Posted: October 20th, 2022

Assignment 1 One of Latoils main products is crude oil

Latoil is a fictional Norwegian multinational energy company headquartered in Stavanger, Norway. The company’s main business is petroleum, and the company operates several oil fields, refineries, petrol stations and other related activities.

Assignment 1
One of Latoils main products is crude oil. Crude oil can be sold directly abroad (exported) or be used as an input in a local refinery. At this local refinery one barrel of oil (which is about 159 liters) can be used as an input to produce the following products:
Diesel
Heating oil
Jet fuel
Heavy fuel oil
Liquified petroleum gases
Gasoline

It takes 1 liter of crude oil as an input to produce 1 liter of each of the above-mentioned products. In addition, each product requires a certain amount of labour, testing and materials to produce 1 liter. The production technology is shown in the table below:

Production technology at the refinery: Requirement to produce 1 liter of each product
Input
(measured in units) Crude oil Diesel Heating oil Jet fuel Heavy fuel oil Liquified pertroleum
gases Gasoline
Crude oil (1 liter) 1,00 1,00 1,00 1,00 1,00 1,00
Labour 0,10 0,20 0,20 0,25 0,50 0,20 0,20
Testing 1 0,05 0,02 0,01 0,05 0,01
Testing 2 0,02 0,02 0,01 0,02 0,01 0,02
Material 1 0,05 0,40 0,05 0,08 0,03
Material 2 0,05 0,05 0,40 0,05 0,05

The price and total availability for each of the inputs is:

Input Cost Maximum amount available
Labour 10,00 NOK 49.500 units
Testing 1 15,00 NOK 13.500 units
Testing 2 20,00 NOK 14.600 units
Material 1 1,00 NOK 12.300 units
Material 2 2,50 NOK 18.900 units

Each product has a minimum amount that must be produced due to contract obligations. There is also a highest possible production, and we assume, for simplicity, that all production is sold. Both the minimum and maximum number of units produced, along with the price of each product (all converted into NOK per liter), is shown in table below:

Product Price in NOK per liter Lowest possible production Highest possible production
Crude oil 6 50.000 liter 310.000 liter
Diesel 15 5.000 liter 48.900 liter
Heating oil 5 1.000 liter 16.800 liter
Jet fuel 24 2.500 liter 21.200 liter
Heavy fuel oil 9 3.000 liter 25.400 liter
Liquified petroleum gas 10 2.000 liter 22.300 liter
Gasoline 16 10.000 liter 42.200 liter

The company wants to maximize its profits.
Open the starting file Assignment1.xls. Construct the model and determine what type of model this is.
Use Solver to find the optimal solution, meaning the production of each product that gives the company the highest profit.
Latoil is considering allocating more inputs (meaning labour, testing 1 etc) to this refinery. Which inputs should the company prioritize the most? Which inputs should the company not prioritize? Explain your answers.
Use a sensitivity analysis to determine how changes in the price of crude oil affects the optimal decision (use at least 10 different values of the crude oil price).

Assignment 2
Latoil must supply several markets with different products. In this assignment only two products are considered (i) Crude oil; and (ii) Processed oil. For oil to be processed, crude oil must be delivered to the refinery. For simplicity, we assume that processing 1 barrel of crude oil gives 1 barrel of processed oil.

Latoil must manage a network with:
8 oil fields: Oil fields extract the crude oil and sends them to onshore installations
5 onshore installations: Receives the crude oil from the oil fields and either sends them to the refineries to be processed or sells the crude oil directly to the market
6 refineries: Receives the crude oil from the installation and processes the oil before selling it to the market
9 markets: Each market must receive a certain amount of crude oil from the installations and a certain amount of processed oil from the refineries

The oil fields have a minimum and maximum capacity, and can only supply certain onshore installations:

Oil field nr. Minimum Maximum Installations that can be supplied
Oil field 1 730 barrels 19.970 barrels Installation 1, 2, 3 and 4
Oil field 2 1.100 barrels 17.700 barrels Installation 1, 2, 3 and 4
Oil field 3 1.220 barrels 7.750 barrels Installation 1, 2, 4 and 5
Oil field 4 2.160 barrels 17.150 barrels Installation 1, 2, 3, 4 and 5
Oil field 5 2.260 barrels 20.940 barrels Installation 4 and 5
Oil field 6 1.860 barrels 11.510 barrels Installation 1, 2, 3, 4 and 5
Oil field 7 660 barrels 7.680 barrels Installation 1, 3, 4 and 5
Oil field 8 2.020 barrels 14.250 barrels Installation 1, 2, 3 and 5

Each onshore installation also has a minimum capacity (the maximum production capacity is considered non-binding), and can only supply certain markets and refineries:

Installation nr. Minimum Refineries that can be supplied Markets that can be supplied
Installation 1 1.270 barrels Refinery 3, 4, 5 and 6 Market 1, 2, 3, 4, 5 and 6
Installation 2 1.910 barrels Refinery 1, 2, 3, 4, 5 and 6 Market 1, 2, 4, 5 and 6
Installation 3 1.680 barrels Refinery 1, 2 and 3 Market 7, 8 and 9
Installation 4 780 barrels Refinery 1, 2, 3, 4, 5 and 6 Market 1, 2, 3, 6, 7, 8 and 9
Installation 5 1.330 barrels Refinery 3, 4, 5 and 6 Market 1, 3, 4, 6, 7, 8 and 9

Each refinery also has a minimum capacity (the maximum production capacity is considered non-binding) and can deliver processed oil to certain markets:

Refinery nr. Minimum Markets that can be supplied
Refinery 1 2.510 barrels Market 1, 3, 4, 7, 8 and 9
Refinery 2 2.160 barrels Market 1 and 9
Refinery 3 2.320 barrels Market 1, 2, 3, 4, 6, 7, 8 and 9
Refinery 4 2.500 barrels Market 1, 2 and 3
Refinery 5 1.960 barrels Market 1, 2, 3, 4 and 9
Refinery 6 2.650 barrels Market 3, 4, 5, 6, 7, 8 and 9

Each final market needs to be supplied with a certain quantity of crude oil and processed oil:

Market nr. Minimum amount of crude oil Minimum amount of processed oil
Market 1 1.780 barrels 6.960 barrels
Market 2 7.170 barrels 7.350 barrels
Market 3 1.740 barrels 4.460 barrels
Market 4 4.240 barrels 1.590 barrels
Market 5 9.540 barrels 12.210 barrels
Market 6 9.570 barrels 8.850 barrels
Market 7 1.540 barrels 1.180 barrels
Market 8 4.680 barrels 7.160 barrels
Market 9 3.300 barrels 8.200 barrels

Open the starting file Assignment2.xls which shows the cost in NOK for each barrel transported. This is a typical example of a network model. Draw the network model with nodes and arcs (either using office, some other software or just by hand and paste it in the excel sheet).
Assume the objective is to minimize costs. Construct the model and find the optimal order plan.
Latoil is considering expanding the production capacity of the oil fields to expand the maximum production. Which oil fields would should be considered and in which order? Explain you answer.

Assignment 3
The research department at Latoil has invested in eight potential new products. A production facility has been allocated some inputs from Latoil to start producing these products. The quantity and the opportunity cost of these inputs are shown below:

Input Quantity allocated Cost per unit
Barrels of crude oil 48.800 640 NOK
Other material 96.400 420 NOK
Labour hours 75.000 250 NOK
Testing hours 55.000 430 NOK

The eight products require a different combination of these inputs to produce one unit. If one product is produced, a fixed cost incurs, but not if the product is not produced. Also, if the product is produced a minimum number must be produced due to economies of scale. There is also a sales constraint for each product. Finally, each of the products has a sales price. All this information is summarized in the table below:

Prod.1 Prod.2 Prod.3 Prod.4 Prod.5 Prod.6 Prod.7 Prod.8
Barrels of crude oil 6 6 4 4 4 4 6 4
Other material 11 8 6 8 8 8 6 12
Labour hours 9 9 3 5 11 9 2 8
Testing hours 5 5 1 1 4 3 5 4
Fixed cost if produced in NOK 500.000 800.000 1.500.000 1.500.000 1.200.000 700.000 600.000 1.400.000
Mininium production if produced 500 300 300 100 50 350 400 350
Maximum sale 2.000 1.500 3.000 1.000 500 5.000 1.000 2.500
Sales unit price in NOK 15.960 15.044 7.046 9.342 17.005 15.261 9.982 11.630

Latoil wants to know which products should be produced and how many of each with the objective being to maximize profits.
Open the starting file Assignment3.xls. Set up the problem in excel.
Find the optimal solution and comment on the results.
The leadership of the company wants to know how the optimal solution changes given the amount of labour allocated to the production of these products. Conduct a sensitivity analysis which allows the allocated labour to vary from 50.000 hours to 100.000 hours.

Assignment 4
The marketing department at Latoil has recently been able to negotiate a deal giving the company sole access to a new market for petrol stations. Latoil will therefore operate as a monopolist and the two main products supplied are diesel and gasoline. It is assumed that the preferences in the market are comparable to Norway, meaning that Norwegian data on prices and demand can be used to estimate the demand function in the market.
Open the starting file Assignment4.xls and answer the following questions
The starting files contains monthly price and sales data for diesel and gasoline for a three-year period. Graph this relationship using a scatterplot. Use the graph to find the equation for a linear demand function and a power demand function.
The company wants to know which demand function describes the data most accurately. One way to calculate the accuracy of the prediction is to calculate the mean absolute percentage error (MAPE). It means that you need to calculate absolute percentage error for each observation given by:

APE=|Observed demand-Predicted demand|/(Observed demand)

The term |Observed demand-Predicted demand| is the absolute value of the difference between observed and predicted demand. The MAPE is the average APE for all observations. The curve with the smallest MAPE is the one that fits the data the best. Based on this criterion, which curve fits the gasoline data the best? Which curve fits the diesel data the best?
Use the demand curves that fit the data the best to be used in the model (if you are unable to answer b) use the linear functions). Latoil has the following cost functions for gasoline and diesel in the new market:

Cost function diesel production : C_D=x_D^2+20x_D+75
Cost function gasoline production : C_G=0,5x_G^2+10x_G+50

Where x_D is the production of diesel in million of liters and x_G is the production of gasoline in million of liters. Assume that the objective is to maximize profit. Set up the model and find the optimal prices for diesel and gasoline and the associated profit.
Latoil must also decide the total capacity it wants to use in the market. Assume that the cost of capacity is 11 NOK per liter of diesel and 9 NOK per liter of gasoline. It is common practice among gas stations to vary the price during the week. Let’s assume that Latoil considers three different periods during the week:

Diesel Gasoline
High demand x_D^H=390-4,6p_D^H+0,2p_D^N+0,1p_D^L x_G^H=135-2,8p_G^H+0,1p_G^N+0,1p_G^L
Normal demand x_D^N=325-4,6p_D^N+0,3p_D^H+0,2p_D^L x_G^N=125-2,8p_G^N+0,1p_G^H+0,1p_G^L
Low demand x_D^L=290-4,6p_D^L+0,1p_D^H+0,3p_D^N x_G^L=85-2,8p_G^L+0,1p_G^H+0,2p_G^N

The notation is as follow:
x_D^H: Diesel demand in millions of liters in the high demand period
x_D^N: Diesel demand in millions of liters in the normal demand period
x_D^L: Diesel demand in millions of liters in the normal demand period
p_D^H: Diesel price in NOK in the high demand period
p_D^N: Diesel price in NOK in the normal demand period
p_D^L: Diesel price in NOK in the low demand period
x_G^H: Gasoline demand in millions of liters in the high demand period
x_G^N: Gasoline demand in millions of liters in the normal demand period
x_G^L: Gasoline demand in millions of liters in the normal demand period
p_G^H: Gasoline price in NOK in the high demand period
p_G^N: Gasoline price in NOK in the normal demand period
p_G^L: Gasoline price in NOK in the low demand period

Let the cost function be the same as above in 4a). Find the optimal price of diesel and gasoline for the three different periods (you can assume all three periods are of the same duration). Also find the associated optimal capacity and profit.

Assignment 5
The Norwegian market for jet fuel is assumed to have six major fictional customers:

Customer Marginal willingness to pay
Brian Air p=115-5x
Sinnair p=105-7x
KEM p=124-4x
Horwegian p=120-3x
MAS p=114-6x
Videre p=100-4x

For the marginal willingness to pay, the price p is per liter whilst the quantity, x, is in millions of liters. The cost function of producing jet fuel (in million NOK) is given by:

Cost function C(x)=10x+100

We make the simplifying assumption that all jet fuel produced is sold, meaning no inventory and no uncertainty. Latoil is evaluating the profit for different pricing strategies. Find the optimal price(s), quantity sold and profit for the following different strategies:
Single unit price for all consumers.
Each consumer pays a separate unit price.
A two-part tariff, meaning all consumers pay the same fixed price and the same unit price.
The government introduces a “jet fuel tax” of 10 NOK per liter in a bid to reduce emissions from aviation. How does the tax change the optimal solution in each of the cases above?

Assignment 6
Latoil wants to adopt a more environmentally friendly approach, and even considers changing its name to Aquanor to highlight an increased emphasis on hydroelectrical power and on windmills constructed at sea. Latoil needs to deliver electricity to the following markets:

Market Demand market
Market 1 351 GWh
Market 2 267 GWh
Market 3 423 GWh
Market 4 416 GWh
Market 5 472 GWh
Market 6 153 GWh
Market 7 343 GWh
Market 8 192 GWh
Market 9 164 GWh

One Giga Watt hour (GWh) is the same as one million Kilo Watt hours (KWh). The unit price for electricity is 477.300 NOK per GWh (equivalent to 0,4773 NOK per KWh). Latoil has interests in three different kinds of electricity producers. First, Latoil controls eight power plants based on fossil fuel technology, which deliver electricity to specific markets. The capacity of each of these is:

Fossil fuel power plant Maximum production capacity
Fossil fuel power plant 1 239 GWh
Fossil fuel power plant 2 279 GWh
Fossil fuel power plant 3 281 GWh
Fossil fuel power plant 4 234 GWh
Fossil fuel power plant 5 207 GWh
Fossil fuel power plant 6 299 GWh
Fossil fuel power plant 7 281 GWh
Fossil fuel power plant 8 200 GWh

Second, Latoil has an interest in six windmill parks with the following production capacity:

Windmill park Maximum production capacity
Windmill park 1 161 GWh
Windmill park 2 119 GWh
Windmill park 3 131 GWh
Windmill park 4 180 GWh
Windmill park 5 117 GWh
Windmill park 6 194 GWh

Finally, Latoil has an interest in two hydroelectrical power plants with the following production capacity:

Hydro power plant Maximum production capacity
Hydro power plant 1 486 GWh
Hydro power plant 2 488 GWh

For each GWh delivered the company incurs a unit cost and a unit of pollution. The unit of pollution is a number that implies that a higher number generates more pollution.

Latoil has two major objectives: (i) To maximize profit; and (ii) To minimize pollution. These objectives cannot be ranked.
Open the starting file to assignment 6. The file contains the unit costs and the pollution for each of the potential delivery routes. Construct the model with two different objectives functions, one to maximize profits and the other to minimize pollution. Solve the model for each objective function and interpret the optimal solution in each case.
Find the efficiency frontier between pollution and costs. Graph this relationship and give an economic interpretation. Explain why any optimal solution must be pareto efficient.
Latoil has until now maximized profits (and not taken pollution into consideration). If Latoil changes its name to Aquanor, it plans to weight pollution twice as important as profit (meaning that the company wants to reduce pollution by 2/3 of its total pollution reduction when moving from the maximum profit to the minimum pollution solution). Which solution will the company chose and what will the change in profit and pollution following the name change?

Assignment 7
The brent oil price is important for Latoils profits. Go to the EIA (US Energy Information Administration) website (https://www.eia.gov/dnav/pet/hist/rbrteD.htm) and download the monthly brent spot price in USD for the period may 1987-february 2019 by using the “downloand data (xls file)” option.
Use the data you downloaded to create your own starting file and save it as “Assignment 7 Starting File.xls”. Use the Pivot Table to answer the following questions:
What is the average oil price for all February months?
For which month is the oil price on average the highest?
For which month is the oil price the most volatile measured by standard deviation?
The price of a forward for January 2020 is 69,27 USD, which is also the expected spot price for January 2020. The company considers three scenarios:
Low oil price: One standard deviation below the expected oil price with a probability of 30 %
Medium oil price: The expected oil price with a probability of 50 %
High oil price: One standard deviation above the expected oil price with a probability of 20 %

The standard deviation used is for the all January months (use the data from a). Latoil has already agreed to a delivery of 25.000 barrels in January 2020, but now needs to decide whether to hedge the price. Only two options are considered, a full hedge (using the forward) or no hedge (not to use the forward). The company wants to have the highest expected revenue possible. Show the decision tree related to this problem and find the expected monetary revenue. Should Latoil hedge the price?
Latoil could hire a firm dealing with oil price forecast, Fystad Energy. Their predictions are considered among the most accurate in the market, and their services will cost the company 10.000 USD. Historically, Fystad has the following accuracy on their predictions:

Fystad predicted a high price Fystad predicted a medium price Fystad predicted a low price
Actual outcome: High price 72,7 % 23,7 % 3,6 %
Actual outcome: Medium price 19,3 % 60,7 % 20,0 %
Actual outcome: Low price 8,0 % 15,6 % 76,4 %

Make a new decision tree which takes this option into account for Latoil. Should the company hire Fystad Energy? When should Latoil use a forward?
Find the expected monetary value of sample information (EVSI) and the expected value of perfect information (EVPI) for c).

Assignment 8
Latoil is considering an investment in a new oil field. There are several uncertain variables that affect the profitability of the project.
Investment cost incurs in year 0 and the installation takes 2 years to construct (this is assumed not uncertain). Production starts in the beginning of the year the installation is completed. In addition, the following is known about the investment cost:
Oil production can start once the installation is completed. The cost of the project is most likely at 80 million NOK. The company knows that the cost will not be below 40 million NOK and could be as high as 200 million NOK.
Additional investments are needed to construct the pipelines for gas transportation to allow natural gas production (investments costs also incur in year 0). These investments are done at the same time as the oil field installation investment above, and the construction time is the same. It means that gas production can start at the same time as oil production. The cost of the project is most likely at 100 million NOK. The company knows that the cost will not be below 20 million NOK and could be as high as 120 million NOK.
All oil and gas production are assumed sold (no inventories). The following information is given for production:
Oil production varies each year. The production is never going to be below than 20.000 barrels and cannot exceed 100.000 barrels. The most likely production is 75.000 barrels.
Gas production also varies each year. Most likely the production is 9,6 million Sm³ (standard cubic meter). Gas production is never below 2,4 million Sm³ and cannot exceed 12 million Sm³.
Prices vary each year and both are assumed to be normally distributed (Hint: In @Risk “RiskNormal(Mean;Standarddeviation)” is used for a normal distribution)
The oil price has a mean of 677 NOK per barrel (80 USD with an exchange rate of 8,5 NOK per 1 USD) and a standard deviation of 223 NOK.
The price of natural gas is measured in Sm³ (standard cubic meter) and has a mean of 5,5 NOK per Sm³. The standard deviation is 1 NOK per Sm³.
Reserves are partially unknown with the following information currently available:
Oil: The known reserves at the present time is 1 million barrels, but there could be a potential maximum of 0,6 million barrels in additional reserves. The most likely is an additional reserve of 0,4 million barrels.
Gas: The known reserves are 120 million Sm³ gas. At most there is an additional 80 million Sm³ more in reserves, with their most likely being 50 million Sm³ more in reserves.
Other relevant information:
Annual fixed cost is known. For oil production it is 1,5 million NOK and for gas production it is 1,0 million NOK.
Unit costs are also known. For oil production it is 400 NOK per barrel and for gas production it is 4,2 NOK per Sm³ (standard cubic meter).
The discount rate is 5 % and is also known in advance

The student version of @Risk only allows 100 inputs, which is less than is needed to run the entire model. Therefore, you should make two different models, one for oil and one for gas. Open the starting files Assignment8 Oil.xls. and Assignment8 Gas.xls. Set up the two models with the net present value of the investment being the objective function (Note: One should only have one excel file open at a time when running @Risk).
Run simulations on both models with @Risk using 1000 iterations with the net present value of profit and the number of years of production being the main outputs (To store the results it is recommended you take a screenshot of the simulation results). Find:
The mean years of production for the oil and gas fields, the standard deviation, the lowest number of years in operation and the highest number of years in production.
Mean net present value, the standard deviation, the lowest net present value and the highest net present value.
Based on the results in b) and c) above, make a recommendation for Latoil on whether to invest in oil production? Should they in addition invest in gas production?

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