MS5220 Homework4 :
Read the Toscosa case and answer the questions:
THE TASCOSA REFINERY (A)
In the fall of 1962 several members of the Process Economics Department of the Tascosa Refinery were engaged in projects involving economic analysis of the operations of the refinery.
One of the projects was specifically concerned with the development of a linear programming model of the refinery which was hoped would be useful in scheduling refinery operations on a week-to-week basis. This particular project was being handled by Charles Henderson, a chemical engineer with some formal training in economic analysis using linear programming.
Several of the members of the operating management group were quite eager for the LP model to be completed quickly, since they believed that other refineries had for several years gained some advantage through the adoption of linear programming and other mathematical techniques.
Little effort had been devoted to this type of project at Tascosa in the past because of the history of the refinery. The refinery had been owned until 1955 by the Palo Duro Oil Company, a small integrated oil company which operated in several southwestern and Rocky Mountain states.
In 1955, which was about the time that the use of linear programming in refinery operations was becoming widespread, Palo Duro was purchased by the Caprock Gas Company and made an operating subsidiary. Almost immediately afterward began a period of great change and expansion at the Tascosa Refinery.
By 1961, when the name of the subsidiary was changed to the Caprock Oil Company, the Tascosa Refinery had nearly tripled its crude refining capacity, and had added a unit for producing high purity aromatic petrochemicals. During this period, the engineering staff had been so concerned with the multitude of problems brought about by rapid expansion that very little effort was directed toward the topic of economic optimization.
By late 1962, however, the refinery had become one of the most modern in the industry and had proved itself capable of producing an output that was often greater than its demand for product at existing prices.
At this time the Process Economics Department was created as a subgroup of the process engineering section and began to emphasize economic optimization of operations.
The Gasoline Blending Program
One of the first projects the group at Tascosa undertook was to develop a gasoline blending program, in the linear programming format, to be run on the refinery's digital computer. The problem was to determine what mixtures or blends of various gasoline blending stocks produced in the refinery should be utilized to provide Caprock with the greatest profit.
Since the Caprock control system made the blending process a profit center, the problem was formulated in terms of the Blending Sections' profit based on inter-section transfer prices which were set by Caprock's Centralized Accounting Department.
The exact formulation also depended upon the demand existing for regular and premium gasoline and the relationship of this demand to the refinery's productive capacity for the components available for blending
Though the gasoline products were blended to meet a large number of specifications such as vapor pressure, sulfur content, gum content, and various octane numbers, the most important specifications for motor gasoline were the octane number and the vapor pressure. Premium gasoline has, in general, a higher octane rating and a lower vapor pressure than regular.
When Henderson began the job he decided that he should first formulate a greatly simplified model much smaller than that which would finally be used in order to allow himself to get a "feel" of the problem.
He felt it would be useful to do this before spending the several weeks' time which would be needed to develop equations representing the various processing units. This approach might allow him to test his ability at problem formulation and possibly save the time which would be wasted if his first formulation proved to be a false start.
He knew that premium and regular gasoline were the two basic products of the blending process. Furthermore, the 99-octane premium (vapor pressure: 6 “pounds per square inch,” or 6 “psi”), which was selling at $4.83/barrel, was considered more profitable than the lower octane regular (vapor pressure: 8 psi).
On the other hand, he was aware that recently the demand for regular had been such that Tascosa could sell at $4.40/barrel all the 92-octane regular they could produce while premium sales had rarely gone much above a rate of 5000 “barrels per day” (hereafter “B/D”).
The refinery produced several components which were blended to yield the company's gasoline. The most important were: (i) reformate (98 octane, 6 psi) from the catalytic reformer; (ii) naphtha (76 octane, 8 psi) straight from the crude processing unit; (iii) raffinate (79 octane, 16 psi), waste from the petrochemical units; (iv) gasoline produced in the catalytic cracking unit, and called cat-cracked or catalytic gasoline (99 octane, 5 psi); and (v) high octane alkylate (103 octane, 4 psi).
Henderson assumed for his simplified problem that the octane number and the vapor pressure of the blended gasoline would be the weighted average of the octane and vapor pressure ratings of the volumetric proportion of the constituents in the blend, e.g., a blend of 50 barrels of alkylate (103 octane, 4 psi) and 50 barrels of catalytic (99 octane, 5 psi) would yield 100 barrels of gasoline with an octane rating of 101 and a vapor pressure of 4. 5 psi.
The blending stocks, their octane ratings, quantities to be used, and variable costs are presented in Exhibit 1.
After the blending program had been developed and debugged he obtained the solution to the problem shown by the computer run (Exhibit 2). This showed a profit of $8,195, utilizing the components shown in Exhibit 1.
He also developed the contribution per barrel for premium and regular gasoline as shown on Exhibit 3.
Henderson was confident that the computer solution was indeed the optimum obtainable under the constraints given, but he was seeking a way to explain his results at a meeting of the operating committee which was scheduled to review the operations of the Process Economics Department.
He was also concerned about a problem which had bothered the Blending Section's manager for some time. The section manager had often complained that he could show a much better profit if he were not forced to use all the blending stocks supplied by the refinery. Henderson was under quite some pressure to use his results to fortify the blending manager's case.
Exhibit 1, Tascosa Refinery (A)
Blending Stock Octane Vapor Pressure Availability Accounting
No. (psi) B/D Price $/B
Reformate 98 6 4800 4.29
Naphtha 76 8 1000 3.07
Raffinate 79 16 4200 3.24
Catalytic 99 5 6800 4.34
Alkylate 103 4 2700 4.57
ANSWER :
LINEAR PROGRAMMING FORMULATION:
Step 1 (define the decision variables):
Define: REFR as the # of B/D of Reformate (REFormate) used to blend Regular (Regular) gasoline,
and REFP as the # of B/D of Reformate used to blend Premium gasoline.
Should there be other decision variables? Please try to define them. I will do this in class. Token points will be given to class members who can help me to do this in class.
Hints: How many decision variables should there be? Can you say what these decision variables represent using one simple English sentence?
Step 2a (state the objective function):
Max or Min (?). Z = ¬???REFR + …. (can you finish this?)
I will do this in class (hopefully with your help).
Step 2b (state the constraints): I will do this in class (hopefully with your help).
s.t. (subject to)
1. REFP + ???????????????????????? 5000 (Premium Market)
2. ??
3. ??
4. ??
5. ??
6. ??
The following 4 constraints are probably too difficult for some of you:
7. -6REFR + 16NAPR + 13RAFR - 7CATR - 11ALKR 0 (ON,Regular)
8. REFP + 23NAPP + 20RAFP - 4ALKP 0 (ON, Premium)
9. -2REFR + 8RAFR - 3CATR – 4ALKR 0 (VP, Regular)
10. 2NAPP + 10RAFP - CATP - 2ALKP 0 (VP, Premium)
Note: There are altogether 10 decision variables and 10 constraints.
LINEAR PROGRAMMING SOLUTION. The optimal solution is:
Z* = $8194.75;
REFR*=4500, NAPR*=1000, etc. (will finish this in class)
Can you say the above in plain English (e.g., does “REFR*=4500” mean “the optimal price of raffinate is $4500”?)
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