Introduction to mathematical programming
Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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Chapter 4, Problem 23RP

Explanation of Solution

Optimal solution:

  • Consider the LP problem of football match between teams Miami dolphins, the Buffalo bills, and the New York Jets.
  • Each team is given these rating:
  • M=Miami rating
  • J=Jets rating
  • B=bills rating
  • Games played between the teams in the year 1972 are given in the table below:
MiamiBillsJets
27 17
28 24
2423 
3016 
 2441
 341
  • From the game board when Miami plays Jets they win by 27-17=10 points in the first game and 28-24=4 points in second game. Hence total points by which Miami wins is 10+4=14.
  • According, to rating system Miami is expected to win by M-B points. Hence, prediction error is |Bj1|0.22. Hence we have,
    • |MB1|14
  • Our predicted value should be close to observed value. Hence, introduce deviation variable as follows:
    • si+= Amount by which we numerically exceed the predicted value.
    • si= Amount by which we are numerically under the predicted value.
  • So,
    • MB1+s1+s1=14
  • Similarly, for match between Miami and Bills, Miami wins by 24-23=1 point in first match and by 30-16=14 points is second match. Miami wins by total of 1+14=15 points. Hence, we have,
    • MJ1+s2+s2=15
  • In the match between Bills and Jets, Jets win by 41-24=17 points in first match and 41-3=38 points in second match.
  • Jet wins by a total of 17+38=55 points and therefore,
    • JB1+s3+s3=55
  • We have to minimize the total deviation. Hence the minimization equation is
  • Minimize z=s1+s1+s2+s2+s3+s3
  • Hence the following LP is formulated.
  • Minimize z=s1++s1+s2++s2+s3++s3
  • Such that
    • MB1+s1+s1=14
    • MJ1+s2+s2=15
    • JB1+s3+s3=55
  • Enter the following LP equations in the file:
    • min sp1+sm1+sp2+sm2+sp3+sm3
    • Subject to
    • mj+sp1sm1=15
    • mb+sp2sm2=16
    • bj+sp3sm3=56
    • End
  • The output is given below:
    • Global optimal solution found.
    • Objective value: 57.00000
    • Infeasibilities: 0.000000
    • Total solver iterations: 4
    • Elapsed runtime seconds: 1.47
    • Model class: LP
    • Total variables: 9
    • Non-linear variables: 0
    • Integer variables: 0
    • Total constraints: 4
    • Non-linear constraints: 0
    • Total non-zeroes: 18
    • Non-linear non-zeroes: 0
VariableValue
sp10.000000
sm10.000000
sp21.000000
sm20.000000
sp356.000000
sm30.000000
m15.000000
j0.000000
b0.000000
RowSlack or Surplus
157.00000
20.000000
30.000000
40.000000
  • Therefore, it can be concluded that the rating of these teams should be
    • M=15
    • J=0
    • B=0
  • From the given results it can be said that this rating system is not accurate and cannot be used to rate teams early in the season.
  • Since Miami plays Jets 27+17=44 times and wins 27 times. So probability of winning is 27/44

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Chapter 4 Solutions

Introduction to mathematical programming

Ch. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.10 - Prob. 1PCh. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Prob. 4PCh. 4.10 - Prob. 5PCh. 4.11 - Prob. 1PCh. 4.11 - Prob. 2PCh. 4.11 - Prob. 3PCh. 4.11 - Prob. 4PCh. 4.11 - Prob. 5PCh. 4.11 - Prob. 6PCh. 4.12 - Prob. 1PCh. 4.12 - Prob. 2PCh. 4.12 - Prob. 3PCh. 4.12 - Prob. 4PCh. 4.12 - Prob. 5PCh. 4.12 - Prob. 6PCh. 4.13 - Prob. 2PCh. 4.14 - Prob. 1PCh. 4.14 - Prob. 2PCh. 4.14 - Prob. 3PCh. 4.14 - Prob. 4PCh. 4.14 - Prob. 5PCh. 4.14 - Prob. 6PCh. 4.14 - Prob. 7PCh. 4.16 - Prob. 1PCh. 4.16 - Prob. 2PCh. 4.16 - Prob. 3PCh. 4.16 - Prob. 5PCh. 4.16 - Prob. 7PCh. 4.16 - Prob. 8PCh. 4.16 - Prob. 9PCh. 4.16 - Prob. 10PCh. 4.16 - Prob. 11PCh. 4.16 - Prob. 12PCh. 4.16 - Prob. 13PCh. 4.16 - Prob. 14PCh. 4.17 - Prob. 1PCh. 4.17 - Prob. 2PCh. 4.17 - Prob. 3PCh. 4.17 - Prob. 4PCh. 4.17 - Prob. 5PCh. 4.17 - Prob. 7PCh. 4.17 - Prob. 8PCh. 4 - Prob. 1RPCh. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - Prob. 18RPCh. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RP
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