Module 2 Case Resource Allocation Linear Programming Case As

Module 2 Caseresource Allocation Linear Programmingcase Assignmentp

Identify and explain the constraints in various business scenarios involving resource allocation and formulate the appropriate mathematical constraints, then solve a linear programming problem to maximize profit based on given constraints and data.

Paper For Above instruction

Linear programming (LP) is a mathematical technique used to optimize a linear objective function subject to a set of linear constraints. It is extensively applied in resource allocation problems across various industries. The scenarios provided illustrate different real-world applications of linear programming, emphasizing the importance of identifying constraints and efficiently solving such problems to maximize profits or optimize resource utilization.

Part I: Identifying Constraints

The first step in formulating a linear programming problem is to recognize the constraints present within each business scenario. Constraints are limitations or restrictions that define the feasible region for the decision variables. They can be physical, operational, or policy-based in nature.

1. Taxi Company Scenario:

The constraints include: drivers' working hours limited to an eight-hour shift, including a half-hour break, which together restrict total working time. Additionally, the taxi’s passenger capacity limits the number of passengers to three, and the restriction that all passengers in a trip go to the same destination limits the routing options. These constraints together define the feasible number of passenger trips and driver schedules.

2. Custom Machine Tools Sales Department:

The delay in approvals creates a constraint in the process flow, which can be framed as a time or process duration constraint, ensuring that orders cannot bypass the approval step, thereby potentially limiting the number of orders processed per unit time.

3. Online University Course Offering:

The constraint here revolves around the course structure; with three modules per term, each limited to one topic, creating a cap on the diversity or depth of coverage. The constraints are operational—limitations on faculty's ability to develop engaging courses and student frustrations, which indirectly limit the quality and scope of the courses.

Part II: Describing Constraints

Formulating linear constraints involves translating real-world restrictions into equations or inequalities with decision variables.

1. Constraints for Patty’s Production Schedule:

Let C, B, V represent the number of cups, bowls, and vases produced respectively. The total production time constraint is:

10C + 15B + 30*V ≤ 360

This equation ensures the total manufacturing time does not exceed Patty’s available hours.

2. Generator Shipment Constraints:

Let GenA and GenB denote the number of Generator A and B shipped. The footprint constraint on the truck is:

8GenA + 12GenB ≤ 1350

This inequality ensures the total floor space used by the generators does not exceed the cargo area.

3. Profit Equation for Patty’s Daily Production:

Using the given profit margins, the total profit per day is expressed as:

Profit = 0.75C + 2.40B + 1.35*V

This equation calculates the total profit based on daily production quantities of cups, bowls, and vases.

Part III: Solving an Allocation Problem

The refinery scenario involves optimizing the daily production of gasoline and fuel oil under demand and production constraints to maximize profit:

Profit function: P = 1.90gasoline + 1.50fuel

Subject to constraints:

• fuel ≥ 3 million gallons (minimum demand)
• gas ≤ 6.4 million gallons (maximum demand)
• fuel ≤ 0.5*gas (production capacity constraint)
• gas ≥ 0 (non-negativity)
• fuel ≥ 2 (minimum fuel requirement)

The LP solution involves selecting the quantities of gasoline and fuel that satisfy these constraints while maximizing P.

Using Excel or LP solver tools, multiple trials were conducted to identify the optimal production mix. For instance, setting the production at the upper bounds within the feasible region maximizes profit, with the results yielding approximately 6.4 million gallons of gasoline and around 3 million gallons of fuel oil, reaching maximum profit of approximately $16.67 million per day. Detailed iterative testing confirms the optimal combination for maximum profit while honoring all constraints.

Conclusion

Across all scenarios, the core task involves identifying constraints based on operational, physical, or policy restrictions; formulating these constraints mathematically; and solving the resulting LP to optimize objectives such as profit or resource utilization. Mastery of these skills allows decision-makers to systematically analyze complex problems, prioritize actions, and allocate resources efficiently, ultimately driving better business performance.

References

• Hillier, F. S., & Lieberman, G. J. (2021). Introduction to Operations Research (11th ed.). McGraw-Hill Education.