The northern regions of Italy are the heartlands of the Italian textile industry, providing textile products for many great Italian (and non-Italian) fashion houses. Most of today’s companies were founded at the turn of the century and have grown and thrived despite a series of twentieth century catastrophes, including the Great Depression and World War II. Sales in the entire Italian textile and apparel industry were about $80 billion in 2007. The Italian textile-fashion industry employs over 400,000 people.
Filatoi Riuniti is a family-owned spinning mill located in northwest Italy. They produce cotton yarn, which is one of the raw materials that is used to produce the fabrics that are then cut and sewn by Italian stylists into the worldwide famous Italian clothes.
Demand for Filatoi Riuniti’s production is strong, but their spinning machine capacity is insufficient to meet its production orders. They decided to outsource part of the spinning production to six local family-owned spinning mills: Ambrosi, Bresciani, Castri, De Blasi, Estensi, and Giuliani. The local mills charge higher prices for spinning finer yarns, so Filatoi Riuniti has decided to spin as much as possible of the finer yarns entirely in-house and to outsource only the spinning of low-end (coarser) yarns. Last month, they faced a total demand of 104,500 kg of cotton and they outsourced 32,000 kg of the low-end sizes.
Filatoi Riuniti hired a consulting firm to see if the outsourcing strategies to the six local mills could be improved. After analyzing the data, they immediately saw the potential for very large savings through more optimal outsourcing strategies. In this problem, we’ll improve the outsourcing strategy of Filatoi Riuniti using linear optimization. (Note: This problem is based off of a real case, but the names have been changed for privacy reasons.)
Solution Spreadsheet available at:
https://github.com/ssindw/MITAnalyticsEdge/blob/master/8LinearProgramming/FilatoiRiuniti.xlsx
One of the earliest and most successful applications of linear optimization was in the oil industry in the 1950s. According to Bill Drew, the former manager of research for Exxon Mobil, the company used linear optimization to “schedule our tanker fleets, design port facilities, blend gasoline, create financial models, you name it.” In this problem, we’ll use optimization to blend gasoline. (While the application of this problem is real, the data that we will be using here has been created for this problem.)
Gasoline blending occurs in oil refineries, where crude oil is processed and refined into more useful products, such as gasoline and diesel fuel. We will consider three products: super gasoline, regular gasoline, and diesel fuel. These can be made by mixing three different types of crude oil: crude 1, crude 2, and crude 3. Each product is distinguished by its octane rating, which measures the quality of the fuel, and its iron content, which is a contaminant in the gas. The crude oils each have an octane rating and iron content as well. The following table shows the required octane ratings and iron contents for each of the products, as well as the known octane ratings and iron contents of each of the crude oils:
Product or Oil Octane Rating Iron Content Super Gasoline at least 10 no more than 1 Regular Gasoline at least 8 no more than 2 Diesel Fuel at least 6 no more than 1 Crude 1 12 0.5 Crude 2 6 2.0 Crude 3 8 3.0 The gasoline produced must meet these standards for octane ratings and iron content. The octane rating and iron content of a product is the weighted average of the octane rating and iron content of the crude oils used to produce it. For example, if we produce regular gasoline using 20 barrels of Crude 1, 5 barrels of Crude 2, and 10 barrels of Crude 3, the Octane Rating of the regular gasoline would be:
(2012 + 56 + 10*8)/35 = 10
The numerator is the number of barrels of Crude 1 used times the octane rating of Crude 1, plus the number of barrels of Crude 2 used times the octane rating of Crude 2, plus the number of barrels of Crude 3 used times the octane rating of Crude 3. The denominator is the total number of barrels used.
Similarly, the iron content of the regular gasoline would be:
(200.5 + 52.0 + 10*3.0)/35 = 1.43
The objective of the oil company is to maximize profit. The following table gives the sales price (revenue) for one barrel of each of the products:
Product Sales Price Super Gasoline $70 Regular Gasoline $60 Diesel Fuel $50 And the following table gives the purchase price for one barrel of each of the crude oils:
Oil Purchase Price Crude 1 $45 Crude 2 $35 Crude 3 $25 We would like to maximize the amount made by selling the products, minus the amount it costs to buy the crude oils.
The company can only buy 5,000 barrels of each type of crude oil, and can process no more than 14,000 barrels total of crude oil. One barrel of crude oil makes one barrel of gasoline or fuel (nothing is lost in the conversion).
How many barrels of each type of crude oil should the company use to make each product?
Solution Spreadsheet available at:
https://github.com/ssindw/MITAnalyticsEdge/blob/master/8LinearProgramming/GasolineBlending.xlsx