Question

Retailing Distribution Center Location Problem A major retail firm is considering a major change in its distribution system. It is considering relocating its distribution centers such that all retail stores can be replenished within 24 hours of a request for merchandise. The table below is a matrix of cover coefficients for 10 stores and 10 potential sites for distribution centers. If store i can be replenished within 24 hours from site j, then a_ij=1; otherwise a_ij-0. The cost of installation of the distributions centers vary from zone to zone and are given in the DC cost row. The firm wants to find the optimal location of the distribution centers.

Answer 1

1) Mathematical model is following

Decision variable: Xj = 1, if distribution center is located at site j, otherwise Xj = 0

Min 193X1+721X2+822X3+156X4+607X5+593X6+246X7+385X8+309X9+867X10

s.t.

X1+X2+X8+X9+X10 >= 1 (coverage for store 1)

X1+X2+X3 >= 1    (coverage for store 2)

X1+X2+X4+X5+X8+X9 >= 1 (coverage for store 3)

X3+X5+X6+X7+X9+X10 >= 1 (coverage for store 4)

X3+X4+X6+X7+X8+X9 >= 1 (coverage for store 5)

X4+X5+X6+X7+X10 >= 1   (coverage for store 6)

X2+X4+X6+X8+X9 >= 1    (coverage for store 7)

X1+X2+X3+X5+X8+X9+X10 >= 1     (coverage for store 8)

X1+X3+X4+X5+X7+X10 >= 1   (coverage for store 9)

X3+X6+X7+X8+X9+X10 >= 1   (coverage for store 10)

Xj $\in$ {0,1}

2)

Decision variable: Xj = 1, if distribution center is located at site j, otherwise Xj = 0

Yi = 1 if store i can be replenished by at least one of the distribution centers, otherwise Yi = 0

Max Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y8+Y9+Y10

s.t.

193X1+721X2+822X3+156X4+607X5+593X6+246X7+385X8+309X9+867X10 <= 300

X1+X2+X8+X9+X10 - Y1 >= 0 (coverage for store 1)

X1+X2+X3 - Y2 >= 0   (coverage for store 2)

X1+X2+X4+X5+X8+X9 - Y3 >= 0   (coverage for store 3)

X3+X5+X6+X7+X9+X10 - Y4 >= 0   (coverage for store 4)

X3+X4+X6+X7+X8+X9 - Y5 >= 0   (coverage for store 5)

X4+X5+X6+X7+X10 - Y6 >= 0   (coverage for store 6)

X2+X4+X6+X8+X9 - Y7 >= 0   (coverage for store 7)

X1+X2+X3+X5+X8+X9+X10 - Y8 >= 0 (coverage for store 8)

X1+X3+X4+X5+X7+X10 - Y9 >= 0 (coverage for store 9)

X3+X6+X7+X8+X9+X10 - Y10 >= 0 (coverage for store 10)

Xj, Yi  $\in$ {0,1}

3)

Decision variable: Xj = 1, if distribution center is located at site j, otherwise Xj = 0

Yi = 1 if store i can be replenished by at least one of the distribution centers, otherwise Yi = 0

Min 193X1+721X2+822X3+156X4+607X5+593X6+246X7+385X8+309X9+867X10

s.t.

Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y8+Y9+Y10 >= 10*0.85

X1+X2+X8+X9+X10 - Y1 >= 0 (coverage for store 1)

X1+X2+X3 - Y2 >= 0   (coverage for store 2)

X1+X2+X4+X5+X8+X9 - Y3 >= 0   (coverage for store 3)

X3+X5+X6+X7+X9+X10 - Y4 >= 0   (coverage for store 4)

X3+X4+X6+X7+X8+X9 - Y5 >= 0   (coverage for store 5)

X4+X5+X6+X7+X10 - Y6 >= 0   (coverage for store 6)

X2+X4+X6+X8+X9 - Y7 >= 0   (coverage for store 7)

X1+X2+X3+X5+X8+X9+X10 - Y8 >= 0 (coverage for store 8)

X1+X3+X4+X5+X7+X10 - Y9 >= 0 (coverage for store 9)

X3+X6+X7+X8+X9+X10 - Y10 >= 0 (coverage for store 10)

Xj, Yi  $\in$ {0,1}

4)

Solution of model 1 using LINDO follows:

Lingo 17.0-Solution Report . Lingo! File Edit Solver Window Help 23Solution Report Lingol Lingo Model-Lingo Min- 193"X1+721"X2 +822-X3+156*X4+607"X5+5931X6 +246 X7+385 X8+309 x9+867 X10 Global optimal1 solution found Objective value: Objective bound: Infeas1bilities: Extended solver steps: Total solver iterations: Elapsed runtime seconds: 595.0000 595.0000 0.000000 X1+X2+X8+X9+X1O > 1; X1+X2+X3l; 0.16 X3 X4+X6+X7+X8+X9 X4+X5+X6+XT+X10 >= 1; X2+X4+X6+X8+X9 -1; X1+X2+X3+X5+X8eX5+X10 Model Class: PILP 1; Total variables: Nonlinear variables: Integer variables: >= 10 10 BIN (x1) BIN (X2) 3BIN (X3) BIN (X4); BIN (X5) BIN (X6); BIN (X7) 3BIN (X8) BIN (X9) BIN (x10): Total constraints: Nonlinear constraints: Total nonzeros: Nonlinear nonzeros: 65 Variable X1 X2 x3 X4 X5 X6 X7 X8 Value 1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 Reduced Cost 193.0000 721.0000 822.0000 156.0000 607.0000 593.0000 246.0000 385.0000 309.0000 867.0000 X10 Row Slack or Surplus 595.0000 0.000000 0.000000 1.000000 Dual Price -1.000000 0.000000 0.000000 0.000000 For Help, press F1 NUM MOD Ln 42, Col 77 6:20 am

X1, X4, X7 = 1

Total cost = 595

____________________

Solution of model 2 using LINDO follows:

Lingo 17.0 Solution Report Lingol File Edit Solver Window Help Solution Report Lingo1 Lingo Model Lin Max = Y1+Y2+73+74+75+Y6+774Y5+YS+710; gol Global optimal solution found Objective value: Objective bound: Infeasibilities: Extended 3ㅇ!ver 3teps: Total solver iterations: Elapsed runtime seconds 193"X1+721*X2 +822-X3+156*X4+607"X5+593"Хб +246"X7+385"X8+309"X9+867"X10 <= 300; 5.000000 5.000000 0.000000 X1+X2+X3-Y2 -0: X3+X5+X64X7+X9+X10-Y40 Model Class: PILP Total variables: Nonlinear variables: Integer variables: X2+X4X6+X8+X9-170: X34X6+XT+X8eX5+X10-Y10 >= 0; BIN (X1) BIN (X2) BIN (X3) 2BIN (X4) BIN (X5) BIN (X6) BIN (X7) BIN X8) BIN (X9): BIN (X10) BIN (T1) BIN (Y2); BIN (Y3) BIN (Y4) BIN (YS): BIN (Y6) BIN (Y7) BIN (T8) BIN (Y9) BIN (Y10) Total constraintS: Nonlinear constraints: 12 Total nonzeros: Nonlinear nonzeros: Variable 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Reduced Cost -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Y2 Y5 Ye Ye Y9 Y10 X1 X2 X3 X4 X6 For Help, press F1 NUM Ln 1, Col1 6:29 am

Solution Report- Lingo1 Variable Yl Y2 Y3 Y4 YS Y6 Y7 Y8 Y9 Y10 X1 X2 X3 X4 X5 X6 X7 Value 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Reduced Cost -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 X9 X10 Dual Price 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Row Slack or Surplus 5.000000 107.0000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 7 10 12

X1 = 1,

Stores 1,2,3,8,9 are covered.

____________________

Solution of model 3 using LINDO follows:

Lingo 17.0 Solution Report Lingol File Edit Solver Window Help le- EZ Solution Report . Lingo! Lingo Model Lin Min 193 X1+721 X2+822 X3+156 X4+607 X5+593 X6 gol Global optimal solution found Objective value: Objective bound: Infeasibilities: Extended з Iver 3teps: Total solver iterations: Elapsed runtime seconds +246 X7+385 X8+309 X9+867 X10 439.0000 439.0000 0.000000 X1+X2+X8+X9+X10-Yl X1+X2 +X3-Y2 >= 0; X1+X2+X4+X5+X8+X9-Y3 - 0: Model Class: PILP Total variables: Nonlinear variables: Integer variables: X1+X2+X3+X5+X3+X9+X10-Y8-0 X3+X6+X7+X8+X9+X10-Y10-0: BIN (X1) BIN (X2) BIN (X3) BIN (X4) BIN (X5) BIN (X6) BIN (X7) BIN (X8) BIN X9) BIN (x10) BIN (Y1) BIN (T2) BIN (Y3) BIN(T4) BIN (YS) BIN(Y6): BIN 7): BIN (Y8) BIN (T9) BIN (Y10) Total constraintS: Nonlinear constraints: 12 Total nonzeros: Nonlinear nonzeros: Variable x1 X2 X3 X4 X5 x6 X7 X8 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 Reduced Cost 193.0000 721.0000 822.0000 156.0000 607.0000 593.0000 246.0000 385.0000 309.0000 867.0000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Yl Y5 For Help, press F1 CAP NUM Ln 1, Col1

Solution Report- Lingo1 Variable X1 X2 X3 X4 X5 X6 X7 X10 Yl Y2 Y3 Y4 YS Y6 17 Y8 Y9 Y10 value 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 1.000000 1.000000 1.000000 Reduced Cost 193.0000 721.0000 822.0000 156.0000 607.0000 593.0000 246.0000 385.0000 309.0000 867.0000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Slack or Surplus 439.0000 0.5000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 Dual Price -1 000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 OW 2 7 10 12

X1, X7 = 1

Total cost = 493

Except store 7, all stores are covered.