By Frederick S. Hillier, Gerald J. Lieberman

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BLFß[ Ÿ %, BLFßJ Ÿ % BWGßQ Ÿ &, BWGßX ? Ÿ &, BWGß[ Ÿ &, BWGßJ Ÿ & BOWßQ Ÿ $, BOWß[ Ÿ $, BOWßX 2 Ÿ ) BR OßX 2 Ÿ ', BR OßJ Ÿ # #&ÐBOGßQ  BOGß[  BOGßJ Ñ  #'ÐBHLßX ?  BHLßX 2 Ñ  #%ÐBLFßQ  BLFßX ?  BLFß[  BLFßJ Ñ  #$ÐBWGßQ  BWGßX ? ÐBR OßX 2  BR OßJ Ñ BOGßQ  BOGß[  BOGßJ ) BHLßX ?  BHLßX 2 ) BLFßQ  BLFßX ?  BLFß[  BLFßJ ) BWGßQ  BWGßX ?  BWGß[  BWGßJ ) BOWßQ  BOWß[  BOWßX 2 ( BR OßX 2  BR OßJ ( BOGßQ  BLFßQ  BWGßQ  BOWßQ œ "% BHLßX ?  BLFßX ?  BWGßX ? œ "% BOGß[  BLFß[  BWGß[  BOWß[ œ "% BHLßX 2  BLFßX 2  BR OßX 2 œ "% BOGßJ  BLFßJ  BWGßJ  BR OßJ œ "% B34 !

The decision variables deal with the cost of transfers between plants, the cost of recipes, and carrying cost- all cost that are key to the common planning unit of tons" [p. 20]. The goal is to find the optimal grape juice quantities shipped to custom ers and transferred between plants over a 12-m onth horizon. , the sum of transportation, recipe and storage costs. They satisfy balance equations, bounds on the ratio of grape juice sold, and limits on total grape juice sold. The JLM resulted in significant savings by preventing unprofitable decisions of the management.

Then the problem can be formulated as follows: ma ximize T œ B"  #B# subject to B"  $B# Ÿ #!! #B"  #B# Ÿ $!! B# Ÿ '! B" ß B# ! 1-9. (a) Let B" be the number of units on special risk insurance and B# be the number of units on mortgages. maximize subject to D œ &B"  #B# $B"  #B# Ÿ #%!! B# Ÿ )!! #B" Ÿ "#!! , B# ! Ñ and ^ ‡ œ $'!! (c) The relevant two equations are $B"  #B# œ #%!! , so B" œ '!! and B# œ "# Ð#%!! D œ &B"  #B# œ $'!! 1-10. Þ"F Ÿ #!! Þ#&L Ÿ )!! $L  #F Ÿ "#ß !!! Lß F ! Ñ and T ‡ œ #*#!

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