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Engineering Computer Science Operations Research : Applications and Algorithms

Formulating an IP: Define a decision variable for each decision that the Cubs must take while signing the pitchers for adding the maximum victories to them. Define 0-1 variables as given below: x 1 = 1 if RS is signed 0 otherwise x 2 = 1 if BS is signed 0 otherwise x 3 = 1 if DE is signed 0 otherwise x 4 = 1 if ST is signed 0 otherwise x 5 = 1 if TS is signed 0 otherwise According to the given information, the objective function for adding the maximum victories to the Clubs can be defined as shown below: max z = 6 x 1 + 5 x 2 + 3 x 3 + 3 x 4 + 2 x 5 a. At most, $12 million can be spent: The first constraint limits the amount to be spent on signing the Pitchers to at most $12 million. Therefore, the constraint for using the cost of signing each Pitcher from the given table will be written as follows: 6 x 1 + 4 x 2 + 3 x 3 + 2 x 4 + 2 x 5 ≤ 12 b. If DE and ST are signed, then BS cannot be signed: The second constraint states that if DE and ST are signed, then BS cannot be signed. Thus, the definition of decision variables defined for DE, ST, and BS implies that, If x 3 = 1 and x 4 = 1 , then x 2 = 0 Therefore, the respective constraint will be written as follows: x 2 + x 3 + x 4 ≤ 2 c. At most two right-handed pitchers can be signed: The third constraint is formed such that at most two right-handed pitchers can be signed. The given table shows that the choices for the right-handed pitchers are RS, BS, DE, and TS. i.e, x 1 , x 2 , x 3 , a n d x 5 in terms of decision variables. Therefore, the respective constraint will be written as follows: x + 1 x 2 + x 3 + x 5 ≤ 2 d. The Cubs cannot sign both BS and RS: The fourth constraint states that the cubs cannot sign both BS and RS. Thus, the definition of decision variables defined for BS and RS implies that, Either x 1 = 1 and x 2 = 0 , OR x 1 = 0 and x 2 = 1 Therefore, the respective constraint will be written as follows: x 1 + x 2 ≤ 1 Hence, the given problem can be represented by using the following 0-1 IP, max z = 6 x 1 + 5 x 2 + 3 x 3 + 3 x 4 + 2 x 5 Subject to the constraints, 6 x 1 + 4 x 2 + 3 x 3 + 2 x 4 + 2 x 5 ≤ 12 x 2 + x 3 + x 4 ≤ 2 x + 1 x 2 + x 3 + x 5 ≤ 2 x 1 + x 2 ≤ 1 All variables are 0 or 1.

Formulating an IP: Define a decision variable for each decision that the Cubs must take while signing the pitchers for adding the maximum victories to them. Define 0-1 variables as given below: x 1 = 1 if RS is signed 0 otherwise x 2 = 1 if BS is signed 0 otherwise x 3 = 1 if DE is signed 0 otherwise x 4 = 1 if ST is signed 0 otherwise x 5 = 1 if TS is signed 0 otherwise According to the given information, the objective function for adding the maximum victories to the Clubs can be defined as shown below: max z = 6 x 1 + 5 x 2 + 3 x 3 + 3 x 4 + 2 x 5 a. At most, $12 million can be spent: The first constraint limits the amount to be spent on signing the Pitchers to at most $12 million. Therefore, the constraint for using the cost of signing each Pitcher from the given table will be written as follows: 6 x 1 + 4 x 2 + 3 x 3 + 2 x 4 + 2 x 5 ≤ 12 b. If DE and ST are signed, then BS cannot be signed: The second constraint states that if DE and ST are signed, then BS cannot be signed. Thus, the definition of decision variables defined for DE, ST, and BS implies that, If x 3 = 1 and x 4 = 1 , then x 2 = 0 Therefore, the respective constraint will be written as follows: x 2 + x 3 + x 4 ≤ 2 c. At most two right-handed pitchers can be signed: The third constraint is formed such that at most two right-handed pitchers can be signed. The given table shows that the choices for the right-handed pitchers are RS, BS, DE, and TS. i.e, x 1 , x 2 , x 3 , a n d x 5 in terms of decision variables. Therefore, the respective constraint will be written as follows: x + 1 x 2 + x 3 + x 5 ≤ 2 d. The Cubs cannot sign both BS and RS: The fourth constraint states that the cubs cannot sign both BS and RS. Thus, the definition of decision variables defined for BS and RS implies that, Either x 1 = 1 and x 2 = 0 , OR x 1 = 0 and x 2 = 1 Therefore, the respective constraint will be written as follows: x 1 + x 2 ≤ 1 Hence, the given problem can be represented by using the following 0-1 IP, max z = 6 x 1 + 5 x 2 + 3 x 3 + 3 x 4 + 2 x 5 Subject to the constraints, 6 x 1 + 4 x 2 + 3 x 3 + 2 x 4 + 2 x 5 ≤ 12 x 2 + x 3 + x 4 ≤ 2 x + 1 x 2 + x 3 + x 5 ≤ 2 x 1 + x 2 ≤ 1 All variables are 0 or 1.

Solution Summary: The author defines a decision variable for each decision that the Cubs must take while signing the pitchers for adding the maximum victories to them.

Operations Research : Applications and Algorithms

Operations Research : Applications and Algorithms

4th Edition

ISBN: 9780534380588

Author: Wayne L. Winston

Publisher: Brooks Cole

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