Quantitative Management Practice Assignment

Part I

Introduction

AMARCO, Inc was formed in 1937 as joint venture between Standard Oil Company of California and Government of Saudi Arabia to explore, produce and market any petroleum found in the country.  Since 1950, the role of company has greatly increased by involving in exploration, drilling and refining production activities. As a result company introduce a Operation Research Department to review the firm’s operations to find other areas where costs might be decreased or profits increased by application of operation research. Now, main focus of the company is to identify the most favorable situation of how to produce the maximum aviation gasoline i.e. A, B & C which are made by blending of four feedstocks, through Operational Research Department to maximize the profits (David, 1982). Now, company is doing a operation research by using Linear Programming model to get the maximum results in the different business situations.

Modeling

Linear programming is a technique which shows practical problems as a series of mathematical equations which can be manipulated to find the optimum or best solution (Adams, 1969). Blending is a graphical approach to linear programming which deals with resource allocation subject to constraints. It is a model which assists firms in deciding the best possible utilization of limited resources. Each resource constraint is represented as a mathematical linear equation. A linear expression is an equation which links two variables, and if plotted on a graph, would be represented by a straight line. By plotting all the equations, the optimal use of the business's resources can be easily identified.

Blending can be useful to firms when deciding how to make the best use of their resources. Businesses can use this method to allocate factors of production so that profits are maximized or costs minimized, depending on the business's objective. Another advantage of blending is that is allows the business to decide a combination of the two goods to produce, as compared to other invest appraisal or decision making techniques where either one or the other option must be selected, but not both (Anderson et al 2002).

In oil refining business, production is done on the basis of certain product mix (Andrade, 1990).  Standardization of product mix is complex task and which is required a lot of analysis and research.  AMARCO Inc has already developed the Linear Programming Models for their organization and getting the better results and as a result company has increased their profits and production (Dantzig, 1963). Here, blending method is used which is a part of LP model.  Mainly, blending problems are identified in gasoline operations.

Examples of blending problems:

 Input Output Meat, Filler, Water Different type of sausage Various type of oil Heating oil, gasolines, aviation fuels

Data Description

In the present situation, AMARCO, Inc has to produce the aviation gasoline, A, B & C which are made by blending four feedstock; Alkylate, Catalytic Cracked Gasoline, Straight Run Gasoline and Isopentane.

Data is classified into two parts i.e. Input Data & Output Data

• Input Data – Four Feedstock i.e. Alkylate / Catalytic Cracked Gasoline / Straight Run Gasoline / Isopentane
• Output – Three types of Aviation Gasoline i.e Gas A, Gas B & Gas C

Results

By using the Blending model, management know about the best combination of production of resources available to get the best outcome and most profitability.  Various type of situation has been applied to get the better result in the different situation with the help of LP Models.  As a result, management get to know which situation will be the best for the company.  In the standard model of LP, the revenue of company was \$ 816200.  As per the situation of Question No. 3, the revenue of company was increased from \$ 816200 to \$ 1632400.  In another situation, as per question 4 with the situation of Question 3, variable has been re-determined to align the requirement with availability and also because of reduction of stock level, revenue was slightly down from \$1632400 to \$1339523.  As per the situation of Question No. 7, the revenue of the company as drastically down and that is because of rate impact.  Stock level should be reduced while we enter in the downward market.  As per the situation of question no. 8, if there is not criteria of Octane level than company will first liquidate the stock of lower value in the downwards market and keep the stock of high value in hand so that the profit of the company will not be reduced.  As per situation of question no. 9, other factors also derive the profit / revenue of the business.  As the Reid Vapor pressure is reduced from 7 to 6, we need to change our variable mix, to achieve the desired level of result of environment.  As per question no. 10, as per market demand of the respective product lead to put your business into profit and loss.  As per this situation, if the order is for less profitable gases than simultaneously, product mix will also be changed to get the better result because market share is also a important factor and not ignorable.

Part – II

 Adjustable Cells Cell Name Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease \$C\$6 Values of Variables X11 2,175.34 - 15.00 0.24 - \$D\$6 Values of Variables X12 - (0.00) 16.00 0.00 1.00 \$E\$6 Values of Variables X13 5,424.66 - 16.50 - 1.00 \$F\$6 Values of Variables X21 - (0.00) 15.00 0.00 1.00 \$G\$6 Values of Variables X22 6,780.82 - 16.00 - 0.00 \$H\$6 Values of Variables X23 6,219.18 - 16.50 0.19 - \$I\$6 Values of Variables X31 8,523.12 - 15.00 - 0.00 \$J\$6 Values of Variables X32 5,476.88 - 16.00 0.00 - \$K\$6 Values of Variables X33 - (0.00) 16.50 0.00 1.00 \$L\$6 Values of Variables X41 2,301.54 - 15.00 - 0.00 \$M\$6 Values of Variables X42 742.29 - 16.00 0.00 - \$N\$6 Values of Variables X43 356.16 - 16.50 - 3.02 \$O\$6 Values of Variables X1 6,400.00 - 17.00 34.91 0.12 \$P\$6 Values of Variables X2 - (1.90) 14.50 1.90 1.00 \$Q\$6 Values of Variables X3 - (3.70) 13.50 3.70 1.00 \$R\$6 Values of Variables X4 7,600.00 - 14.00 3.00 0.75 Constraints Cell Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease \$S\$10 Gas A LHS 0.00 0.20 - 32,630.14 34,523.12 \$S\$11 Gas B LHS (0.00) 0.20 - 11,108.40 3,255.71 \$S\$12 Gas C LHS - 0.20 - 79,200.00 5,379.31 \$S\$14 Alkalyte LHS 14,000.00 17.00 14,000.00 1.00 6,400.00 \$S\$15 Cat Crack Gas LHS 13,000.00 16.40 13,000.00 925.70 5,697.49 \$S\$16 Str Run Gas LHS 14,000.00 17.20 14,000.00 694.27 6,000.00 \$S\$17 Isopentane LHS 11,000.00 14.00 11,000.00 1.00 7,600.00 \$S\$19 Gas A LHS 13,000.00 - 12,000.00 1,000.00 1.00 \$S\$20 Gas B LHS 13,000.00 (2.20) 13,000.00 3,692.31 456.30 \$S\$21 Gas C LHS 12,000.00 (0.10) 12,000.00 7,384.62 1,786.15 \$S\$22 Marketing LHS (0.00) (1.60) - 7,384.62 979.19 \$S\$24 Gas A LHS 13,839.21 - - 13,839.21 1.00 \$S\$25 Gas B LHS 4,273.12 - - 4,273.12 1.00 \$S\$26 Gas C LHS (0.00) - - 13,515.22 26,000.00

As per above LP spreadsheet model, maximum values of the different criteria are considered as constraints and resources is considered as Adjustment cells.  As product mix impact the profitability of the business but while allocating the mix, constraints should also be taken care as if constraints are not followed than the quality of the product will not be adequate.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 2175 0 5425 0 6781 6219 8523 5477 0 2302 742 356 6400 0 0 7600 Objective function 15 16 17 15 16 17 15 16 17 15 16 17 17 14.5 14 14 816200 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 13839 ≥ 0 Gas B 17 2 -4 17 4273 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

As per the above spreadsheet, total revenue of the company will be \$ 816200.  This revenue is as per current mix plan.  The every constraint is as per required level.  So, by above LP spreadsheet, it is a good production mix.

There are two situation of this question.

Situation No. 1-

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 0 4844 9156 9750 3250 0 3250 3583 7167 0 1322 3062 0 0 0 6615 Objective function 30 32 33 30 32 33 30 32 33 30 32 33 17 14.5 14 14 1538308 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 19385 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 9750 ≥ 0 Gas B 17 2 -4 17 94577 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

As per above spreadsheet, a product mix has changed because of this profit has been increased from \$816200 to \$1538308.  So, in the demanding market, profits are always seen in the organisation because organisation got the profit on the material which was produced from stock.  In the above example also, revenue has been increased because the sold material was produced with low cost.

Situation No. 2-

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 2175 0 5425 0 6781 6219 8523 5477 0 2302 742 356 6400 0 0 7600 Objective function 30 32 33 30 32 33 30 32 33 30 32 33 34 29.0 27 28 1632400 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 13839 ≥ 0 Gas B 17 2 -4 17 4273 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

Total revenue is increased from will be increased \$816200 to \$1632400.  But profit has not been increased as per revenue increased because due to increase in price, cost of production has also been increased.  Product mix has not been changed in the above spreadsheet.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 6775 0 5425 0 4781 6219 3117 5883 0 1762 990 356 0 0 0 7892 Objective function 30 32 33 30 32 33 30 32 33 30 32 33 34 29.0 27 28 1339523 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 12200 ? 14000 Cat Crack Gas 1 1 1 1 11000 ? 13000 Str Run Gas 1 1 1 1 9000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 11654 ≥ 12000 Gas B 1 1 1 1 11654 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 102578 ≥ 0 Gas B 17 2 -4 17 2857 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

As per question no. 4, availability of raw material has been decreased, but the prices are considered as per question no. 3, so revenue has been decreased because per day stock has been reduced.  As per above product mix revenue has been decreased from \$1632400 to \$1339523.

As per the situation mentioned in question no. 4, the availability ration of raw material has been decreased.  Cost of processing will be increased because the availability of Straight Run Gasoline is decreased @ 35% which is almost double the rest other products.  And the cost of Straight Run Gasoline is cheaper than other products.  So, increasing the production in the present situation will lead to increase the cost of production.  So, profit will be decreased.  So, in this situation, product increase should be on the cost of increase the price of Gases also.

As per current standard production mix with current demand, company is earning around 8% profit.  The declining and growing markets affect the profit of the company.  Increase in demand of one product and decline in demand of another product cal also lead to change the profit ration of the company.  The entire LP study is based on production mix.

It is sensitive analysis that how change in market demand will impact the profit of company.  20% of increase or 20% decrease will change the profit of the company.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 2175 0 5425 0 6781 6219 8523 5477 0 2302 742 356 6400 0 0 7600 Objective function 10 11 12 10 11 12 10 11 12 10 11 12 17 14.5 14 14 626200 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 13839 ≥ 0 Gas B 17 2 -4 17 4273 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

As per above spreadsheet, suddenly drop in aviation gasoline price lead to loss to company because stock in hand was bought by company at the higher price.  Now, It has not been told about the prices impact Alkylate / Catalytic Cracked Gasoline / Straight Run Gasoline / Isopentane.  If the price of raw material has also been decreased than it’s a one time loss, but if the prices of raw material has not been decreased than company needs to seriously think about the continuation of business and needs to take the hard step regarding continuation of business.  Higher raw material cost could not be compensated with any method or technique.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 0 0 7600 3250 9750 0 8125 3250 2625 1625 0 1775 6400 0 0 7600 Objective function 10 11 12 10 11 12 10 11 12 10 11 12 17 14.5 14 14 626200 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0

As per above spreadsheet, it is clear that there is no revenue impact because of octane constraint.  Octane level acceptance was so high in the product mix because of that we have found any revenue impact.  So, from AMARCO’s point, it is not impacting in the current production mix.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 3133 684 6316 2250 6329 4421 6750 5987 1263 867 0 0 3867 0 0 10133 Objective function 15 16 17 15 16 17 15 16 17 15 16 17 17 14.5 14 14 808600 Constraints Reid Vapor Pressure Gas A -1 2 -2 14 0 ≤ 0 Gas B -1 2 -2 14 0 ≤ 0 Gas C -1 2 -2 14 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 23117 ≥ 0 Gas B 17 2 -4 17 0 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0

As per above spreadsheet, it is clear because of change in Reid Vapor pressure, product mix is also required to change.  Due to change product mix, revenue is also been decreased.  It is a loss due to product mix which were changed due to change in Reid Vapor Pressure norm from 7 to 6.  Change in constraints lead to change in adjustment cell.

 Particulars Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane LHS Sign RHS blended in gasoline blended in gasoline blended in gasoline blended in gasoline Feedstock A B C A B C A B C A B C not blended in gasoline Variables X11 X12 X13 X21 X22 X23 X31 X32 X33 X41 X42 X43 X1 X2 X3 X4 Values of Variables 2175 0 5425 0 6781 6219 8523 5477 0 2302 742 356 6400 0 0 7600 Objective function 15 16 17 15 16 17 15 16 17 15 16 17 17 14.5 14 14 816200 Constraints Reid Vapor Pressure Gas A -2 1 -3 13 0 ≤ 0 Gas B -2 1 -3 13 0 ≤ 0 Gas C -2 1 -3 13 0 ≤ 0 Availability Alkalyte 1 1 1 1 14000 ? 14000 Cat Crack Gas 1 1 1 1 13000 ? 13000 Str Run Gas 1 1 1 1 14000 ? 14000 Isopentane 1 1 1 1 11000 ? 11000 Demand Gas A 1 1 1 1 13000 ≥ 12000 Gas B 1 1 1 1 13000 ≥ 13000 Gas C 1 1 1 1 12000 ≥ 12000 Marketing 1 -1 1 -1 1 -1 1 -1 0 ≥ 0 Octane Gas A 14 3 -6 15 13839 ≥ 0 Gas B 17 2 -4 17 4273 ≥ 0 Gas C 8 -7 -13 8 0 ≥ 0 New MKT constraint 1 1 -1 1 1 -1 1 1 -1 1 1 -1 14000 ≥ 0

As per above spreadsheet, the current marketing mix generate around 8% of profit.  But as per the price difference between Gas A, Gas B & Gas C it is around 9%, so increase in market share with the new contract will always a burden on the company with the current product mix. Further, as per above product mix, it is not understandable about the fix and variable cost of the cost.  As per above spreadsheet, it is assumed as variable cost.  S o, in my view, additional order at the cost of loss in the current product mix is not advisable.

References

• Adams, W. J. (1969). Elements of linear programming. Van Nostrand Reinhold Publishing Company International.
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• Dantzig, G. B. (1963). Linear programming and extension. Priceton University Press.
• David, C.H. (1982). Introduction concept and application of operation research