moreover, redundant constraints must be eliminated to streamline the solution space (Bertsimas & Tsitsiklis, 1997). Thus, the design of an effective LP model encompasses an array of intricacies, requiring comprehensive assessments to ensure that only relevant variables inform the decision-making process.

In applied contexts, managerial insights emerge when businesses actively outline their production costs and revenue streams. For example, in a watch-making firm, if the cost of production is defined by the function C = 100 + Q^2, wherein Q represents the output level and C reflects total costs, one can derive insights into output maximization and profit levels. Calculating the marginal cost as 2Q reflects a pertinent relationship between cost and output—a pivotal indicator for production decisions. If watches are priced at $60, the immediate query becomes the quantity that maximizes profit. This presentation demands analysis through marginal cost equal to marginal revenue, guiding production levels (Varian, 2010).

To determine how many watches to produce, we can equate the marginal cost to the price of the watch: 2Q = 60, yielding Q = 30 watches. To find profit, we must subtract total costs from total revenue, thereby providing a holistic view of the firm's financial status. The total revenue (TR) equals price times quantity (TR = 60 * 30 = $1800), while total cost (TC = 100 + (30)^2 = $100 + 900 = $1000), thus, profit becomes: Profit = TR - TC = $1800 - $1000 = $800.

In evaluating market dynamics, one must also consider the conditions of entry and exit relative to profitability. If the price reaches a threshold below average variable costs, firms may exit the market, emphasizing the necessity for understanding price ceilings that establish a positive output (Pindyck & Rubinfeld, 2017). Moreover, to ascertain the short-run supply curve for the firm, the price must align at least with the average variable cost, defining a minimal price point that allows the continuation of production. The characterization of this supply curve ultimately rests upon fluctuating prices and existing production costs.

Through this exploration, the ratio test and foundational principles of the simplex algorithm create a robust foundation for understanding linear programming in practical applications. Many firms face similar situations regarding demand and cost structures, necessitating adept resource allocation and output determination strategies that integrate mathematical modeling with real-world financial scenarios.

References

• Bertsimas, D., & Tsitsiklis, J. N. (1997). Introduction to linear optimization. Athena Scientific.
• Chvátal, V. (1983). Linear programming. W. H. Freeman.
• Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
• Korte, B., & Vygen, J. (2018). Combinatorial Optimization: Theory and Algorithms. Springer.
• Pindyck, R. S., & Rubinfeld, D. L. (2017). Microeconomics. Pearson.
• Taha, H. A. (2017). Operations Research: An Introduction. Pearson.
• Varian, H. R. (2010). Intermediate Microeconomics: A Modern Approach. W.W. Norton & Company.
• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Brooks/Cole.
• Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-Hill.
• Smith, R. (2019). Linear Programming in Practice. Prentice Hall.