Abstract
This case study demonstrates the value of classical analysis and to a lesser degree, system decomposition for finding a global optimum missed by a sequential linear programming scheme which converges to a non-global local minimum. The example is a 20 variable steelmaking problem in which the variable annual cost to be minimized is linear, as are all constraints except a non-convex one in each blast furnace. The sequential linear programming method gives a provenlocal minimum, although the non-convex nonlinearity prevents any proof of global optimality. The provenglobal minimum found here has a 4% lower cost. The local minimum costs only 0.2% per annum less than the rather flat global maximum, so the original local minimization only achieved about 5% of the economy possible. In the overall plant, the cost saving is over three million US$ (1972) annually.
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Abbreviations
- H :
-
combined hot iron rate (X 4 + X 4)
- r 1,r 2 :
-
sinter/iron ratio for BF1 and BF2
- u 1,u 2 :
-
unit composition cost variation for BF1 and BF2
- v 1,v 2 :
-
variable feed cost for BF1 and BF2
- v b(H):
-
total variable feed cost for both blast furnaces
- v s (H):
-
total variable cost for both steel furnaces
- x 1 :
-
sintered iron ore rate into BF1
- x 2 :
-
pelleted iron ore rate into BF1
- x 3 :
-
coke rate into BF1
- X 4 :
-
hot iron rate from BF1
- x 5 :
-
sintered iron ore rate into BF2
- x 6 :
-
pelleted iron ore rate into BF2
- x 7 :
-
coke rate into BF2
- X 8 :
-
hot iron rate from BF2
- x 9 :
-
hot iron to basic oxygen furnace (BOF)
- x 10 :
-
home scrap to BOF
- x 11 :
-
bought scrap to BOF
- x 12 :
-
silicon carbide to BOF
- x 13 :
-
crude steel from BOF
- x 14 :
-
home scrap to open-hearth furnace (OH)
- x 15 :
-
bought scrap to OH
- x 16 :
-
hot iron to OH
- x 17 :
-
crude steel from OH
- x 18 :
-
total crude steel
- x 19 :
-
total home scrap
- x 20 :
-
total bought scrap
- Y 1,Y 2 :
-
additional hot iron from BF1 and BF2
References
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Papalambros, P. and D. J. Wilde (1988),Principles of Optimal Design, Cambridge University Press, New York.
Ray, W. H. and J. Szekeley (1973),Process Optimization, Wiley-Interscience, New York, pp. 299–310.
Wilde, D. J. and C. S. Beightler (1967),Foundations of Optimization, Prentice-Hall, Englewood Cliffs, NJ, pp. 398–406.
Wilde, D. J. (1978),Globally Optimal Design, Wiley-Interscience, New York.
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Wilde, D.J. Convexity analysis in detecting a steel plant hidden global optimum. J Glob Optim 3, 117–131 (1993). https://doi.org/10.1007/BF01096733
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DOI: https://doi.org/10.1007/BF01096733