Test 1 DMT Spring 2012 Instructions Format Times New Roman 1

Test 1 Dmt Spring 2012instructionsformat Times New Roman 12pts

Test 1 Dmt Spring 2012instructionsformat Times New Roman 12pts

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Analyze the provided situations and formulate appropriate mathematical models to optimize the given objectives, considering the constraints specified in each scenario. Develop linear programming models for situations involving resource allocation, production scheduling, and facility planning. Construct network models for routing problems. Formulate integer programming models for staffing and advertising decisions. For each situation, clearly define decision variables, objective functions, and constraints based on the descriptive parameters provided. Solve each model using suitable optimization techniques or computational tools, and interpret the results to offer concrete, actionable recommendations aligned with the strategic goals outlined in the scenarios.

Situation 1: Pizza Production Optimization

Edda's Inc. produces two types of frozen pizzas, Regular and Deluxe, with specific selling prices, costs, and ingredient requirements. The company has limited weekly resources: 70 pounds of dough, 25 pounds of topping, and a $320 budget. The demand constraints are at least 40 Regular and 35 Deluxe pizzas weekly. Decision variables are the number of Regular (x) and Deluxe (y) pizzas to produce. The objective is to maximize net income: (10 - 6) x + (15 - 9) y. Constraints include ingredient usage: 1 pound of dough per pizza, 16 ounces of topping per pizza, availability limits, demand minima, and budget caps. The model includes all these constraints, and the solution identifies the optimal production mix.

Situation 2: Optimal Assembly Process Selection

RT produces Standard and Trimline telephone models on one of two assembly lines. Each model's assembly times vary by process, and each process has associated costs. The company employs 11 workers, each working 7.5 hours daily, with costs per minute for manufacturing, assembly, and quality control that depend on the process chosen. The sale prices are $7 for Standard and $10 for Trimline models. Decision variables include the number of Standard and Trimline units produced, and the selection between Process 1 and Process 2, modeled by binary variables or alternative formulations. The goal is to maximize profit, calculated as revenue minus process-driven costs, under constraints of labor hours, process costs, and demand or capacity considerations.

Situation 3: Shortest Path for Road Trip

José and Maria aim to reach Denver in the shortest possible time, traveling between multiple cities with known travel times. This is a classic shortest path problem in a network graph where nodes are cities and edge weights are travel times. Model the problem as a shortest path network, with decision variables indicating whether a route between cities is part of the trip. Constraints ensure a continuous path from the starting city to Denver without cycles or repeated segments, and the objective minimizes total travel time. Solving this network model yields the most time-efficient route.

Situation 4: Faculty Hiring Optimization

The Polytechnic University of PR plans to hire up to 20 faculty members within a budget of $1,450,000. The ranks are assistant, associate, and full professors, with specific salary and experience data. The constraints include minimum assistant hires (at least half of new hires), maximum full professors (at least 3), and that at least 70% of hires are below the full professor rank. Decision variables are the number of faculty at each rank, with objectives to maximize total experience. This problem is formulated as an integer linear programming model, with constraints capturing hiring limits, salary budgets, and rank proportions. Solving the model provides the optimal faculty composition to maximize total experience within constraints.

Situation 5: Resource Allocation for Timber Products

Tiny Timber Company seeks to allocate its timber resources between lumber and plywood production. The resource constraints are 32,000 board feet of spruce and 72,000 of Douglas fir. Each product's raw materials requirements are specified, along with minimum production levels. Decision variables are the quantities of lumber (L) and plywood (P), expressed in thousands of board feet or square feet. The objective is to maximize profit: $45 per 1000 board feet of lumber and $60 per 1000 square feet of plywood, subject to resource constraints, minimum production levels, and non-negativity. Formulating this as a linear programming model enables optimal resource utilization.

Situation 6: Product Mix for PRComputers

PRComputers needs to determine optimal quarterly production quantities for notebook and desktop computers, considering resource limitations: chip supply, memory, and assembly time. Variables are the number of notebooks and desktops produced. Constraints include chip limit (10,000 chips), memory sets (15,000 available), and assembly time (20,000 minutes). The objective is to maximize profit: $210 per notebook and $300 per desktop. The model involves linear inequalities reflecting resource limits, with an objective function maximizing total profit. Sensitivity analysis can assess the impact of resource changes or profit variations.

Situation 7: Advertising Campaign Optimization

Carmen aims to optimize advertising investments across radio, newspaper, and television to maximize reach while satisfying policy constraints. Decision variables are the number of ads for each medium, with costs and audience reach specified. Constraints include the number of radio ads being at least twice the sum of newspaper and TV ads, minimum total reach of 120,000 customers, a proportionality constraint for young and senior audiences, and a minimum percentage of women viewers. The formulation involves linear inequalities and an objective of maximizing or minimizing total advertising cost or reach. Solving the model identifies the optimal ad mix that meets all policies and constraints, guiding effective advertising expenditure.

References

  • Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear Programming and Network Flows. Wiley.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson/Brooks/Cole.
  • Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-Hill Education.
  • Nemhauser, G. L., & Wolsey, L. A. (1999). Integer and Combinatorial Optimization. Wiley.
  • Schrader, R. (2011). Network Flow Models: Shortest Path & Transportation Problems. Springer.
  • Garey, M. R., & Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman.
  • Hodgson, M. (2014). Introduction to Linear Optimization. Wiley.
  • Wernicke, H. (2018). Resource Allocation and Scheduling Models. Springer.
  • Williams, H. P. (2013). Model Building in Mathematical Programming. Wiley.
  • Greenberg, H. J. (2015). Advanced Optimization Techniques. Springer.

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