The IRS has determined that during each of the next 12 months it will need the n
ID: 3143271 • Letter: T
Question
The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in Table 1. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $1,000 to rent a supercomputer for one month, $1, 800 for two months, and $2, 500 for three months. At the beginning of month 1, the IRS has no supercomputers. 1. Formulate an LP to determine the rental plan that meets the next 12 months requirements at minimum total cost 2. Give the dual of the LP in item 1.Explanation / Answer
(1)
Let JAN1 number of computers rented at beginning of January for one month, and so on. Also define IJAN number of computers available to meet January demand, and so on. The appropriate LP is
min z = 1000(JAN1 + FEB1 + MAR1 + APR1 + MAY1 + JUN1+ JULY1+AUG1+SEP1+OCT1+NOV1+DEC1) + 1800(JAN2 + FEB2 + MAR2 + APR2 + MAY2 + JUN2+ JULY2+AUG2+SEP2+OCT2+NOV2+DEC2) + 2500(JAN3 + FEB3 + MAR3 + APR3 + MAY3 + JUN3+ JULY3+AUG3+SEP3+OCT3+NOV3+DEC3) – 833.33 NOV3 - 1666.67 DEC3 - 900DEC2
s.t.
IJAN = JAN1 + JAN2 + JAN3
IFEB = IJAN - JAN1 + FEB1 + FEB2 + FEB3
IMAR = IFEB - JAN2 - FEB1 + MAR1+ MAR2 + MAR3
IAPR = IMAR - JAN3 - FEB2 - MAR1 + APR1 + APR2 + APR3
IMAY = IAPR - FEB3 - MAR2 - APR1+ MAY1 + MAY2 + MAY3
IJUN = IMAY - MAR3 - APR2 - MAY1+ JUN1 + JUN2 + JUN3
IJUL = IJUN - APR3 - MAY2 - JUN1+ JUL1 + JUL2 + JUL3
IAUG = IJUL - MAY3 - JUN2 - JUL1+ AUG1 + AUG2 + AUG3
ISEP = IAUG - JUN3 - JUL2 - AUG1+ SEP1 + SEP2 + SEP3
IOCT = ISEP - JUL3 - AUG2 - SEP1+ OCT1 + OCT2 + OCT3
INOV = IOCT - AUG3 - SEP2 - OCT1+ NOV1 + NOV2 + NOV3
IDEC = INOV - SEP3 - OCT2 - NOV1+ DEC1 + DEC2 + DEC3
IJAN 800
IFEB 1000
IMAR 500
IAPR 600
IMAY 1200
IJUN 400
IJUL 800
IAUG 600
ISEP 400
IOCT 500
INOV 800
IDEC 600
All variables 0
(2)
Thus, Dual program of above LP is
max z = 800(1JAN) + 1000(1FEB) + 500(1MAR) + 600(1APR) + 1200(1MAY) + 400(1JUN) + 800(1JUL) + 600(1AUG) + 400(1SEP) + 500(1OCT) + 800(1NOV) + 600(1DEC)
s.t.
JAN1 + FEB1 + MAR1 + APR1 + MAY1 + JUN1+ JULY1+AUG1+SEP1+OCT1+NOV1+DEC1 1000
JAN2 + FEB2 + MAR2 + APR2 + MAY2 + JUN2+ JULY2+AUG2+SEP2+OCT2+NOV2+DEC2 1800
JAN3 + FEB3 + MAR3 + APR3 + MAY3 + JUN3+ JULY3+AUG3+SEP3+OCT3+NOV3+DEC3 2500
All variables 0
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