Question 1: An Optimal Solution Is Any Set Of Decision Varia

Question 1an Optimal Solution Is A Any Set Of Decision Varia

Question 1: An optimal solution is ________. a. any set of decision variable values that maximizes or minimizes the objective function b. the quantity that we seek to minimize or maximize c. the limitation or requirement that decision variables must satisfy d. also known as the constraint function

Question 2: In the expression 3.0A + 3.5B + 2.3C ≥ 850, where A, B, and C are decision variables of a problem, the constraint function is the ________. a. right-hand side of the expression b. contradiction of the inequality given in the expression c. left-hand side of the expression d. inequality in the expression

Question 3: A binding constraint is one ________. a. for which the Cell Value is equal to the right-hand side of the value of the constraint b. for which the Cell Value is greater than the right-hand side of the value of the constraint c. for which the Cell Value is not equal to the right-hand side of the value of the constraint d. for which the Cell Value is lesser than the right-hand side of the value of the constraint

Question 4: The ________ is the difference between the right- and left-hand sides of a constraint. a. shadow price b. allowable increase c. optimal solution d. slack

Question 5: The shadow price ________. a. tells how much the value of the allowable decrease will change as the right-hand side of a constraint is reduced to 0 b. tells how much the objective coefficient needs to be reduced in order for a nonnegative variable that is zero in the optimal solution to become positive c. tells how much the value of the allowable increase will change as the left-hand side of a constraint is reduced to 0 d. tells how much the value of the objective function will change as the right-hand side of a constraint is increased by 1

Question 6: Consider the scenario given below. Use Excel Solver to answer the following question(s). Peca Inc. is a small manufacturer of two types of office chairs, the swivel and no-swivel models. The manufacturing process consists of two principal departments: fabrication and finishing. The fabrication department has 24 skilled workers, each of whom works 7 hours per day. The finishing department has 6 workers, who also work a 7-hour shift. A swivel type requires 7 labor hours in the fabricating department and 2 labor hours in finishing. The no-swivel model requires 8 labor hours in fabricating and 3 labor hours in finishing. Peca Inc. makes a net profit of $100 on the swivel model, and $130 on the no-swivel model. The company anticipates selling at least twice as many no-swivel models as swivel models. The company wants to determine how many of each model should be produced on a daily basis to maximize net profit. Let X1 be the amount of swivel model to be produced in a day, and X2 be the amount of no-swivel model. Determine the objective function? a. Minimize profit = 5.375X1 + 10.75X2 b. Minimize profit = 130X1 + 100X2 c. Maximize profit = 100X1 + 130 X2 d. Maximize profit = 24X1 + 14X2

Question 7: Please use Solver to calculate the number of swivel chairs produced in a day. a. 30 units b. 5.25 units c. 10.5 units d. 15 units

Question 8: Using the data, calculate the number of no-swivel chairs produced in a day. a. 10.5 units b. 5.25 units c. 15 units d. 30 units

Question 9: Read the Answer Report generated by Solver, the difference between the right- and left-hand sides of the fabrication labor constraint is ________. a. 10.5 units b. 0 units c. 47.25 units d. 5.25 units

Question 10: The optimal values of the decision variables will change if the ________. a. unit profit for swivel chairs either increases by more than 45 or decreases by more than 21 b. unit profit for swivel chairs either increases by more than 80 or decreases by more than 20 c. unit profit for no-swivel chairs either increases by more than 45 or decreases by more than 21 d. unit profit for no-swivel chairs either increases by more than 20 or decreases by more than 180

Question 11: What is the allowable decrease per unit of swivel chair produced? a. $13.33 b. $42 c. $47.25 d. $120.75

Question 12: In the finishing constraint, the shadow price of 45 indicates ________. a. that if there are 45 additional hours of finishing available, then the total profit will change by $1 b. that if an additional hour of finishing time is available, then the total profit will change by $45 c. that if the allowable decrease 42 changes to 43, then the total profit will decrease by $45 d. that if the allowable decrease 42 changes to 41, then the total profit will increase by $45

Question 13: If the limitation in the finishing department is changed to 43 labor hours, the total profit ________. a. will increase by $45 b. will decrease by $5 c. will decrease by $45 d. will increase by $5

Question 14: If 7.2 additional hours of finishing time were available, the profit would change by ________. a. $45 b. $324 c. $369 d. $450

Question 15: If two workers in the finishing department were ill for a day and could not report to work, the overall profit will reduce by ________. a. $315 b. $1,260 c. $180 d. $630

Question 16: If a worker in the finishing department falls ill for a day and is unable to report to work, the overall profit will reduce by ________. a. $1,260 b. $180 c. $315 d. $630

Paper For Above instruction

Optimization problems in management science primarily involve determining the best possible decision variables to maximize or minimize a specific objective, subject to various constraints. Understanding the foundational concepts of optimal solutions, formulation of constraint functions, binding constraints, slack, and shadow prices is essential for effective decision-making in resource allocation and production scheduling.

Introduction

Optimization models, especially linear programming (LP), serve as critical tools for businesses to optimize operations, maximize profits, and efficiently utilize resources. An optimal solution refers to the decision variable values that yield the best outcome given the constraints, whether that outcome is maximized profit, minimized costs, or other objectives. Grasping the fundamental concepts associated with these solutions—such as the nature of constraints, binding status, slack, and shadow prices—enables managers and analysts to interpret model results accurately and make informed decisions.

Understanding Optimal Solutions

An optimal solution in linear programming is characterized by a set of decision variable values that result in the best feasible value of the objective function. This solution is not merely any feasible point; rather, it specifically maximizes or minimizes the objective, depending on the problem's goal. For example, in profit maximization, these decision variables might specify the quantity of products to produce. The key is that these variables satisfy all constraints while achieving the goal. The conceptual understanding aligns with choice (a), which states that an optimal solution is any set of decision variable values that maximize or minimize the objective function (Winston, 2020).

The Role of Constraints and the Constraint Function

Constraints in an LP model limit the feasible region and are expressed in terms of decision variables. The constraint function includes the functional form, such as inequalities or equations, representing resource limitations or requirements. In the expression 3.0A + 3.5B + 2.3C ≥ 850, the left-hand side reflects the total resource utilization, while the right-hand side (850) signifies the either capacity or minimum requirement. The constraint function encompasses the entire inequality, which includes both sides, but when referring to the specific functional component, the left side often represents the constraint function (Hillier and Lieberman, 2021).

Binding Constraints and Slack

A bound constraint is termed binding when it actively restricts the decision variables, typically observed when the constraint's value equals its limit—i.e., the left side equals the right side, as in choice (a). Non-binding constraints are those where the actual value is less than or greater than the limit, implying they do not affect the optimal solution. The slack is the difference between the two sides of a constraint; it indicates unused resources. When slack is zero, the constraint is binding, and the resource is fully utilized, which is critical for understanding resource constraints at optimality (Zimmermann, 2022).

Shadow Price and Its Significance

The shadow price represents the rate of improvement in the objective function per unit increase in the right-hand side of a constraint, holding other factors constant. For example, a shadow price of 45 in a finishing constraint indicates that adding one additional hour would increase profit by $45. Conversely, it also informs about the limits to resource flexibility; beyond certain thresholds, additional resource units may not improve the outcome (Bazaraa et al., 2019). This insight assists managers in resource planning and prioritization.

Application: Production Scheduling in a Chair Manufacturing Scenario

The scenario of Peca Inc., a manufacturer of swivel and no-swivel chairs, exemplifies applying LP in real-world decision-making. The problem involves defining the objective function—maximizing profit—and formulating constraints based on labor hours in fabrication and finishing departments. The production quantities, decision variables (X1 for swivel, X2 for no-swivel), and resource constraints form the core of the LP model (Hiller & Lieberman, 2021). Solving this using Excel Solver enables optimal production levels to maximize profit while respecting resource limitations.

Objective Function Formulation and Solver Application

The objective function, in this case, is to maximize total profit: Profit = 100X1 + 130X2. The resource constraints include labor hours in fabrication and finishing departments, with the total hours available set by the number of workers and their shifts. Additionally, demand constraints such as the requirement to produce at least twice as many no-swivel chairs as swivel chairs must be incorporated. Using Excel Solver, decision variables are adjusted within these boundaries to find the optimal product mix that yields the highest profit.

Interpreting Solver Results and Constraints

Once the solver identifies a solution, the output reveals the number of chairs to produce (X1, X2), the slack in resource constraints, and the shadow prices. If the slack in fabrication labor is zero, it indicates that this resource is fully utilized, and the shadow price implies the incremental profit associated with adding more hours. Conversely, if slack exists, adding hours may not improve profit until the slack is exhausted.

Impacts of Variations in Parameters

Changes in profit coefficients (unit profit per chair) influence the optimal solution. For instance, if the profit margin on swivel chairs increases substantially, it may become more beneficial to allocate more resources to produce swivels. Similarly, the allowable decrease or increase in coefficients defines the robustness of the solution—certain thresholds beyond which the current production plan is no longer optimal (Hillier & Lieberman, 2021). Understanding such sensitivities enables better strategic planning, resource allocation, and risk assessment.

Conclusion

Linear programming provides a structured and quantifiable framework for solving complex resource allocation and production problems. Recognizing the significance of concepts such as optimal solutions, constraints, binding status, slack, and shadow prices equips decision-makers with insights necessary for optimizing operational efficiency. The practical scenario of chair manufacturing underscores the relevance of these principles in real-world business contexts, demonstrating how mathematical tools facilitate strategic and operational decision-making.

References

• Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2019). Nonlinear programming: Theory and algorithms. Springer.
• Hillier, F. S., & Lieberman, G. J. (2021). Introduction to operations research (11th ed.). McGraw-Hill Education.
• Winston, W. L. (2020). Operations research: Applications and algorithms. Cengage Learning.
• Zimmerrmann, A. (2022). Foundations of management science. McGraw-Hill.
• Hiller, F. S., & Lieberman, G. J. (2021). Introduction to Operations Research (11th ed.). McGraw-Hill Education.