ROTHSCHILD and BALSIGER: LINEAR-PROGRAMMING SOLUTION 



effective. If this low-valued fish is released on 

 a day on which the cannery constraint was bind- 

 ing, then the cannery constraint is no longer 

 binding and a fish of another entity can be 

 caught on that day if a fish of that other entity 

 is available. For example, 



f(SPii = tij - Clio + C.I. 10 - C'3 11 



= $1,362 - $1,270 + $1,371 - $1,356 

 = $0,107, 



which is the value shown in Figure 9. The run 

 shadow prices for entities 2 to 4 on days 1 to 3 

 are simply calculated as the value of the entity 

 on that day minus the value of that entity on 

 the lowest priced, and hence, last day on which 

 it is included in the catch scheme. For example, 



RSP}j = C3 I - Oil 



= $1,453 - $1,356 



= $0,097. 



Checking Figure 5, we see that day 11 was the 

 last day or which entity 3 fish were caught, and 

 Figure 9 shows that the value ($0,097) is 

 correct. 



Run constraints for entity 3 are not binding 

 after day 3, so run shadow prices associative are 

 zero. 



To obtain the run shadow price for entity 1 

 for days 4 to 6, one may, for example, subti-act 

 from the value of entity 1 day 4, the value of 

 entity 3 day 4 (which must be released to main- 

 tain the daily cannery constraints) , subtract the 

 value of entity 1 day 10 (which must be released 

 to maintain the season entity 1 limit), and add 

 the value of entity 3 day 10 (which can be 

 caught since an entity 1 has been released on 

 day 10); or 



RSP\,4.(, = C\4 - Ct,4 - f I 10 + C3,io 



= $1,353 - $1,445 - $1,270 + $1,371 

 = $0,009, 



which can be seen in Figure 9. 



Run shadow prices for entities 2 and 4 for 

 days 4 to 6 are similar. For illustration, entity 



4 day 4 will be calculated: 



RSP44 = ('4 4 - £-3,4 - c-4 13 + f3,ll- 



That is, the run shadow price of entity 4 day 

 4 equals the value of entity 4 day 4 less the value 

 of entity 3 day 4 (to preserve the daily cannery 

 constraint) less the value of entity 4 day 13 (to 

 preserve the seasonal entity 4 limit) plus the 

 value of entity 3 day 11 (since an entity 3 day 

 4 was excluded, this will be within the entity 3 

 seasonal limit constraint) . Thus 



RSP4_4 = $1,930 - $1,445 - $1,780 + $1,356 

 = $0,061. 



This is the- value shown in Figure 9. 



The run shadow prices for entity 1 after day 

 7 are zero since the daily run constraints are no 

 longer binding. For entities 2 and 4 days 7 to 

 10, the run shadow prices can be calculated sim- 

 ilarly. For example, 



RSP 



2,10 



C2.\0 - C3.10 - Q.I I + O.ll. 



That is, the run shadow price of entity 2 day 

 10 equals the value of entity 2 day 10 less the val- 

 ue of entity 3 day 10 (to preserve daily cannery 

 constraints) less the value of entity 2 day 11 (the 

 last day on which entity 2's are caught and, 

 hence, the cheapest entity 2 which can be re- 

 leased to preserve the seasonal limit) plus the 

 value of entity 3 day 11 (since an entity 3 was 

 released on day 10, this will not fracture the 

 seasonal entity 3 limit constraint). Thus 



RSP2.\o = $1,836 - $1,371 - $1,812 + $1,356 

 = $0,009, 



which is shown as the correct value in Figure 9. 

 The run shadow prices for entity 4 on days 11 

 and 12 are easily calculated since the cannery 

 constraints are no longer binding and this is 

 the only entity being caught after day 12. Hence, 



RSP4,l\ = C4J1 - C4,|3 



= $1,808 - $1,780 

 = $0,028. 



133 



