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Integer Programming Model Formulation. Coach Night Is Trying

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Integer Programming Model Formulation. Coach Night Is Trying Integer Programming Model Formulation. Coach Night is trying to choose the starting line-up for the basketball team. The team consists of seven players who have been rated (on a scale of 1=poor to 3=excellent) according to their ball-handling, shooting, rebounding, and defensive abilities. The positions that each player is allowed to play (G=guard, C=center, F=forward) and the player's abilities are: PlayerPositionBall-handling Shooting ReboundingDefense 1 G 3 3 1 3 2 C 2 1 3 2 3 G-F 2 3 2 2 4 F-C 1 3 3 1 5 G-F 1 3 1 2 6 F-C 3 1 2 3 7 G-F 3 2 2 1 ILP Model (Basketball Team) page 1 of 7 The five-player starting line-up must satisfy the following restrictions: (i) At least 3 members must be able to play guard, at least 2 members must be able to play forward, and at least one member must be able to play center. (ii)The average ball-handling, shooting, and rebounding level of the starting lineup must each be at least 2. (iii) If player 3 starts, then player 6 cannot start. (iv) If player 1 starts, then players 4 and 5 must both start. (v) Either player 2 or player 3 (or both) must start. Define variables: Xi = 1 if player # i is selected for the line-up, else 0 Given these constraints, Coach wants to maximize the total defensive ability of the starting team. Formulate an integer LP that will help him choose his starting team, and use LINDO (or other software) to find the optimal solution. ILP Model (Basketball Team) page 2 of 7 Integer Programming Model Formulation (Selecting basketball team) Solution: Define variables: 1 if player i is selected X = j 0 otherwise Constraints: The number of players selected must be exactly five: X1 + X2 + X3 + X4 + X5 + X6 + X7 = 5 At least 3 members must be able to play guard: X1 + X3 + X5 + X7 ³ 3 At least 2 members must be able to play forward: X3 + X5 + X7 ³ 2 At least one member must be able to play center: X2 + X4 + X6 ³ 1 The average ball-handling level of the starting lineup must be at least 2: 3X + 2X + 2X + X + X + 3X + 3X 1 2 3 4 5 6 7 ³ 2 Þ 3X + 2X + 2X + X + X + 3X + 3X ³ 10 5 1 2 3 4 5 6 7 The average shooting level of the starting lineup must be at least 2: 3X + X + 3X + 3X + 3X + X + 2X 1 2 3 4 5 6 7 ³ 2 Þ 3X + X + 3X + 3X + 3X + X + 2X ³ 10 5 1 2 3 4 5 6 7 The average rebounding level of the starting lineup must be at least 2: ILP Model (Basketball Team) page 3 of 7 X + 3X + 2X + 3X + X + 2X + 2X 1 2 3 4 5 6 7 ³ 2 Þ X + 3X + 2X + 3X + X + 2X + 2X ³ 10 5 1 2 3 4 5 6 7 If player 3 starts, then player 6 cannot start: 1 – X3 ³ X6 Û X3 + X6 £ 1 If player 1 starts, then players 4 and 5 must both start: X1 £ X4 & X1 £ X5 Þ 2X1 £ X4 + X5 Note: the single inequality on the right is equivalent the pair of inequalities on the left if all variables are binary. However, if they are continuous variables restricted to the interval [0,1], the single inequality is implied by the pair on the left, but not vice-versa. In ILP, it is better for the sake of computational efficiency to use the pair of inequalities, which gives a smaller feasible region for the LP obtained by relaxing the integer restrictions. Either player 2 or player 3 (or both) must start: X2 + X3 ³ 1 Objective: Maximize the total defensive ability of the team: Maximize 3X1 + 2X2 + 2X3 + X4 + 2X5 + 3X6 + X7 LINDO output: MAX 3 X1 + 2 X2 + 2 X3 + X4 + 2 X5 + 3 X6 + X7 SUBJECT TO ILP Model (Basketball Team) page 4 of 7 2) X1 + X2 + X3 + X4 + X5 + X6 + X7 = 5 3) X1 + X3 + X5 + X7 >= 3 4) X3 + X5 + X7 >= 2 5) X2 + X4 + X6 >= 1 6) 3 X1 + 2 X2 + 2 X3 + X4 + X5 + 3 X6 + 3 X7 >= 10 7) 3 X1 + X2 + 3 X3 + 3 X4 + 3 X5 + X6 + 2 X7 >= 10 8) X1 + 3 X2 + 2 X3 + 3 X4 + X5 + 2 X6 + 2 X7 >= 10 9) X3 + X6 <= 1 10) X1 - X4 <= 0 11) X1 - X5 <= 0 12) X2 + X3 >= 1 END INTE 7 Solution: LP OPTIMUM FOUND AT STEP 17 OBJECTIVE VALUE = 9.71428585 SET X3 TO <= 0 AT 1, BND=9.000 TWIN=-0.1000E+31 28 NEW INTEGER SOLUTION OF 9.00000000 AT BRANCH 1 PIVOT 28 BOUND ON OPTIMUM: 9.000000 DELETE X3 AT LEVEL 1 ENUMERATION COMPLETE. BRANCHES= 1 PIVOTS= 28 LAST INTEGER SOLUTION IS THE BEST FOUND ILP Model (Basketball Team) page 5 of 7 RE-INSTALLING BEST SOLUTION... OBJECTIVE FUNCTION VALUE 1) 9.000000 VARIABLE VALUE REDUCED COST X1 1.000000 -3.000000 X2 1.000000 -2.000000 X3 0.000000 -2.000000 X4 1.000000 -1.000000 X5 1.000000 -2.000000 X6 0.000000 -3.000000 X7 1.000000 -1.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.000000 3) 0.000000 0.000000 4) 0.000000 0.000000 5) 1.000000 0.000000 6) 0.000000 0.000000 7) 2.000000 0.000000 8) 0.000000 0.000000 9) 1.000000 0.000000 10) 0.000000 0.000000 11) 0.000000 0.000000 12) 0.000000 0.000000 ILP Model (Basketball Team) page 6 of 7 NO. ITERATIONS= 28 BRANCHES= 1 DETERM.= 1.000E 0 That is, the starting lineup should consist of players 1, 2, 4, 5, and 7. ILP Model (Basketball Team) page 7 of 7.
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