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Back SCMA $350,$ Assignment $5.$docx

SCMA $350,$ Group Assignment $5,$ Instructor: Botan Citil

Question $1$: Artisan Furniture Production Problem

A craftsman at Lincoln Artisan Works will produce two types of custom furniture: Modern Minimalist Tables and Rustic Wooden Chairs, for an upcoming exhibition next month.

Each Modern Minimalist Table requires $50$ board feet of wood and $15$ hours of labor.

Each Rustic Wooden Chair requires $30$ board feet of wood and $10$ hours of labor.

Each Modern Minimalist Table can be sold for $$2,000,$ and each Rustic Wooden Chair can be sold for $$1,200.$

The craftsman has $400$ board feet of wood and $100$ hours of labor available before the exhibition and wants to maximize the total profit from the sales of the furniture. What is the maximum profit that the craftsman can earn?

Tasks:

A$.$ Define the objective and constraints mathematically based on the information provided above and provide your mathematical model and solve the model. Report on the optimal production amounts and the total profit. Note that production amounts should be integer.

Question $2$: Traditional Italian Cooking Optimization Problem

You are deciding how many pizzas and lasagnas to prepare, two popular Italian dishes.

You have a budget of $$250,$ with each pizza costing $$20$ and each lasagna costing $$25.$

You have $180$ minutes available for cooking, with each pizza taking $18$ minutes and each lasagna taking $25$ minutes.

Your friends enjoy pizza at a rate of $2\mathrm{.}1$ units of enjoyment per unit and lasagna at a rate of $1\mathrm{.}7$ units of enjoyment per unit.

Only integer amounts of pizzas and lasagnas can be prepared.

Tasks:

A$.$ Define the objective and constraints mathematically based on the information provided above and provide your mathematical model and solve the model. Report the optimal amounts of pizza and lasagna to maximize total enjoyment $($p$.$s$.,$ optimal amounts can be decimal$).$

B$.$ If you realize that cooking a pizza takes $20$ minutes per unit, how does this change in data affect the optimal solution?Question $3$: Portfolio Optimization Problem

Determine the dollar amount to invest in SPY$,$ GOOGL, and NVDA.

Objective:

Maximize the total one$-$year return of the portfolio. The total one$-$year return is calculated by summing the returns of all three stocks. For each stock, the return is the product of its return rate, and the amount invested in that stock.

Constraints:

The combined dividend yield of the three stocks must exceed $$900,000.$ For each stock, the dividend yield is calculated by multiplying its dividend yield rate by the amount invested in that stock.

The total amount invested across all three stocks must equal $$200,000,000.$

The combined beta volatility of the three stocks must be less than $$170,000,000.$ For each stock, the beta volatility is determined by multiplying its beta by the amount invested in that stock.

Stock Data Table:

$\backslash$table$[[$Stock$,$Return Rate Dividend Yield Rate Beta,$],[$SPY$,0\mathrm{.}14,0\mathrm{.}022,0\mathrm{.}95],[$GOOGL$,0\mathrm{.}12,0\mathrm{.}015,1\mathrm{.}15],[$NVDA$,0\mathrm{.}20,0\mathrm{.}010,1\mathrm{.}30]]$

Tasks:

A$.$ Define the objective and constraints mathematically based on the information provided above and provide your mathematical model. Solve the problem to determine the amounts to invest in SPY$,$ GOOGL, and NVDA to achieve the maximum one$-$year return while satisfying all constraints and report how much capital will you end up at the end of the year.