Suppose that, last year, you purchased 150 shares of eight different stocks (for a total of 1200 shares). The spreadsheet Investment.ods for LibreOffice or OpenOffice, and Investment.xlsx for Microsoft Excel, lists the stocks that you purchased, the price you purchased them for last year, the current price, and the price estimate for next year.
If you sell any shares, you have to pay a transaction cost of 1% of the amount transacted.
In addition, you must pay a capital-gains tax at the rate of 30% on any capital gains at the time of the sale. For example, suppose that you sell 100 shares of a stock today at $50 per share, which you originally purchased for $30 per share. You would receive $5,000. However, you would have to pay capital-gains taxes of:
0.30×($5,000−$3,000)=$600,
and you would have to pay:
0.01×$5,000=$50
in transaction costs. Therefore, by selling 100 shares of this stock, you would have a net cashflow of
$5,000−$600−$50=$4,350.
Note that none of the stocks decreased in value since the time of purchase, so we don’t have to deal with capital losses.
You would like to sell enough shares of stock today to generate $10,000 to use as part of a down payment on a new home. You need to decide how many shares of which stocks to sell in order to generate $10,000, after taxes and transaction costs, while maximizing the estimated value of your stock portfolio next year. Let’s formulate this as a linear optimization problem.
How many decision variables should your model have?
Ans: 8
We’ll assume for this problem that you can’t sell more shares of stock than you own, and you can’t buy additional shares. What is the maximum value your decision variables can be?
Ans: 150 and 0 (minimum)
Your objective is to maximize the estimated value of your stock portfolio next year. To do this, you should sum the estimated value of each stock next year. Suppose you sell x shares of your stock in Microsoft. What is the estimated value of your Microsoft stock next year?
Ans: 34.55 x (150 -x)
You need to make sure you get $10,000 in cash from selling your stocks, after taxes and transaction costs. How much would you get in cash, after taxes and transaction costs, if you sell 50 shares of your Intel stock?
Ans: 1154.715
In the optimal solution, which stocks do you sell at least one share of? Select all that apply.
Ans: Yahoo, Microsoft, Intel
What is the objective value of the optimal solution (the estimated value of your portfolio of stocks next year)?
Ans: 26773.66271
How many shares of stock in total should you sell to make sure you have enough cash, according to the optimal solution? (Assume that you can sell fractional shares.)
Ans: 367.7232981
As an invester, you like having a portfolio of eight different stocks because it diversifies your investment. If one or two stocks do poorly this year, you won’t worry as much because you have many other stocks. In the optimal solution for this problem, you sold all of your shares of some stocks, but you would like to keep at least half of the shares of each of your stocks.
Adjust the formulation so that you sell no more than 75 shares of each stock, and solve it again.
In the optimal solution, you sell at least one share of which of your stocks? Select all that apply.
Ans: Yahoo, GE, Microsoft, Cisco, Intel, Pfizer
The value: 26468.54116
Even though your investment is worth a bit less next year by diversifying, you expect this diversification to help you long term.
However, you notice that you expect the Yahoo! stock to decrease in value next year. So, while you would like to sell no more than 75 shares of your other stocks, you would like to sell exactly 100 shares of your Yahoo! stock. Adjust your formulation in LibreOffice again, and re-solve to get the new optimal solution.
You now sell at least one share of how many different stocks?
Ans: 5 Stocks, with value = 26507.52535
The objective is to minimize costs. The prices charged by the six local mills for production, as well as the production cost at Filatoi Riuniti, are provided in the spreadsheet. Additionally, the yarn that is spun by the six local mills needs to be transported. The transportation costs per kg of yarn are also provided in the spreadsheet.
Our model has two types of constraints: capacity constraints and demand constraints. The spreadsheet contains a table showing the production capacity and production rate per product for each of the mills (including Filatoi Riuniti). For example, at the Bresciani mill, it takes 0.70 hours to produce one kg of extra fine yarn, and there are a total of 3,000 machines hours per month available at this mill. There is also a table that estimates the demand for the four yarn sizes in the current month. We should produce at least the demand for each yarn type.
Formulate and solve this problem in LibreOffice (or in the spreadsheet software you are using). The objective should minimize the sum of the production and transportation costs. You should have the following constraints: non-negative decision variables, a capacity constraint for each mill, and a demand constraint for each type of yarn. Remember that Ambrosi and De Blasi can’t produce extra fine yarn, so you should also account for this.
What is the objective value of the solution?
Ans: 1382544.334
Ans: 9182.982595
Ans: 6250
Ans: Every mill except De Blasi
Ans: No, don’t rent
Ans: Yes, rent
Suppose that the shadow price of the medium yarn demand constraint is $12.30, with an allowable increase of 5388. Recall that the shadow price is the amount that the objective increases per unit increase in the right hand side of the constraint. So a shadow price of $12.30 for the medium yarn demand constraint means that the cost increases by $12.30 for every unit increase in the medium yarn demand. The allowable increase of 5388 means that this shadow price hold up to an increase of 5388 in the right hand side of this constraint. If we increase the right hand side by more than 5388, the shadow price will change, and it is impossible to know how it changes without re-solving the model.
A new client is interested in purchasing up to 5,000 kg/month of medium size yarn. What is the minimum price per kg of yarn that Filatoi Riuniti should quote to this new client? (In answering this question, assume that Filatoi has not decided to increase its spinning machine capacity, and that Filatoi does not want to change the prices that they currently charge their existing clients.) Ans: 12.3
Now suppose that the client wants to purchase 6,000 kg/month of medium size yarn. Now what is the minimum price per kg of yarn that Filatoi Riuniti should quote to this new client? (In answering this question, assume that Filatoi has not decided to increase its spinning machine capacity, and that Filatoi does not want to change the prices that they currently charge their existing clients.)
Ans: 12.44892473
In many situations, the problem data is estimated but is not known for sure. In this problem, the optimization model is based in part on the prices charged by the local mills to Filatoi Riuniti and on an estimate of Filatoi Riuniti’s internal production costs. The plant manager, the accounting department, and you estimate that Filatoi Riuniti’s internal production costs could vary within a 5% range of the numbers given.
If Filatoi Riuniti’s production costs increase by 5%, by how much does the objective value of the solution change? You should adjust and re-optimize the model to answer this question.
Ans: 48924.49062
If Filatoi Riuniti’s production costs decrease by 5%, by how much does the objective value of the solution change (in absolute difference)? You should adjust and re-optimize the model to answer this question.
Ans: same as above = 48924.49062
Now you think that the production capacity of one of your local mills, De Blasi, could vary within a 20% range of the estimated value. In the current solution, De Blasi produces 2040.125 kg of medium yarn, and none of the other types of yarn. If De Blasi’s capacity is actually 20% higher or lower than the estimated value, will the solution change? HINT: Think about De Blasi’s capacity constraint.
Ans: No. Nothing changes
One of the earliest and most successful applications of linear optimization was in the oil industry in the 1950s. According to Bill Drew, the former manager of research for Exxon Mobil, the company used linear optimization to “schedule our tanker fleets, design port facilities, blend gasoline, create financial models, you name it.” In this problem, we’ll use optimization to blend gasoline. (While the application of this problem is real, the data that we will be using here has been created for this problem.)
Gasoline blending occurs in oil refineries, where crude oil is processed and refined into more useful products, such as gasoline and diesel fuel. We will consider three products: super gasoline, regular gasoline, and diesel fuel. These can be made by mixing three different types of crude oil: crude 1, crude 2, and crude 3. Each product is distinguished by its octane rating, which measures the quality of the fuel, and its iron content, which is a contaminant in the gas. The crude oils each have an octane rating and iron content as well. The following table shows the required octane ratings and iron contents for each of the products, as well as the known octane ratings and iron contents of each of the crude oils:
The company can only buy 5,000 barrels of each type of crude oil, and can process no more than 14,000 barrels total of crude oil. One barrel of crude oil makes one barrel of gasoline or fuel (nothing is lost in the conversion).
How many barrels of each type of crude oil should the company use to make each product? Formulate this problem as a linear optimization problem, and solve it in LibreOffice (or in the spreadsheet software you are using). The first problem below asks about the formulation, and the second problem asks about the solution. So if you get stuck in formulating the problem, try working through Problem 1 below.
Ans: We have 9 constraints (3 for each gasoline type and crude oil) Plus, we also have 10 constraints: - crude limits (3) - total crude (1) - 6 more constraints for the octane and lead threshold values
Solve your optimization model in LibreOffice (or in the spreadsheet software you are using). What is the objective value of the solution?
Ans: 375000
In the optimal solution, we produce the MOST barrels of which gasoline?
Ans: Regular Gasoline
In the optimal solution, how many barrels of diesel fuel are produced?
Ans: 0
The company is unhappy with this solution, because they are exceeding the customer demand for regular gasoline. They estimate that the customer demand for super gasoline is 3,000 barrels, the customer demand for regular gasoline is 2,000 barrels, and the customer demand for diesel fuel is 1,000 barrels. They don’t want to produce more than the customer demand, since they will lose revenue. Add constraints to your model to make sure that the solution produces no more than the customer demand for each of the products, and re-solve your model.
What is the objective value now?
Ans: 150000
The following are the shadow prices for each of the demand constraints:
Super gasoline demand shadow price = 29
Regular gasoline demand shadow price = 27
Diesel fuel demand shadow price = 9
The super gasoline shadow price holds to an allowable increase of 1250, the regular gasoline shadow price holds to an allowable increase of 2500, and the diesel fuel shadow price holds to an allowable increase of 1250. The “allowable increase” is the amount that you can increase the right-hand side and still use this shadow price. For a larger increase, the shadow price will change, and you have to resolve the model to get the new shadow price.
Please answer the following questions without re-solving the optimization model.
What does a shadow price of 29 for the super gasoline demand constraint mean?
Ans: For one additional barrel of demand of super gasoline, the total profit will increase by 29.
According to the shadow prices, which type of gasoline should the company advertise to increase demand? Suppose that advertising costs $2 per unit increase in demand regardless of the type of gasoline being advertised. (Also suppose the company can only choose one type of gasoline to advertise.)
Ans: Super gasoline’s shadow price is the highest –> choose this to promote
How much additional profit would you gain if the super gasoline demand increased by 500?
(You can assume that the demand increased on its own and you didn’t have to pay anything in advertising for this increase).
Ans: 29*500
29*500## [1] 14500How much additional profit would you gain if you increased the super gasoline demand by 1500?
(You can assume that the demand increased on its own and you didn’t have to pay anything in advertising for this increase).
Ans: We can’t answer this without resolving the model. The allowable increase = 1250, whereas in the question we have 1500