# IE311 Operation Research

**FACULTY OF ENGINEERING**

**Internet Case Assignment: Agri-Chem Corporation**

**Case Assignment**

**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.

| |

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.

| |||

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 10
^{3}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 10^{3}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 Assignment: Agri Chem Corporation__

This case demonstrates an interesting use of LP in a production setting. Let X_{1} = tons of ammonia to produce, X_{2} = tons of ammonium phosphate to produce, X_{3} = tons of ammonium nitrate to produce, X_{4} = tons of urea to produce, X_{5} = tons of hydrofluoric acid to produce, X_{6} = tons of chlorine to produce, X_{7} = tons of caustic soda to produce, and X_{8} = tons of vinyl chloride monomer to produce.

Objective function: Maximize profit = 80X_{1} + 120X_{2} + 140X_{3} + 140X_{4} + 90X_{5} + 70X_{6} + 60X_{7} + 90X_{8}

Subject to:

X_{1} ≤ 1,200 (= 80% of 1,500) X_{2} ≤ 540 X_{3} ≤ 490

X_{4} ≤ 160 X_{5} ≤ 560 X_{6} ≤ 1,200

X_{7} ≤ 1,280 X_{8} ≤ 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: 8X_{1} + 10X_{2} + 12X_{3} + 12X_{4} + 7X_{5} + 18X_{6} + 20X_{7} + 14X_{7} ≤ 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: 8X_{1} + 10X_{2} + 12X_{3} + 12X_{4} + 7X_{5} + 18X_{6} + 20X_{7} + 14X_{7} ≤ 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.