Example 13.2 Generalized Assignment Problem

The generalized assignment problem (GAP) is that of finding a maximum profit assignment from tasks to machines such that each task is assigned to precisely one machine subject to capacity restrictions on the machines. With each possible assignment, associate a binary variable , which, if set to , indicates that machine is assigned to task . For ease of notation, define two index sets and . A GAP can be formulated as a MILP as follows:

In this formulation, constraints (assignment) ensure that each task is assigned to exactly one machine. Inequalities (knapsack) ensure that for each machine, the capacity restrictions are met.

Consider the following example taken from Koch et al. (2011) with tasks to be assigned to machines. The data set profit_data provides the profit for assigning a particular task to a particular machine:

%let NumTasks    = 24;
%let NumMachines = 8;

data profit_data;
datalines;
25 23 20 16 19 22 20 16 15 22 15 21 20 23 20 22 19 25 25 24 21 17 23 17
16 19 22 22 19 23 17 24 15 24 18 19 20 24 25 25 19 24 18 21 16 25 15 20
20 18 23 23 23 17 19 16 24 24 17 23 19 22 23 25 23 18 19 24 20 17 23 23
16 16 15 23 15 15 25 22 17 20 19 16 17 17 20 17 17 18 16 18 15 25 22 17
17 23 21 20 24 22 25 17 22 20 16 22 21 23 24 15 22 25 18 19 19 17 22 23
24 21 23 17 21 19 19 17 18 24 15 15 17 18 15 24 19 21 23 24 17 20 16 21
18 21 22 23 22 15 18 15 21 22 15 23 21 25 25 23 20 16 25 17 15 15 18 16
19 24 18 17 21 18 24 25 18 23 21 15 24 23 18 18 23 23 16 20 20 19 25 21
;


The data set weight_data provides the amount of resources used by a particular task when assigned to a particular machine:

data weight_data;
datalines;
8 18 22  5 11 11 22 11 17 22 11 20 13 13  7 22 15 22 24  8  8 24 18  8
24 14 11 15 24  8 10 15 19 25  6 13 10 25 19 24 13 12  5 18 10 24  8  5
22 22 21 22 13 16 21  5 25 13 12  9 24  6 22 24 11 21 11 14 12 10 20  6
13  8 19 12 19 18 10 21  5  9 11  9 22  8 12 13  9 25 19 24 22  6 19 14
25 16 13  5 11  8  7  8 25 20 24 20 11  6 10 10  6 22 10 10 13 21  5 19
19 19  5 11 22 24 18 11  6 13 24 24 22  6 22  5 14  6 16 11  6  8 18 10
24 10  9 10  6 15  7 13 20  8  7  9 24  9 21  9 11 19 10  5 23 20  5 21
6  9  9  5 12 10 16 15 19 18 20 18 16 21 11 12 22 16 21 25  7 14 16 10
;


Finally, the data set capacity_data provides the resource capacity for each machine:

data capacity_data;
input b @@;
datalines;
36 35 38 34 32 34 31 34
;


The following PROC OPTMODEL statements read in the data and define the necessary sets and parameters:

proc optmodel;
/* declare index sets */
set MACHINES = 1..&NumMachines;

/* declare parameters */
num capacity {MACHINES};

/* read data sets to populate data */
read data capacity_data into [_n_] capacity=b;


The following statements declare the optimization model:

   /* declare decision variables */

/* declare objective */
max TotalProfit =
sum {i in MACHINES, j in TASKS} profit[i,j] * Assign[i,j];

/* declare constraints */
sum {i in MACHINES} Assign[i,j] = 1;

con KnapsackCon {i in MACHINES}:
sum {j in TASKS} weight[i,j] * Assign[i,j] <= capacity[i];


The following statements use two different decompositions to solve the problem. The first decomposition defines each assignment constraint as a block and uses the pure network simplex solver for the subproblem. The second decomposition defines each knapsack constraint as a block and uses the MILP solver for the subproblem.

   /* each assignment constraint defines a block */
AssignmentCon[j].block = j;

solve with milp / logfreq=1000
decomp        =()
decomp_subprob=(algorithm=nspure);

/* each knapsack constraint defines a block */
AssignmentCon[j].block = .;
for{i in MACHINES}
KnapsackCon[i].block = i;

solve with milp / decomp=();
quit;


The solution summaries are displayed in Output 13.2.1.

Output 13.2.1: Solution Summaries

The OPTMODEL Procedure

Solution Summary
Solver MILP
Algorithm Decomposition
Objective Function TotalProfit
Solution Status Optimal within Relative Gap
Objective Value 563

Relative Gap 0.0000994814
Absolute Gap 0.0560135845
Primal Infeasibility 0
Bound Infeasibility 0
Integer Infeasibility 0

Best Bound 563.05601358
Nodes 8809
Iterations 9366
Presolve Time 0.01
Solution Time 44.06

Solution Summary
Solver MILP
Algorithm Decomposition
Objective Function TotalProfit
Solution Status Optimal
Objective Value 563

Relative Gap 0
Absolute Gap 0
Primal Infeasibility 0
Bound Infeasibility 0
Integer Infeasibility 0

Best Bound 563
Nodes 7
Iterations 44
Presolve Time 0.01
Solution Time 0.97

The iteration log for both decompositions is shown in Output 13.2.2. This example is interesting because it shows the tradeoff between the strength of the relaxation and the difficulty of its resolution. In the first decomposition, the subproblems are totally unimodular and can be solved trivially. Consequently, each iteration of the decomposition algorithm is very fast. However, the bound obtained is as weak as the bound found in direct methods (the LP bound). The weaker bound leads to the need to enumerate more nodes overall. Alternatively, in the second decomposition, the subproblem is the knapsack problem, which is solved using MILP. In this case, the bound is much tighter and the problem solves in very few nodes. The tradeoff, of course, is that each iteration takes longer because solving the knapsack problem is not trivial. Another interesting aspect of this problem is that the subproblem coverage in the second decomposition is much smaller than that of the first decomposition. However, when dealing with MILP, it is not always the size of the coverage that determines the overall effectiveness of a particular choice of decomposition.

Output 13.2.2: Log

 NOTE: There were 8 observations read from the data set WORK.PROFIT_DATA. NOTE: There were 8 observations read from the data set WORK.WEIGHT_DATA. NOTE: There were 8 observations read from the data set WORK.CAPACITY_DATA. NOTE: Problem generation will use 4 threads. NOTE: The problem has 192 variables (0 free, 0 fixed). NOTE: The problem has 192 binary and 0 integer variables. NOTE: The problem has 32 linear constraints (8 LE, 24 EQ, 0 GE, 0 range). NOTE: The problem has 384 linear constraint coefficients. NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range). NOTE: The MILP presolver value AUTOMATIC is applied. NOTE: The MILP presolver removed 0 variables and 0 constraints. NOTE: The MILP presolver removed 0 constraint coefficients. NOTE: The MILP presolver modified 0 constraint coefficients. NOTE: The presolved problem has 192 variables, 32 constraints, and 384 constraint coefficients. NOTE: The MILP solver is called. NOTE: The Decomposition algorithm is used. NOTE: The Decomposition algorithm is executing in single-machine mode. NOTE: The DECOMP method value USER is applied. NOTE: The subproblem solver chosen is an LP solver but at least one block has integer variables. NOTE: The decomposition subproblems consist of 24 disjoint blocks. NOTE: The decomposition subproblems cover 192 (100.00%) variables and 24 (75.00%) constraints. NOTE: The deterministic parallel mode is enabled. NOTE: The Decomposition algorithm is using up to 4 threads. Iter         Best       Master         Best       LP       IP   CPU  Real Bound    Objective      Integer      Gap      Gap  Time  Time NOTE: Starting phase 1. 1       0.0000       8.9248            . 8.92e+00        .     0     0 4       0.0000       0.0000            .    0.00%        .     0     0 NOTE: Starting phase 2. 5     574.0000     561.1588            .    2.24%        .     0     0 6     568.8833     568.5610            .    0.06%        .     0     0 7     568.6464     568.6464            .    0.00%        .     0     0 .     568.6464     568.6464     562.0000    0.00%    1.17%     0     0 NOTE: Starting branch and bound. Node  Active   Sols         Best         Best      Gap     CPU    Real Integer        Bound             Time    Time 0       1      1     562.0000     568.6464    1.17%       0       0 1000     860      1     562.0000     565.1615    0.56%       5       4 2000    1534      1     562.0000     564.5238    0.45%      10       9 3000    1986      1     562.0000     564.1515    0.38%      16      14 4000    2258      1     562.0000     563.8829    0.33%      21      19 5000    2270      1     562.0000     563.6617    0.29%      27      24 6000    2104      1     562.0000     563.4776    0.26%      32      29 7000    1694      1     562.0000     563.3076    0.23%      38      34 8000    1092      1     562.0000     563.1618    0.21%      44      39 8242     884      2     563.0000     563.1363    0.02%      46      41 8808     318      2     563.0000     563.0560    0.01%      49      44 NOTE: The Decomposition algorithm used 4 threads. NOTE: The Decomposition algorithm time is 44.19 seconds. NOTE: Optimal within relative gap. NOTE: Objective = 563. NOTE: The MILP presolver value AUTOMATIC is applied. NOTE: The MILP presolver removed 0 variables and 0 constraints. NOTE: The MILP presolver removed 0 constraint coefficients. NOTE: The MILP presolver modified 0 constraint coefficients. NOTE: The presolved problem has 192 variables, 32 constraints, and 384 constraint coefficients. NOTE: The MILP solver is called. NOTE: The Decomposition algorithm is used. NOTE: The Decomposition algorithm is executing in single-machine mode. NOTE: The DECOMP method value USER is applied. NOTE: The decomposition subproblems consist of 8 disjoint blocks. NOTE: The decomposition subproblems cover 192 (100.00%) variables and 8 (25.00%) constraints. NOTE: The deterministic parallel mode is enabled. NOTE: The Decomposition algorithm is using up to 4 threads. Iter         Best       Master         Best       LP       IP   CPU  Real Bound    Objective      Integer      Gap      Gap  Time  Time NOTE: Starting phase 1. 1       0.0000       7.0000            . 7.00e+00        .     0     0 5       0.0000       0.0000            .    0.00%        .     0     0 NOTE: Starting phase 2. 6     672.9221     499.7295            .   25.74%        .     0     0 8     653.1045     520.3731            .   20.32%        .     0     0 9     607.0000     528.1905            .   12.98%        .     0     0 10     607.0000     539.1556            .   11.18%        .     0     0 .     607.0000     547.7045     547.0000    9.77%    9.88%     0     0 12     605.4928     548.8623     547.0000    9.35%    9.66%     0     0 13     603.2619     552.0952     547.0000    8.48%    9.33%     0     0 14     595.3226     555.3548     547.0000    6.71%    8.12%     0     0 15     590.4167     557.7500     547.0000    5.53%    7.35%     0     0 16     579.6429     558.5893     547.0000    3.63%    5.63%     0     0 17     576.5000     560.8750     547.0000    2.71%    5.12%     0     0 18     570.0000     563.4286     547.0000    1.15%    4.04%     0     0 19     566.2500     563.5000     547.0000    0.49%    3.40%     0     0 20     565.0000     564.0000     547.0000    0.18%    3.19%     0     0 .     565.0000     564.0000     562.0000    0.18%    0.53%     0     0 22     564.0000     564.0000     562.0000    0.00%    0.35%     0     0 .     564.0000     564.0000     562.0000    0.00%    0.35%     0     0 NOTE: Starting branch and bound. Node  Active   Sols         Best         Best      Gap     CPU    Real Integer        Bound             Time    Time 0       1      3     562.0000     564.0000    0.35%       0       0 5       1      4     563.0000     563.5000    0.09%       1       0 6       0      4     563.0000     563.0000    0.00%       1       0 NOTE: The Decomposition algorithm used 4 threads. NOTE: The Decomposition algorithm time is 0.90 seconds. NOTE: Optimal. NOTE: Objective = 563.