Coastal Refining Company operates a refinery with a distillation capacity of 12,000 barrels per day. As a new member of Coastal's management team, you have been given the task of developing a production schedule for the refinery, i.e., determining how much of which products to produce and how.
In simplified form, the refinery process starts with distillation, which feeds crude oil into a pipestill, as it is commonly called. Here the crude is heated, and as the temperature rises, different products are given off in vapor form. These products can be collected separately and sold as produced, or blended and sold, or else processed further in a catalytic cracker. In the cracker, less profitable products such as heating oil can be converted to more profitable products such as naphtha. Finally, the various products of the pipestill and the cracker are blended and sold on the marketplace.
To streamline the analysis, the lighter distillation products have been grouped under the single category of naphtha. Similarly, the lighter streams from the catalytic cracker have been grouped under the single category of catalytic naphtha. Based on present operation, a satisfactory gasoline blend can be obtained by combining the straight-run naphtha (directly from the pipestill) and catalytic naphtha in the ratio of at most 5 parts straight run to 4 parts catalytic.
Two crude oil sources are available to the refinery, in quantities up to 10,000 barrels per day for each source of crude. The yields from distillation and the delivered cost of the inputs are as follows:

The yields for each of the crude oil sources add to less than one because there is always some residue from the process, averaging 10 percent.
Heating oil can be used as feedstock to the catalytic cracker, or alternatively, it can be treated and sold in the form produced by the pipestill. Gasoil can also be fed to the cracker, or, with some blending, it can be sold as equipment fuel. The blending process requires that at least 1 part diesel fuel must be mixed with 4 parts gasoil.
In the catalytic cracking process, the feedstock is recycled through the cracker until fully converted. For example, each barrel of gasoil originally fed into the catalytic cracker uses an average of 4 barrels of capacity when fully cycled. The products of this process will be catalytic naphtha, catalytic heating oil, and pitch. When heating oil is used as feedstock, the possibilities are a bit more complicated. In normal mode, each barrel of heating oil originally fed into the catalytic cracker uses an average of 2.5 barrels of capacity when fully cycled; but there is also a "high-severity" mode of operation, which uses an average of 2.0 barrels of capacity. The capacity of the catalytic cracker is 15,000 barrels of throughput per day and its yields are as follows:

The sums exceed 100 percent because the cracking process reduces the density of the output.
Current prices (in dollars per barrel) for the major final products are as follows: gasoline $42, diesel fuel $38, catalytic heating oil $36, straight-run heating oil $35, equipment fuel $32 and pitch $25. The marketing department has indicated that the company could probably not sell more than 4,000 barrels per day each of diesel fuel and catalytic heating oil, but it could sell as much of the other products as the refinery is able to produce.
A recent internal study has reported that the direct processing cost per barrel of crude oil going into the pipestill is $1.20 and the direct cost for the catalytic cracker is $1.50 per barrel of input, with a 15 percent premium for the high-severity mode of operation.
a. What is the optimal schedule for Coastal Refining?
b. What is the optimal total cost?
c. What are the shadow prices for pipestill capacity and cracker capacity? Over what ranges do these figures hold? What do they tell us about the value of capacity expansion?

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Crude 2 Crude 1 Naphtha Diesel Fuel Gasoil Heating Oil Pitch Cost/barrel 0.20 0.25 0.15 0.20 0.40 0. 10 0.30 0.20 0.05 $34 0.10 $32 Heating Oil- Normal Heating Oil- Hi-severity Gasoil Catalytic naphtha Catalytic heating oil Pitch 35% 80% 15% 30% 50% 50% 70% 10% 15%