Linear Programming Estimation Problem - Rensvold Mar 2017

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Identify and formulate the constraints for a series of business scenarios involving resource allocation and production planning. Explain the constraints in each case, using appropriate mathematical notation, and develop the relevant equations or inequalities that define the feasible solution space. Additionally, solve an optimization problem for a refinery producing gasoline and fuel oil, maximizing profit subject to demand and production constraints by testing various production levels and calculating the resulting profit. Provide comprehensive explanations and accurate modeling of each scenario for effective linear programming application.

Paper For Above instruction

Introduction

Linear programming (LP) is a mathematical technique used extensively in operations research to optimize a particular objective, such as profit or cost, subject to a set of constraints. Constraints depict limitations or requirements within a real-world problem, shaping the feasible region from which an optimal solution is chosen. This paper discusses the identification and formulation of constraints for various business scenarios, and applies linear programming to solve a refinery production problem aimed at maximizing profit within given resource and demand limitations.

Part I: Identifying Constraints in Business Scenarios

1. Taxi Company Operations

The taxi company's operational constraints include limits on driver working hours and passenger capacity. Firstly, drivers are restricted to one eight-hour shift per day, with a mandatory half-hour break, which translates into a maximum shift duration constraint: Shift hours: 8 hours. Additionally, since drivers can only work for one shift, this limit forms a maximum working time constraint. Secondly, the taxi cannot carry more than three passengers in the back seat, establishing a capacity constraint on the number of passengers: Passengers per ride: ≤ 3. Lastly, the taxi does not allow for combining stops, meaning all passengers in a trip must have the same destination, which limits route flexibility but doesn't directly translate into a numerical constraint unless route or destination variables are introduced.

2. Approval Delay in Custom Machine Tool Sales

In Acme Unlimited's sales process, each customer order must be sent to Engineering for approval. This introduces a constraint of sequential processing or approval delay, represented as a process delay constraint such that the total processing time per order exceeds or equals a fixed minimum, or the process flow must pass through the approval stage. The constraint could be modeled as Approval time per order: ≥ a certain threshold, or as a wait time constraint, reflecting bottleneck capacity in processing flow.

3. University Course Module Structuring

The university offers six-week terms with three two-week modules, each supposed to focus on one topic. The constraint here is related to course content diversity and workload balance. To prevent overlaps or excessively dense courses, a constraint can be formulated as the total number of topics per module or per term, for example: Number of topics per module: ≤ 1. This ensures focus and depth in each module, reinforcing curriculum design constraints that limit content scope within each period.

Part II: Formulating Mathematical Constraints

1. Space Constraints for Office Cabinets

Given the maximum storage volume and cabinet sizes:

  • Variables: A = number of Model A cabinets, B = number of Model B cabinets

The total storage space must be at least 17 cubic feet:

A 3 + B 4.5 ≥ 17

This inequality represents the space constraint ensuring sufficient capacity.

2. Production Time Constraints in Pottery

Pattie's joint production constraint, considering total time:

10C + 15B + 30*V ≤ 360

Where C, B, and V are the number of cups, bowls, and vases produced, respectively, and total time cannot exceed the 6 hours available.

3. Generator Shipment Constraints

Variables: GenA = number of Generator A units, Gen B = number of Generator B units

The footprint constraint based on cargo space:

8GenA + 12GenB ≤ 1350

4. Profit Calculation for Patty's Daily Production

Using per-unit profits, the daily profit function becomes:

Profit = 0.75C + 2.40B + 1.35*V

This equation sums the profit contributions from each product based on quantities produced.

Part III: Solving the Refinery Production Optimization Problem

The refinery's goal is to maximize daily profit: P = (1.90)gas + (1.5)fuel, subject to the constraints:

  • Fuel demand: fuel ≥ 3 million gallons
  • Gasoline demand: gas ≤ 6.4 million gallons
  • Production relation: fuel ≤ 0.5 * gas
  • Non-negativity constraints: gas ≥ 0, fuel ≥ 0

To explore this, multiple test trials are conducted by selecting different combinations of gas and fuel that satisfy constraints, and calculating the corresponding profit:

For example, suppose gas = 5 million gallons, fuel = 3 million gallons (which satisfies fuel ≤ 0.5*gas and other constraints). The profit would be:

P = 1.90 5 + 1.5 3 = 9.5 + 4.5 = 14 million dollars.

Repeating this process with varying levels (e.g., gas = 6.4, fuel = 3; gas = 4, fuel = 2.5) helps identify the maximum profit point. Optimally, LP methods like graphical solution or simplex algorithm are implemented for precise results, but manual trial provides initial insights into feasible solution ranges.

Conclusion

In conclusion, effective linear programming modeling involves defining variables, formulating constraints that reflect resource, time, or Capacity limitations, and establishing an objective function to maximize or minimize. The scenarios discussed demonstrate diverse applications, from transportation limits to production capacity and profit optimization. Properly formulating these problems allows for the utilization of LP algorithms to find optimal solutions efficiently. The refinery example illustrates how constraint testing and profit calculations help approximate the optimal production mix before applying formal LP methods.

References

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