3.4.3

3.5.5   [ Also solve the problem using LINDO or Linear. Note that the 6 pm to 12 am and 12 am to 6 am shifts are consecutive ]

3.8.2   [ Also solve the problem using LINDO or Linear. Note that it is possible that not all the oranges will be sold. ]

Oil Refinery Problem

An oil refinery purchases two grades of crude oil (crude 1 and crude 2). These crude oils are put through four processes: distillation, reforming, cracking, and blending, to produce gasoline and other fuels which are sold.

Distillation

Distillation separates each crude oil into fractions known as light naphtha, medium naphtha, heavy naphtha, light oil, heavy oil, and residuum according to their boiling points. Light, medium and heavy naphthas have octane numbers of 90, 80 and 70 respectively. The fractions into which one barrel of each crude splits are given in the table:

	Light naptha	Medium naphtha	Heavy naphtha	Light oil	Heavy oil	Residuum
Crude 1	0.10	0.20	0.20	0.12	0.20	0.13
Crude 2	0.15	0.25	0.18	0.08	0.19	0.12

Note: the numbers in each row do not add up to 1, as there is a small amount of wastage in distillation.

Reforming

The naphthas can be immediately used for blending into different grades of gasoline, or can go through a process known as reforming. Reforming produces a product known as reformed gasoline with an octane number of 115. The yields of reformed gasoline from each barrel of the different naphthas are as follows: light naphtha 0.6, medium naphtha 0.52, heavy naphtha 0.45.

Cracking

The light and heavy oils can either be used directly for blending into jet fuel or fuel oil, or can be put through a process known as catalytic cracking. The catalytic cracker produces cracked oil and cracked gasoline. Cracked gasoline has an octane number of 105. The yields are as follows:

	Cracked oil	Cracked gasoline
Light oil	0.68	0.28
Heavy oil	0.75	0.20

Blending

Regular gasoline and premium gasoline are obtained by blending the naphthas, reformed gasoline, and cracked gasoline. The only stipulations concerning them are that regular gasoline must have an octane number of at least 84 and that premium must have an octane number of at least 94. It is assumed that octane numbers blend linearly by volume.

Jet fuel is blended from light, heavy and cracked oils and residuum. Its vapour pressure must be between 8 and 10. The vapour pressures for the ingredients are as follows:

	Light oil	Heavy oil	Cracked oil	Residuum
Vapour pressure	10.0	6.0	15.0	0.5

It may be assumed that vapour pressures blend linearly by volume.

Fuel oil is produced by blending light oil, cracked oil, heavy oil and residuum in the ratio 10:4:3:1.

Lube oil is produced from residuum. One barrel of residuum yields 0.5 barrels of lube oil.

Availabilities and capacities

20000 barrels a day of crude 1 are available (you can buy any amount up to this).

30000 barrels a day of crude 2 are available.

At most 45000 barrels a day of crude can be distilled.

At most 10000 barrels a day of naphtha can be reformed.

At most 8000 barrels a day of oil can be cracked.

The daily production of lube oil must be between 500 and 1000 barrels.

Costs and revenues

Crude 1 costs $24 per barrel. Crude 2 costs $26 per barrel. Cracking and reforming each cost $1 per barrel of input. The revenues from each product are as follows.

	Premium gasoline	Regular gasoline	Jet fuel	Fuel oil	Lube oil
$/barrel	35	34	29	28.5	26

How should the operations of the refinery be planned in order to maximize profit?

Note: You will want lots of variables and constraints here: my formulation involves 40 decision variables and 37 constraints. You should have at least one decision variable for each input, each intermediate product and each final product. When a product can be used for different purposes, the amount used for each should again be a decision variable, and there will be a material balance constraint.