Joint assignment, scheduling and routing models to Home Care optimization: a pattern based approach Online Supplement

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1 Joint assignment, scheduling and routing models to Home Care optimization: a pattern based approach Online Supplement Paola Cappanera Maria Grazia Scutellà March 17, Details on symmetry management efficiency Tables 1 and 2, respectively for the maxmin and the maxmin with EC cases, report for the Heur, the ImplSol and two flow based pattern generation policies, the node of the branch and bound tree where the best integer solution is found with respect to the total number of nodes explored: as an example, 0/20 reports that 20 nodes have been overall explored within the time limit, and that the best solution has been found at root node (node number 0). These information allow to measure the efficiency of each approach: the bigger the number of explored nodes the faster the computing at each branch and bound node; likewise, the smaller the node where the best solution is found the bigger the capability of a truncated branch and bound approach, based on the ILP models proposed in this paper, to provide solutions of good quality. In particular, Tables 1 and 2 reveal that the symmetry management generally allows one to explore more nodes, and that the very good solutions found by policy FB-0.50 are computed quite early within the branch and bound tree. Table 1: Maxmin - a measure of efficiency FlowBased Heur ImplSol January 2006 NoSymm 0/20 0/1 821/63253 n.a. Symm 837/ / /2382 n.a. April 2007 NoSymm 180/189 0/3 824/ /16083 Symm 390/905 n.a. 5846/ /5473 Dipartimento di Ingegneria dell Informazione, Università degli Studi di Firenze, Firenze, Italy - paola.cappanera@unifi.it Dipartimento di Informatica, Università di Pisa, Pisa, Italy - scut@di.unipi.it 1

2 Table 2: Maxmin with EC - a measure of efficiency FlowBased Heur ImplSol January 2006 NoSymm n.a. 0/ /1884 n.a. Symm 824/824 n.a. 1729/14850 n.a. April 2007 NoSymm n.a. n.a. 4/ /1545 Symm n.a. n.a. 2129/ /21447 Table 3: Impact of symmetry management on LP dimension Heur ImplSol FlowBased 0.50 LPRows LPCols LPRows LPCols LPRows LPCols January 2006 NoSymm Symm April 2007 NoSymm Symm Details on symmetry management and LP dimension Here we provide some comments and explanations in terms of dimension of the solved problems. To this end, Table 3 presents the dimension of the LP problems in terms of constraints (columns LPRows) and variables (columns LPCols) for the three policies Heur, ImplSol and FB-0.50 when the maxmin objective function is used. The data refer to the LP dimension reported by CPLEX after the preprocessing phase. Similar results hold for the alternative objective function, i.e. minmax. Results relative to the symmetry management are given in rows labeled Symm while the rows NoSymm report data relative to the base model. The symmetry management allows to reduce the LP dimension and the three policies exhibit a quite similar behavior in terms of reduction. In fact, for the selected week in January 2006 the average reduction in the number of constraints when the model is equipped with the symmetry management is equal to 5.51% while the percentage reduction of the variables is equal to For April 2007 the figures are respectively 11.72% for the rows and 14.04% for the columns. Despite the reduction in terms of constraints and variables, the time required to solve the LP may increase when the symmetries are treated. However, as observed in the paper, the symmetry management is crucial almost everywhere either to halve the ILP gap or to find feasible solutions in the most critical cases. 3 Details on minmax efficiency Table 4, referred to minmax, gives for each of the three pattern generation policies Heur, ImplSol and FB-0.50 the value of the best integer solution (IPValue) and information about the total number of nodes explored in the branch and bound tree with the node where the best solution is found (NodeInf). 2

3 Table 4: minmax - efficiency results Heur ImplSol FB-0.50 IPValue NodeInf IPValue NodeInf IPValue NodeInf / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / n.a. n.a. n.a. n.a / n.a. n.a. n.a. n.a / n.a. n.a. n.a. n.a / n.a. n.a. n.a. n.a. n.a. n.a n.a. n.a. n.a. n.a / n.a. n.a. n.a. n.a / Details on care continuity constraints Columns in Table 5 report, for the minmax case, respectively the LPTime, the IPTime, the optimality gap, the IPValue and the node where the best integer solution is found with respect to the total number of nodes explored. For each instance, results relative to the case where care continuity is assured and to the case where care continuity is relaxed are reported in consecutive rows. In particular, a suffix CC is appended to the instance name in the former case whereas suffix NoCC is used in the latter case. In general, we observe that relaxing the care continuity constraints has no impact on the value of the solutions except in few cases; on the other hand, disregarding the care continuity, the model seems to be faster in finding feasible solutions with respect to the case where the care continuity is ensured. In regards to the IPTime, we observe a non monotonic behavior since the relaxation of the care continuity constraints can involve either a reduction or an increase of computational time with respect to the more constrained model, at least on instances with up to 60 patients. Results are a little bit clearer on the most critical instances in the test bed, i.e. on instances characterized by 80 patients. In such a case we report either a valuable reduction of the IPTime or of the optimality gap; indeed, the elimination of care continuity constraints allows to find a feasible solution also for instance whereas no feasible solution is found in the other case. For some instances, the IPValue is smaller for the NoCC variant than in the CC counterpart. Similar results are obtained in the maxmin case. Furthermore, information on the quality of the solutions under maxmin and minmax for the restricted instances related to April 2007 are reported in Figures 1 and 2. There, 3

4 Table 5: minmax - impact of care continuity on efficiency and efficacy FB-0.50 LPTime IPTime %Gap IPValue NodeInf CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC / NoCC / CC n.a. n.a. n.a NoCC / CC / NoCC / CC / NoCC /556 4

5 as defined throughout the paper, UF refers to the operator utilization factor, while T T F gives the fraction of travel time with respect to the maximum weekly workload. Here and in the following figure as well, for each instance, information relative to the case where care continuity is assured and to the case where care continuity is relaxed are reported consecutively so as to facilitate their comparison. Finally, see Figures 3, 4, 5 and 6 for an analysis of the number of operators per patient in both scenarios (maxmin and minmax), under the care continuity constraints and their removal. In these figures the percentage of patients that have been assigned to one, two, three, four and five operators is given (in the legend Op stands for Operator ). We observe that the number of patients assigned to more than two operators is quite low when the care continuity constraints are ignored, for both the alternative objective functions. For restricted instances referring to January 2006 this behavior is more evident than for instances related to April Metrics QoS Ind min UF max UF avg UF min TTF max TTF avg TTF Instances Figure 1: maxmin - solution quality analysis on 0407 instances 5

6 Metrics QoS Ind min UF max UF avg UF min TTF max TTF avg TTF Instances Figure 2: minmax - solution quality analysis on 0407 instances 100% 90% 80% 70% 60% 5 Op 4 Op 3 Op 2 Op 1 Op 50% 40% 30% 20% 10% 0% Figure 3: maxmin - frequency analysis on 0106 instances 6

7 100% 90% 80% 70% 60% Op 5 Op 4 Op 3 Op 2 Op 1 50% 40% 30% 20% 10% 0% Figure 4: maxmin - frequency analysis on 0407 instances 100% 90% 80% 70% 60% Op 5 Op 4 Op 3 Op 2 Op 1 50% 40% 30% 20% 10% 0% Figure 5: minmax - frequency analysis on 0106 instances 7

8 100% 90% 80% 70% 60% Op 5 Op 4 Op 3 Op 2 Op 1 50% 40% 30% 20% 10% 0% Figure 6: minmax - frequency analysis on 0407 instances 8