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IE311 Operation Research

FACULTY OF ENGINEERING

Internet Case Study: Agri-Chem Corporation

Case Study

Operation Research IE 311

Introduction

In June 1980 the general manager of Texas Harry Sinclair received a notification from Ben Elliot that there is a shortage of natural gas in the Houston area due to the heat wave. The manager producer of the natural gas decided that in the case of a shortage in natural gas he will allocate gas to its customers under the following priorities:  

  • First Priority: Residential and commercial heating and cooling
  • Second Priority: Commercial and industrial firms that use natural gas as a source of raw material
  • Third Priority: Industrial firms that use natural gas as a boiler fuel

According to Ben Elliot Agri-Chem Corporation’s uses were in the second and third priority.

Therefore Enerco decided to enforce “rolling browouts” which is temporary and periodic curtailment of natural gas supplies.

Now we need to study and analyze the information and decide which complex should the company produces that will be less affected by a natural gas curtailment and decide what production level the company should use to maximize the profit of the company.

Discussion and Results

First of all we will identify the X's of the products:

Let:

X1: The amount of Ammonia produced

X2: The amount of Ammonium Phosphate produced

X3: The amount of Ammonium Nitrate produced

X4: The amount of Urea produced

X5: The amount of Hydrofluoric acid produced

X6: The amount of Chlorine produced

X7: The amount of Caustic soda produced

X8: The amount of Vinyl chloride monomer produced

The table below will show the contribution of the profit and overhead for each product which will help us in concluding the objective function.

TABLE 1 Contribution to profit and overhead

Product

$/Ton

Ammonia

80

Ammonium phosphate

120

Ammonium nitrate

140

Urea

140

Hydrofluoric acid

90

Chlorine

70

Caustic soda

60

Vinyl chloride monomer

90

For 20% natural gas curtailment:

Objective Function will be:

Maximize profit 80 +120 +140 +140 +90 +70 +60 +90

Before writing the constraints, we should take the percentage or capacity under consideration, as the table below shows the percentage % of the capacity of each product.

TABLE 2 Operational data

Product

Capacity (tons/day)

Production Rate (% of capacity

Natural Gas Consumption

(1,000 cu. ft./ton)

Ammonia

1,500

80

8.0

Ammonium phosphate

600

90

10.0

Ammonium nitrate

700

70

12.0

Urea

200

80

12.0

Hydrofluoric acid

800

70

7.0

Chlorine

1,500

80

18.0

Caustic soda

1,600

80

20.0

Vinyl chloride monomer

1,400

60

14.0

Calculations for of the Capacity:

  • Ammonia:
  • Ammonium phosphate:
  • Ammonium nitrate:
  • Urea:
  • Hydrofluoric acid:
  • Chlorine:
  • Caustic soda:
  • Vinyl chloride monomer:
  • The current natural gas usage is 85,680 cu. ft. x 103 per day. Enerco projects curtailments in the range of 20 to 40 percent. So, 20% from 85,680 = , then subtract it from 85,680= 68,544 cu. ft. x 103 per day.

Constraints:

  • ≤1200
  • ≤540
  • ≤490
  • ≤160
  • ≤560
  • ≤1200
  • ≤1280
  • ≤840
  • 8 +10 +12 +12 +7 +18 +20 +14 ≤ 68,544

The figures below show the results by using QM. First, we chose linear program then we chose 9 constraints and 8 variables, then we inserted the data into the table from QM.

Calculations for 20% natural gas curtailment:

Figure (1): Inserting the data in QM table.

Results for 20% natural gas curtailment:

After inserting the data, we pressed the Solve button to solve the linear program problem; the figure below shows the result of the case study.

Figure (2): The result in QM.

The optimal solution is :

  • =1200 ton/day
  • =540 ton/day
  • =490 ton/day
  • =160 ton/day
  • =560 ton/day
  • =1200 ton/day
  • =423.2 ton/day
  • =840 ton/day
  • 487,192 ton/day

We can see that the Caustic soda production is reduced from 1200 to 423.2 tons/day.

For 40% natural gas curtailment:

Objective Function will be:

Maximize profit 80 +120 +140 +140 +90 +70 +60 +90

Current natural gas usage = 85,680 cu. ft. *103/day.

40 percent curtailment = 51,408 cu. ft. * 103/day.

Constraints:

  • ≤1200
  • ≤540
  • ≤490
  • ≤160
  • ≤560
  • ≤1200
  • ≤1280
  • ≤840
  • 8 +10 +12 +12 +7 +18 +20 +14 ≤ 51,408

Calculations for 40% natural gas curtailment:

Figure (3): Inserting the data in QM table.

Results for 40% natural gas curtailment:

After inserting the data, we pressed the Solve button to solve the linear program problem; the figure below shows the result of the case study.

Figure (4): The result from QM.

The optimal solution is :

  • =1200 ton/day
  • =540 ton/day
  • =490 ton/day
  • =160 ton/day
  • =560 ton/day
  • =718.2222 ton/day
  • =0 ton/day
  • =840 ton/day
  • 428,075.6 ton/day

After solving the case study with linear program problem and QM software, we can now answer the question: "Which of Agri-Chem’s complexes would be least affected by a gas curtailment?"

It shows that the Caustic soda would be products that is affected by the gas curtailment. Since its production is zero. And the chlorine production is reduced from 1200 to 718.22 tons/day.

Conclusion

After reading and studying the case study, we analyzed the information and constructed the objective function and the constraints. The objective of the case study was maximize profit and by using QM software, we found the solution of the problem; the least product that is affected by gas curtailment, which is Caustic soda. By solving this case study we got many benefits, we learned how to convert the information in a case study to a linear problem. In addition, we learned how to use the QM software to find the solution of the problem, actually, QM software was very helpful, and it makes it very easy to find the solution rather than solving it manually.

LINEAR PROGRAMMING MODELS: GRAPHICAL AND COMPUTER METHODS

Internet Case Study: Agri Chem Corporation

This case demonstrates an interesting use of LP in a production setting. Let X1 = tons of ammonia to produce, X2 = tons of ammonium phosphate to produce, X3 = tons of ammonium nitrate to produce, X4 = tons of urea to produce, X5 = tons of hydrofluoric acid to produce, X6 = tons of chlorine to produce, X7 = tons of caustic soda to produce, and X8 = tons of vinyl chloride monomer to produce.

Objective function: Maximize profit = 80X1 + 120X2 + 140X3 + 140X4 + 90X5 + 70X6 + 60X7 + 90X8

Subject to:

X1 ≤ 1,200 (= 80% of 1,500)                 X2 ≤ 540                                  X3 ≤ 490

X4 ≤ 160                                              X5 ≤ 560                                  X6 ≤ 1,200

X7 ≤ 1,280                                           X8 ≤ 840                                  All variables ≥ 0

The current natural gas usage is 85,680,000 cu. ft. per day (= 1,200 × 8 + 540 × 10 + 490 × 12 + …).

A 20% curtailment implies availability is 0.8 × 85,680 = 68,554,000 cu. ft. per day. Hence, the gas constraint is: 8X1 + 10X2 + 12X3 + 12X4 + 7X5 + 18X6 + 20X7 + 14X7 ≤ 68,544

The solution is shown in file Ch02-Internet Case Agri Chem.XLS, sheet 20%. The production schedule (tons/day) is:

 

Ammonia

Amm. Phosphate

Amm. Nitrate

Urea

Hydro Acid

Chlorine

Caustic Soda

Vinyl Chloride

 

Tons/Day

1,200.0

540.0

490.0

160.0

560.0

1,200.0

423.2

840.0

 

Profit

$80

$120

$140

$140

$90

$70

$60

$90

$487,192.00

Because of the natural gas curtailment, the caustic soda production is reduced from 1,280 tons/day to only 423.2 tons/day.

A 40% curtailment implies availability is 0.6 × 85,680 = 51,408,000 cu. ft. per day. Hence, the gas constraint is: 8X1 + 10X2 + 12X3 + 12X4 + 7X5 + 18X6 + 20X7 + 14X7 ≤ 51,408

The solution is shown in file Ch02-Internet Case Agri Chem.XLS, sheet 40%. The production schedule (tons/day) is:

Ammonia

Amm Phosphate

Amm Nitrate

Urea

Hydro Acid

Chlorine

Caustic Soda

Vinyl Chloride

Tons/Day

1,200.00

540.00

490.00

160.00

560.00

718.22

0.00

840.00

Profit

$80

$120

$140

$140

$90

$70

$60

$90

$428,075.56

Because of the natural gas curtailment, the caustic soda production is eliminated completely and the chlorine production is reduced from 1,200 to 718.2 tons/day.

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