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:

Miami	Bills	Jets
27		17
28		24
24	23
30	16
	24	41
	3	41

• 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 $\left|B-j-1\right|\approx 0.22$. Hence we have,
  • $\left|M-B-1\right|\approx 14$
• Our predicted value should be close to observed value. Hence, introduce deviation variable as follows:
  • $s_i^{+}=$ Amount by which we numerically exceed the predicted value.
  • $s_i^{-}=$ Amount by which we are numerically under the predicted value.
• So,
  • $M-B-1+s_1^{+}-s_1^{-}=14$
• 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,
  • $M-J-1+s_2^{+}-s_2^{-}=15$
• 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,
  • $J-B-1+s_3^{+}-s_3^{-}=55$
• We have to minimize the total deviation. Hence the minimization equation is
• Minimize z=$s_1^{+}-s_1^{-}+s_2^{+}-s_2^{-}+s_3^{+}-s_3^{-}$
• Hence the following LP is formulated.
• Minimize z=$s_1^{+}+s_1^{-}+s_2^{+}+s_2^{-}+s_3^{+}+s_3^{-}$
• Such that
  • $M-B-1+s_1^{+}-s_1^{-}=14$
  • $M-J-1+s_2^{+}-s_2^{-}=15$
  • $J-B-1+s_3^{+}-s_3^{-}=55$
• Enter the following LP equations in the file:
  • min $sp1+sm1+sp2+sm2+sp3+sm3$
  • Subject to
  • $m-j+sp1-sm1=15$
  • $m-b+sp2-sm2=16$
  • $b-j+sp3-sm3=56$
  • 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

Variable	Value
sp1	0.000000
sm1	0.000000
sp2	1.000000
sm2	0.000000
sp3	56.000000
sm3	0.000000
m	15.000000
j	0.000000
b	0.000000

Row	Slack or Surplus
1	57.00000
2	0.000000
3	0.000000
4	0.000000

• 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