Mat540 Homework Week 9 Chapter 51 Row
Mat540 Homework Week 9chapter 51 Row
Rowntown Cab Company has 70 drivers that it must schedule in three 8-hour shifts. The demand for cabs varies throughout the day with specific minimum staffing requirements and different fare averages depending on the shift start time. The company aims to assign drivers optimally to maximize fares while satisfying staffing constraints and additional shift limitations. The problems include formulating and solving integer programming models for driver scheduling, including restrictions on late shift staffing and limits on day shift drivers. Additionally, the assignment involves solving a knapsack problem to optimize item selection, a project selection problem with resource and dependency constraints, a staffing and error minimization problem for mortgage operators, and an investment decision model for asset sales over three years.
Paper For Above instruction
This paper addresses several complex optimization problems faced by different organizations, primarily utilizing integer programming models to optimize resource allocation, scheduling, and profit maximization. The first problem focuses on driver scheduling for Rowntown Cab Company, where the goal is to assign 70 drivers across three shifts to maximize fare revenue while satisfying minimum staffing requirements and specific shift restrictions. The subsequent cases explore a knapsack problem for Juan Hernandez’s item packing to maximize profit within weight limits, a project selection problem balancing costs, resources, and dependencies for Texas Consolidated Electronics, a staffing and error minimization problem for Corsouth Mortgage Associates, and finally, an investment asset sale planning model for Globex Investment Corporation.
Driver Scheduling for Rowntown Cab Company:
The primary goal is to determine the optimal number of drivers to assign to each shift, given varying demand and fare revenue estimates. The constraints include minimum staffing requirements for six different time periods, a total of 70 drivers, and shift-specific fare averages. To model this, decision variables are introduced for drivers starting in each time period, with an objective function that maximizes total fare revenue across all shifts. Additional constraints enforce staffing requirements, shifts limitations, and total drivers. The model includes special restrictions for late-night shifts with a maximum of 15 drivers and for the day shift, limited to 20 drivers.
Formally, the decision variables \( D_i \) represent the number of drivers starting in period \( i \). The objective function maximizes the sum of fare revenue:
\[
\text{Max } Z = 80 D_1 + 500 D_2 + 420 D_3 + 300 D_4 + 270 D_5 + 210 D_6
\]
subject to staffing constraints:
\[
D_1 \geq 10,\quad D_2 \geq 12,\quad D_3 \geq 20,\quad D_4 \geq 25,\quad D_5 \geq 32,\quad D_6 \geq 18
\]
\[
D_1 + D_2 + D_3 + D_4 + D_5 + D_6 \leq 70
\]
with additional limitations for shifts, such as \( D_1 + D_2 \leq 15 \) and \( D_4 + D_5 + D_6 \leq 20 \) for the specified restrictions.
Knapsack Problem of Juan Hernandez:
Juan needs to carry items with specified weights and profits without exceeding a 5-pound limit, aiming to maximize total profit. The decision variables indicate whether each item is included:
\[
x_{\text{jeans}}, x_{\text{CD}}, x_{\text{discs}} \in \{0,1\}
\]
The objective:
\[
\text{Maximize } Z = 90 x_{\text{jeans}} + 150 x_{\text{CD}} + 30 x_{\text{discs}}
\]
subject to the weight constraint:
\[
2 x_{\text{jeans}} + 3 x_{\text{CD}} + x_{\text{discs}} \leq 5
\]
and binary restrictions on variables representing item selection.
Project Selection for Texas Consolidated Electronics:
The company has eight projects with different costs, resource needs, and profits. The model aims to maximize total profit by selecting projects within the management science and budget limits, with dependencies such as project 5 only being selected if project 2 is selected. The decision variables are binary, indicating whether each project is undertaken. The key constraints are the resource and budget limits, along with dependency constraints:
\[
\sum_{i=1}^{8} \text{Cost}_i x_i \leq 300
\]
\[
\sum_{i=1}^{8} \text{Scientists}_i x_i \leq 40
\]
\[
x_5 \leq x_2
\]
\[
x_i \in \{0,1\}, \quad \text{for } i=1,\dots,8
\]
Mortgage Processing Operators Problem:
This problem involves determining the minimum number of permanent and temporary operators to process at least 6,300 accounts daily with a maximum of 32 workstations and error restrictions. The decision variables \( P \) (permanent) and \( T \) (temporary) must satisfy account processing requirements:
\[
220 P + 140 T \geq 6300
\]
and error constraints:
\[
0.4 P + 0.9 T \leq 15
\]
with staffing constraints:
\[
P + T \leq 32
\]
and costs to minimize:
\[
\text{Minimize } 120 P + 75 T
\]
Asset Sale Planning for Globex:
The firm plans to sell each of six companies over three years, with minimum sales targets each year. Decision variables \( C_{i,j} \) represent whether company \( i \) is sold in year \( j \), with the goal of maximizing total return:
\[
\text{Maximize } Z = \sum_{i=1}^6 \sum_{j=1}^3 R_{i,j} \times C_{i,j}
\]
subject to minimum sales each year:
\[
\sum_{i=1}^6 C_{i,j} \geq \text{minimum targets}
\]
and each company sold only once, with binary constraints on \( C_{i,j} \).
Overall, these problems exemplify the utility of linear and integer programming in operational decision-making across transportation, logistics, project management, staffing, and investment. Solving each model provides optimal or near-optimal solutions that improve organizational efficiency, profitability, and resource utilization.
References
- Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall.
- Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-Hill.