Part I: Rockhill Shipping & Transport Company Allen, a manager of the South-Atlantic office of the...

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Part I: Rockhill Shipping & Transport Company Allen, a manager of the South-Atlantic office of the Rockhill Shipping & Transport Company is negotiating a new shipping contract with Chimotoxic, a company that manufactures chemicals for industrial use. Chimotoxic wants Rockhill to pick up and transport waste products from its six plants to three waste disposal sites. Allen is very concerned about this proposal arrangement. The chemical wastes that will be hauled can be hazardous to humans or the environment if they leak. In addition, some of the communities in the regions where the plants are located may prohibit hazardous materials from being shipped through their municipal limits. Thus, not only the shipments have to be handled carefully and transported at reduced speeds, but they may also have to traverse in circuitous routes in some cases. Allen has estimated the cost of shipping a barrel of waste from each of the six plants to each of the three waste disposal sites as shown in the following table. Plant: Denver Orangeburg $12 Morganton 14 Morrisville 13 Pineville 17 Rockhill 7 Statesville 22 Waste Proposal Site Florence Macon $15 $17 9 10 20 || 16 19 14 12 16 18 Table 1: Shipping costs, per barrel of waste from six plants to three waste disposal sites The plants generate the following amounts of waste products each week: Plant: Waste per Week (bbl) Denver 45 Morganton Morrisville 26 42 Pineville 53 Rockhill Statesville 29 38 Table 2: Total Waste generated by each plant The three waste disposal sites at Orangeburg, Florence, and Macon can respectively accommodate a maximum of 65, 80, and 105 barrels per week. In addition to shipping directly from each of the six plants to one of the three waste disposal sites, Allen is also considering using each of the plants and the waste disposal sites as intermediate shipping points. In this case, trucks would be able to drop a load at a plant or a disposal site to be picked up and carried on to the final destination by another truck. Furthermore, Rockhill would not incur any handling costs because Chimitoxic has agreed to take care of all the handling costs at the plants and at the disposal sites. In other words, Rockhill's only cost will be the transportation cost. Therefore, Allen wants to be able to consider the possibility that it may be cheaper to drop and pick up loads at intermediate points rather than ship them directly. Allen has estimated the shipping costs per barrel between each of the six plants to be as follows: Plant: Denver Morganton Plant IIIIII Denver $--- Morganton $3 6 Morrisville $4 7 Morrisville 5 7 Pineville 5 4 3 5 9 5 4 7 || Rockhill Statesville Pineville $9 Rockhill Statesville $5 6 9 $4 4 3 4 9 3 3 14 12 8 Table 3: Shipping costs, per barrel of waste from each plant to another plant The estimated shipping cost per barrel between each of the waste disposal sites is as follows: Waste Disposal Site: Orangeburg Florence Orangeburg $-- 12 10 Waste Proposal Site Florence $12 --- Macon $10 15 Macon 15 Table 4: Shipping costs, per barrel of waste between the three waste disposal sites --- Allen wants to determine the shipping routes that will minimize Rockhill's total cost in order to develop a contract proposal to submit to Chimotoxic for waste proposal. He particularly wants to know if it would be cheaper to ship directly prom the plants to the waste sites or if he should drop and pick up some loads at the various plants and waste sites. In the word document, explain the details of the solutions obtained for the optimal routes and their respective optimal costs for both cases. In particular for the case when loads are dropped and picked up at various plants and waste sites, explain how many barrels, in total, will be transported each week from a source to a destination. Part 2: Investment Allocations An investor has selected the following asset types in his portfolio. The expected return for each asset type has been estimated by using the historical data: Expected Returns Bonds 7% High tech stocks 12% Foreign stocks 11% Call options 14% Put options 14% Gold 9% Table 5: Expected returns of Investments The following table indicates the covariance matrix of the assets' returns. Each diagonal entry is the variance of an asset and non-diagonal entries are the covariances between any pairs of assets. Bonds High tech stocks Foreign stocks Call options Put options Gold Bonds High tech stocks 0.001 0.0003 -0.0003 0.00035 -0.00035 0.0004 0.009 0.0004 0.0016 -0.0016 0.0006 Foreign stocks 0.008 0.0015 -0.0055 -0.0007 Call options 0.012 -0.0005 0.0008 Put options Gold 0.012 -0.0008 0.005 (i) (ii) Table 6: The Covariance matrix of assets' returns Suppose that our investor wishes to invest $10,000 in this portfolio. Determine how he should allocate this investment to the individual assets in his portfolio in order to have a minimum baseline expected return of 11%, and at the same time, at a minimum risk. Let the solution pair be denoted by (r, e), where r denotes the minimized risk and "e" denotes the expected portfolio return after the problem is solved. Use successive values of 10%, 10.5%, 11%, 11.5%, 12%, 12.5%, 13% and 13.5% as the baseline return values to obtain eight pairs of solutions (r, e). Plot e versus r. Explain whether there exists a pattern in this plot. In other words, explain, in your opinion, the type of mathematical relationship that "r" and "e" may have. Part I: Rockhill Shipping & Transport Company Allen, a manager of the South-Atlantic office of the Rockhill Shipping & Transport Company is negotiating a new shipping contract with Chimotoxic, a company that manufactures chemicals for industrial use. Chimotoxic wants Rockhill to pick up and transport waste products from its six plants to three waste disposal sites. Allen is very concerned about this proposal arrangement. The chemical wastes that will be hauled can be hazardous to humans or the environment if they leak. In addition, some of the communities in the regions where the plants are located may prohibit hazardous materials from being shipped through their municipal limits. Thus, not only the shipments have to be handled carefully and transported at reduced speeds, but they may also have to traverse in circuitous routes in some cases. Allen has estimated the cost of shipping a barrel of waste from each of the six plants to each of the three waste disposal sites as shown in the following table. Plant: Denver Orangeburg $12 Morganton 14 Morrisville 13 Pineville 17 Rockhill 7 Statesville 22 Waste Proposal Site Florence Macon $15 $17 9 10 20 || 16 19 14 12 16 18 Table 1: Shipping costs, per barrel of waste from six plants to three waste disposal sites The plants generate the following amounts of waste products each week: Plant: Waste per Week (bbl) Denver 45 Morganton Morrisville 26 42 Pineville 53 Rockhill Statesville 29 38 Table 2: Total Waste generated by each plant The three waste disposal sites at Orangeburg, Florence, and Macon can respectively accommodate a maximum of 65, 80, and 105 barrels per week. In addition to shipping directly from each of the six plants to one of the three waste disposal sites, Allen is also considering using each of the plants and the waste disposal sites as intermediate shipping points. In this case, trucks would be able to drop a load at a plant or a disposal site to be picked up and carried on to the final destination by another truck. Furthermore, Rockhill would not incur any handling costs because Chimitoxic has agreed to take care of all the handling costs at the plants and at the disposal sites. In other words, Rockhill's only cost will be the transportation cost. Therefore, Allen wants to be able to consider the possibility that it may be cheaper to drop and pick up loads at intermediate points rather than ship them directly. Allen has estimated the shipping costs per barrel between each of the six plants to be as follows: Plant: Denver Morganton Plant IIIIII Denver $--- Morganton $3 6 Morrisville $4 7 Morrisville 5 7 Pineville 5 4 3 5 9 5 4 7 || Rockhill Statesville Pineville $9 Rockhill Statesville $5 6 9 $4 4 3 4 9 3 3 14 12 8 Table 3: Shipping costs, per barrel of waste from each plant to another plant The estimated shipping cost per barrel between each of the waste disposal sites is as follows: Waste Disposal Site: Orangeburg Florence Orangeburg $-- 12 10 Waste Proposal Site Florence $12 --- Macon $10 15 Macon 15 Table 4: Shipping costs, per barrel of waste between the three waste disposal sites --- Allen wants to determine the shipping routes that will minimize Rockhill's total cost in order to develop a contract proposal to submit to Chimotoxic for waste proposal. He particularly wants to know if it would be cheaper to ship directly prom the plants to the waste sites or if he should drop and pick up some loads at the various plants and waste sites. In the word document, explain the details of the solutions obtained for the optimal routes and their respective optimal costs for both cases. In particular for the case when loads are dropped and picked up at various plants and waste sites, explain how many barrels, in total, will be transported each week from a source to a destination. Part 2: Investment Allocations An investor has selected the following asset types in his portfolio. The expected return for each asset type has been estimated by using the historical data: Expected Returns Bonds 7% High tech stocks 12% Foreign stocks 11% Call options 14% Put options 14% Gold 9% Table 5: Expected returns of Investments The following table indicates the covariance matrix of the assets' returns. Each diagonal entry is the variance of an asset and non-diagonal entries are the covariances between any pairs of assets. Bonds High tech stocks Foreign stocks Call options Put options Gold Bonds High tech stocks 0.001 0.0003 -0.0003 0.00035 -0.00035 0.0004 0.009 0.0004 0.0016 -0.0016 0.0006 Foreign stocks 0.008 0.0015 -0.0055 -0.0007 Call options 0.012 -0.0005 0.0008 Put options Gold 0.012 -0.0008 0.005 (i) (ii) Table 6: The Covariance matrix of assets' returns Suppose that our investor wishes to invest $10,000 in this portfolio. Determine how he should allocate this investment to the individual assets in his portfolio in order to have a minimum baseline expected return of 11%, and at the same time, at a minimum risk. Let the solution pair be denoted by (r, e), where r denotes the minimized risk and "e" denotes the expected portfolio return after the problem is solved. Use successive values of 10%, 10.5%, 11%, 11.5%, 12%, 12.5%, 13% and 13.5% as the baseline return values to obtain eight pairs of solutions (r, e). Plot e versus r. Explain whether there exists a pattern in this plot. In other words, explain, in your opinion, the type of mathematical relationship that "r" and "e" may have.