You Can Use A Calculator To Do Numerical Calculations No Gra ✓ Solved

You Can Use A Calculator To Do Numerical Calculations No Graphing Cal

You can use a calculator to do numerical calculations. No graphing calculator is allowed. Please DO NOT USE ANY COMPUTER SOFTWARE to solve the problems.

1. (a) What is an assignment problem? Briefly discuss the decision variables, the objective function and constraint requirements in an assignment problem. Give a real world example of the assignment problem. (b) What is a diet problem? Briefly discuss the objective function and constraint requirements in a diet problem. Give a real world example of a diet problem. (c) What are the differences between QM for Windows and Excel when solving a linear programming problem? Which one you like better? Why? (d) What are the dual prices? In what range are they valid? Why are they useful in making recommendations to the decision maker? Give a real world example. Questions 2 and 3 are based on the following LP problem: Let P1 = number of Product 1 to be produced, P2 = number of Product 2 to be produced, P3 = number of Product 3 to be produced, P4 = number of Product 4 to be produced. Maximize 80P1 + 100P2 + 120P3 + 70P4, representing total profit. Constraints: 10P1 + 12P2 + 10P3 + 8P4 ≤ 3200 (Production budget), 4P1 + 3P2 + 2P3 + 3P4 ≤ 1000 (Labor hours), 5P1 + 4P2 + 3P3 + 3P4 ≤ 1200 (Material), P1 ≥ 100 (minimum quantity), and P1, P2, P3, P4 ≥ 0 (non-negativity). The QM for Windows output is provided for analysis.

Sample Paper For Above instruction

Linear programming (LP) plays a pivotal role in optimizing decision-making across various fields such as operations management, logistics, finance, and nutrition. Among the foundational problems within LP are the assignment problem and the diet problem, each serving unique practical purposes. This paper explores these LP problems, compares tools used for solving them, and analyzes the concept of dual prices, providing insights for decision-makers with real-world applications.

Assignment Problem: Definition, Components, and Real-World Example

The assignment problem is a specialized LP problem where the objective is to assign resources to tasks in the most cost-effective or efficient manner. It involves decision variables that typically denote whether a particular assignment is made (binary variables) or the number of units assigned. The primary goal is to optimize the objective function, such as minimizing total cost or maximizing efficiency, subject to constraints ensuring each resource or task is assigned exactly once or within capacity limits.

A typical real-world example is assigning workers to jobs. Suppose a company has five workers and five tasks. The cost of assigning each worker to each task varies, and the goal is to assign each worker to exactly one task such that the total assignment cost is minimized. Decision variables could be binary, indicating the assignment status, and constraints ensure no worker is assigned to more than one task while each task receives exactly one worker.

Diet Problem: Objectives, Constraints, and Illustration

The diet problem is a classic LP problem aimed at minimizing the cost of a diet while meeting nutritional requirements. The objective function quantifies the total cost of selected foods, and constraints ensure nutritional needs such as calories, vitamins, minerals, and other dietary factors are satisfied.

A typical example is determining the least expensive combination of foods that meet daily nutritional standards. Decision variables represent quantities of food items, the objective function sums the costs, and constraints guarantee minimum nutritional intakes. For example, a school cafeteria might need to provide a lunch that supplies at least 500 calories, 20 grams of protein, and 200 milligrams of calcium at minimum cost.

Tool Comparison: QM for Windows vs. Excel

QM for Windows and Excel are common tools for solving LP problems. QM for Windows offers specialized features tailored to LP, including graphical interfaces, sensitivity analysis, and dual price computations. Excel’s Solver add-in, integrated within a versatile spreadsheet environment, provides flexibility and ease of use for less complex problems. While QM is more user-friendly for complex LP models, Excel is more accessible and widely used for simpler, quick analyses. Personally, I prefer Excel due to familiarity and its versatility across different types of data analysis, despite QM's more robust LP-specific functionalities.

Understanding Dual Prices and Their Ranges

Dual prices, also known as shadow prices, indicate the change in the objective function value per unit increase in the right-hand side (RHS) of a constraint. These prices are valid within a specific range—the allowable increase or decrease of RHS—beyond which the dual price may no longer hold. They are invaluable for decision-making as they quantify how much additional profit or cost can be expected by relaxing or tightening a constraint.

For example, in a manufacturing context, the dual price of a resource constraint (say, labor hours) reveals how much additional profit could be generated if more labor hours were available. However, these prices are only valid within the specified ranges, and exceeding those can alter the problem’s optimal solution, highlighting the importance of sensitivity analysis in LP.

Analysis of the Provided LP Problem

Optimal Solution and Interpretation

The given LP problem seeks to maximize profit based on the production quantities of four products subject to resource constraints. The QM for Windows output indicates that the optimal production levels are P1 = 100, P3 = 220, with P2 and P4 not produced (non-basic variables). The optimal profit is $34,400, representing the maximum achievable profit under the given constraints. The solution suggests that focusing on producing products P1 and P3 yields the highest profit efficiently while respecting the resource limits and minimum production stipulation for P1.

Slack and Surplus Analysis

Slacks are unused resources in the constraints. In this case, the slack values are 1 for the labor hours constraint, 0 for the production budget, and 40 for the material constraint. A slack of 1 in the labor constraint shows there is almost no excess capacity, indicating efficient utilization of labor. Zero slack in the budget constraint suggests full utilization of the financial resources allocated for production. A slack of 40 in material means there is unused material capacity, which could potentially allow increased production without violating constraints.

Optimality Ranges and Dual Prices

Optimality Ranges

The ranges over which the profit coefficients for products P1, P2, P3, and P4 produce the same optimal solution are derived from the sensitivity analysis (ranging results). For example, product P1’s profit contribution can vary from $120 to $144 without changing the optimal solution, meaning the decision remains optimal within this range.

Dual Prices and Interpretation

The dual prices for the constraints provide insights into the value of relaxing each resource. For multipliers (shadow prices), the range in which they stay valid offers the decision-maker information on how sensitive the optimal solution is to changes in resource availability. If the profit contribution of Product 2 increases from $100 to $130, based on the ranging results, the optimal solution remains unchanged, but the total profit will increase proportionally, reflecting the increased profit per unit in the solution.

Resource Priority

The resource with the highest dual price should be prioritized for additional procurement to maximize profit, given that additional units of this resource increase total profit the most. From the sensitivity analysis, the resource constraint with the highest dual price or the tightest slack should be targeted first for expansion to enhance profitability.

Investment Portfolio Optimization

For the Charm City Pension Planners’ portfolio, the decision variables are the amounts invested in each of the five mutual funds. The objective function maximizes total annual return, which depends on the investment amount and annual return rate, i.e., sum of the product of investment in each fund and its respective return rate. Constraints include the total investment amount ($1,000,000), the maximum total risk amount ($200,000), and minimum investments in funds 2 and 3. These constraints ensure diversified, risk-managed, and compliant investment allocations, focused on maximizing returns without exceeding risk limits or investment minimums.

Donation Collection Optimization

The LP formulation for the charity’s contact strategy involves decision variables representing the number of contacts made in each mode and time of day. The objective function maximizes total donations, summing each contact type’s average donation multiplied by the number of contacts. Constraints include volunteer hours for each contact type (converted into minutes) and minimum contact targets. These constraints reflect operational limitations and charity goals, ensuring efficient use of volunteer time and achieving minimum outreach efforts.

Truck Transfer Problem: Cost Minimization

The transportation problem involves decision variables representing the number of trucks transferred from detailed outlets with either surplus or shortages. The objective function aims to minimize total transfer costs, summing product of trucks transferred and transportation costs between outlets. Constraints include the total number of extra trucks available, shortages to be filled, and the physical limits on transfers from each source. This optimization ensures effective redistribution of trucks to meet operational needs at the lowest cost.

Conclusion

Linear programming is a versatile tool for solving complex optimization problems across diverse operational scenarios. From assignment and diet problems to investment planning and resource allocation, LP provides structured methods to attain optimal solutions while considering organizational constraints. Understanding the components, sensitivity analysis, and tools such as dual prices enhances decision-making by quantifying the value of resources and the impact of potential changes. Practitioners benefit from these insights, enabling better strategic and operational choices that maximize efficiency and profitability.

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