Activity 66 Module Problems Complete The Following
Activity 66 Module Problemscomplete The Following Problems And Subm
Complete the following problems and submit the results in either a Microsoft Word document or a Microsoft Excel spreadsheet. Each problem has specific requirements, including setting up mathematical models, linear programming formulations, or optimization solutions related to various operational, transportation, or scheduling problems.
Paper For Above instruction
The set of problems presented involves diverse scenarios requiring analytical and mathematical modeling techniques such as linear programming, transportation algorithms, and optimization methods. These problems are designed to develop skills in formulating real-world issues into mathematical models, solving them computationally or manually, and interpreting the results for practical decision-making.
Problem 6-1: Staffing Schedule Optimization for Y. S. Chang Restaurant
Y. S. Chang Restaurant operates 24 hours daily with six distinct staffing periods: 3am-7am, 7am-11am, 11am-3pm, 3pm-7pm, 7pm-11pm, and 11pm-3am. Each shift is 8 hours long. The objective is to determine the number of waiters and busboys to report for duty at each time period to minimize total staffing requirements, considering minimum staffing needs specified for each period. Variables Xi represent the number of staff beginning work in each period (i=1 through 6). The LP model should capture the staffing overlaps and constraints.
Additionally, the problem asks to identify during which period a reduction of one staff member would be most beneficial, implying an analysis of marginal impacts within the staffing schedule.
Problem 6-2: Student Bus Routing and Allocation
The superintendent of Arden County, Maryland, aims to assign students to three high schools based on geographic sectors within the county, minimizing total student miles traveled. The county is divided into five sectors (A, B, C, D, E), with three schools located in sectors B, C, and E. The data include student populations per sector and distances from each sector to each school.
The LP model will define variables representing the number of students assigned from each sector to each school, with constraints ensuring capacity limits (700 to 900 students per school) and that students from a sector assigned to the school within their sector can walk (resulting in zero transportation). The objective function minimizes total miles traveled by bus, computed as the sum over all sectors and schools of the product of assigned students and their respective distances.
Problem 6-3: Fuel Optimization for Coast-to-Coast Airlines
The airline evaluates fuel loads for a flight circuit starting and ending in Atlanta, with legs between Atlanta-Los Angeles, Los Angeles-Houston, Houston-New Orleans, and New Orleans-Atlanta. Fuel quantities are constrained between specified minimums and maximums at each city, and fuel costs vary per city.
The model must determine how many gallons of fuel to purchase at each city, considering fuel consumption and additional costs for extra fuel, which increases due to a 6% loss per 1,000 gallons above the minimum. The LP formulation involves variables representing fuel purchased at each city, with the objective to minimize total fuel cost while satisfying fuel constraints and accounting for consumption and losses.
Problem 6-4: Least-Cost Shipping for Air Conditioners
An air conditioning manufacturer ships units from three plants (Houston, Phoenix, Memphis) to three regional distributors (Dallas, Atlanta, Denver). Each shipping route has an associated cost, and supply and demand constraints must be satisfied. The LP's decision variables specify the number of units shipped along each route, with the goal to minimize total transportation costs while meeting demands and not exceeding plant supplies.
Problem 6-5: Distribution Planning for Finnish Furniture
Finnish Furniture manufactures tables in Reno, Denver, and Pittsburgh, shipping to three retail stores in Phoenix, Cleveland, and Chicago. The problem involves determining the optimal number of units to ship along each route to minimize costs considering supply limits (Reno: 130, Denver: 200, Pittsburgh: 160) and store demands (Phoenix: 140, Cleveland: 160, Chicago: 200). The LP variables are the units shipped from each plant to each store.
Problem 6-6: Advertising Network Scheduling
Gleaming Company plans a TV advertising campaign, scheduling one commercial during each hour from 1pm to 5pm across four networks. The exposure ratings per hour for each network are given. The task is to assign each network to a distinct hour to maximize total viewer exposure, ensuring that each hour and each network is used exactly once. The solution is a matching problem—an assignment problem—solved via optimization methods.
Conclusion
These problems exemplify diverse applications of linear programming and optimization in operational planning, resource allocation, transportation, and scheduling. Proper formulation and solution of these LP models facilitate efficient decision-making, cost minimization, and resource utilization across various industries and scenarios, demonstrating the practical significance of quantitative methods in management science.
References
- Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network Flows. Prentice Hall.