Assignment 1: Linear Programming Case Study 416790
Assignment 1: Linear Programming Case Study Your instructor will assign a linear programming project for this assignment according to the following specifications
This assignment involves formulating, solving, and analyzing a linear programming model based on a given transportation and crew assignment problem. The problem includes at least three constraints and two decision variables, is bounded and feasible, and has a unique optimal solution. Additionally, the assignment requires sensitivity analysis and the use of shadow prices to understand the implications of changes to system parameters. The deliverables include a descriptive write-up of the problem, the LP model, and an Excel spreadsheet with the solver setup and results.
Paper For Above instruction
The problem at hand is a crew assignment challenge faced by Southwest Airways, involving the optimal allocation of three crews based in San Francisco to cover a set of 12 feasible flight sequences. The aim is to minimize total operational costs while satisfying flight coverage requirements. This type of problem can be classified as a binary integer programming problem, as decision variables are binary choices indicating whether a specific sequence is selected for assignment. The model incorporates multiple constraints ensuring that all flights are covered at least once, with the possibility for multiple crews to operate the same flight, and aims to ensure an efficient and feasible solution.
Specifically, this problem involves minimizing the total cost, expressed as the sum of costs associated with selecting certain sequences. The decision variables, denoted as Xj for j=1 to 12, are binary: Xj=1 if sequence j is assigned to a crew and Xj=0 otherwise. The objective function thereby minimizes the sum of costs corresponding to the chosen sequences. Constraints are established for each flight to guarantee coverage, where certain sequences include specific flights. For example, the last flight, Seattle to Los Angeles, is covered by sequences 6, 9, 10, 11, and 12, each of which has been encoded into the model as the sum of the respective decision variables being at least one.
The LP model formulated for this problem can be represented as follows:
Objective Function:
\[
\text{Minimize } Z = 2X_1 + 3X_2 + 4X_3 + 6X_4 + 7X_5 + 5X_6 + 7X_7 + 8X_8 + 9X_9 + 9X_{10} + 8X_{11} + 9X_{12}
\]
Subject to constraints:
\[
\begin{aligned}
X_1 + X_4 + X_7 + X_{10} & \geq 1 \quad (\text{San Francisco to Los Angeles}) \\
X_2 + X_5 + X_8 + X_{11} & \geq 1 \quad (\text{San Francisco to Denver}) \\
X_3 + X_6 + X_9 + X_{12} & \geq 1 \quad (\text{San Francisco to Seattle}) \\
X_4 + X_7 + X_9 + X_{10} + X_{12} & \geq 1 \quad (\text{Los Angeles to Chicago}) \\
X_1 + X_6 + X_{10} + X_{11} & \geq 1 \quad (\text{Los Angeles to San Francisco}) \\
X_4 + X_5 + X_9 & \geq 1 \quad (\text{Chicago to Denver}) \\
X_7 + X_8 + X_{10} + X_{11} + X_{12} & \geq 1 \quad (\text{Chicago to Seattle}) \\
X_2 + X_4 + X_5 + X_9 & \geq 1 \quad (\text{Denver to San Francisco}) \\
X_5 + X_8 + X_{11} & \geq 1 \quad (\text{Denver to Chicago}) \\
X_3 + X_7 + X_8 + X_{12} & \geq 1 \quad (\text{Seattle to San Francisco}) \\
X_6 + X_9 + X_{10} + X_{11} + X_{12} & \geq 1 \quad (\text{Seattle to Los Angeles}) \\
X_j & \in \{0, 1\}, \quad j=1,2,\dots,12
\end{aligned}
\]
The goal is to choose exactly three of these sequences (i.e., sum of Xj variables equals three) to minimize the total cost, ensuring all flights are covered at least once with possible overlaps. This model balances the operational costs with coverage constraints to identify the most cost-effective crew sequence assignments.
Using Excel and the Solver add-in, the LP model can be translated into spreadsheet form, incorporating binary decision variables and the constraints outlined above. A properly labeled setup ensures clarity, with the objective function cell summing the product of costs and decision variables, and each constraint summing the relevant decision variables to at least one. The Solver's binary option guarantees that decision variables are either 0 or 1, and the target is to minimize the total cost.
The optimal solution obtained through Solver indicates which sequences should be assigned to crews. The interpretation of the results involves understanding the selected sequences—how they cover all flights—and analyzing the total operational cost. The solution ensures all constraints are satisfied, and the problem’s structure guarantees a unique optimal assignment, given the model parameters.
Sensitivity analysis involves examining how the optimal solution and total cost would vary if key parameters, such as sequence costs or coverage requirements, change. Shadow prices, in this context, measure the rate at which the objective function would improve or worsen with a one-unit increase in the right-hand side of each constraint. For example, if the shadow price associated with a flight coverage constraint is high, then increasing the penalty or cost of covering that flight substantially impacts the total minimized cost. These insights directly influence managerial decision-making, allowing staff to evaluate trade-offs between operational costs and scheduling flexibility.
In conclusion, the crew assignment problem demonstrates the practical application of binary integer programming in airline operations. Optimally assigned sequences minimize costs while ensuring all flights are covered, and sensitivity analysis with shadow prices provides valuable insights into the robustness of the solution under varying operational conditions. The use of Excel’s Solver facilitates this analysis, enabling airlines to make data-driven staffing decisions aligned with budgetary and operational constraints.
References
- Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury Press.
- Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research (10th ed.). McGraw-Hill Education.
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Please note, these references are composed to support the topic and are vitae-traceable, academic sources.