Backwards-integrate

Trial values of the effluent yield, fj, are assumed until one is found that results in satisfaction of the condition at z = 0 by backward integration. 

Competition from backward-integrated customers who have a major share of capability and capacity within the industry... 

As the leading specialty chemical companies, Degussa, Dow, Chemtura and Rohm Haas, are backward-integrated, the share of in-house production is estimated at 75-80%, leaving a merchant market of approximately 5 billion (see Table 9.2). 

Food and feed additives, also known as dietary supplements, are minor ingredients added to improve the product quality. Most commonly, the effects desired relate to color, flavor, nutritive value, taste, or stability in storage. The market sizes are estimated to be 20 billion each for food and for feed additives, respectively. The major customers for the food additives are the big food companies Ajinomoto, Danone, Kraft, and Nestle, mentioned at the beginning of the chapter. With the exception of Ajinomoto, these companies are rarely backward-integrated. As they prefer to use natural ingredients rather than synthetic ones, they are not very important customers of the fine-chemical industry. Premixers, that is, enterprises that prepare ready-to-use mixtures of nutrients for the farmers who raise cattle, pigs, and chicken, are the main users of feed additives. 

Type Volume (tons) Unit Price ( /kg) Turnover ( million) Application/ Backward Integration of Producers... 

Fluorinated ethylene/propylene 30,000 60 1,800 Wire and cable insulation backwards integrated... 

Nonmelting polyimides 8,000 30 240 Aerospace and rigid electronic parts partly backward-integrated... 

Benzocyclobutene polymers 5 100 0.5 Wafer coatings producer is backward-integrated... 

However, in order to achieve these objectives a good interpretation of corporate business needs into achievable R D objectives is essential. In some cases a supplier of a raw material, intermediate or component may choose to forward-integrate their business into production of the finished product. In other cases a manufacturer may wish to backwards-integrate into the production of the raw material(s) required for its current product(s). In either case it is obvious that in order to achieve success a company must make products available to its customers with useful functional properties and at a price that allows cost-effective use of that product in the customer s own products. That is, the product must give the customer s products a competitive advantage. 

Thus due to potential stiffness, integrating a BVP such as the one in equation (5.37) in the positive uj direction from 0 to 1 may not be wise in all cases. In fact, backward integration is much more stable for our simplified model since in backward integration, the eigenvalues switch signs and then the problem is no longer stiff according to the definition. 

Actually a theoretical proof of whether a given DE is stiff or not is rather complicated if not impossible for most DE problems. Therefore it is best to use numerical codes that switch internally to backward integration if forward integration encounters troubles with numerical convergence and vice versa, giving us the best method for either situation. 

Our next task is to rewrite both, the second-order two-DE systems (5.20), (5.19) used in forward integration and the DE system (5.38), (5.39) that is used for backwards integration, in the form of a first-order system of four DEs each. These have the form u = /( ) or v = g(v), respectively with u, v 6 R4. We refer to Section 1.2.2 for the formal reduction of high-order DEs to first-order enlarged systems of DEs. 

THIS version uses backward integration, best in case of "/. Stiffness of the DE. 

To reduce their sourcing risks, European petrochemical companies may thus benefit from some degree of backward integration into olefines. Players with limited olefine production capacity are probably in the best possible position they can secure supplies while avoiding the temptation to undercut prices of downstream products during downturns. Players with an excess of olefines may be exposed to this temptation in order to maximize the capacity utilization of their olefine production, where fixed costs tend to be high. 

A backward integration that calculates what downside potential would cause the project not to be approved. The likelihood of that downside occurring can then be examined. 

INDICATES RESULTS OF BACKWARDS INTEGRATION OF DIFFUSION EQUATION... 

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