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# Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints

This research was supported by the National Natural Science Foundation of China (Grant Number: 71272086), the National Science and Technology supporting Program of China (Grant Number: 2015BAF05B01), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20120191110042).
• A product service supply chain (PSSC) supplies customers with product-service systems (PSS) consist of integrated products and services. The product manufacturing should match the service supply in the order delivery planning. For PSS orders are usually delivered under time window constraints, this paper is concerned with the integrated order acceptance and scheduling (OAS) decision of the PSSC. Defined the PSS orders by their revenues, product processing times, serving offering times and hard time window constraints, we formulate the OAS problem as a MILP model to optimize total revenue of PSSC and propose two effective value for big-M to solve the problem with small size optimally. The simulated annealing algorithm based on the priority rule of servable orders first (SOF-SA) and the dynamic acceptance and scheduling heuristic (DASH) algorithm are presented. The performance of the model and the two algorithms are proved through simulating instances with different order sizes. Computational tests show that the SOF-SA algorithm is more effective when used for small size problems while the DASH algorithm is more effective for problems with larger size; negotiating with customers to make reasonable delivery time windows should be beneficial to increasing total revenue and improving the decision efficiency.

Mathematics Subject Classification: Primary: 49K30, 49K35; Secondary: 90C11.

 Citation:

• Figure 1.  Graphical representation of the OAS problem in PSSC

Table 1.  The performance of MA

 $\tau$ $R$ $M=\sum_{i=1, ..., n} {(t_{pi} +t_{si} )}$ CPU(s) $\widehat{M_{ij} }=d_{li} +t_{sj}$ CPU(s) $n\text{=}10$ $n\text{=}12$ $n\text{=}10$ $n\text{=}12$ 0.1 0.2 19.45 684.96 19.55 675.23 0.1 0.4 1.28 104.66 1.19 103.99 0.1 0.6 0.36 24.52 0.34 24.75 0.2 0.2 2.45 149.15 2.30 126.21 0.2 0.4 0.41 11.61 0.39 11.5 0.2 0.6 0.13 0.07 0.28 0.07 0.3 0.2 1.20 4.57 1.19 4.08 0.3 0.4 1.97 0.16 1.91 0.23 0.3 0.6 0.06 0.54 0.06 0.54

Table 2.  The algorithms' performance for $n = 10$

 $n$ $\tau$ $R$ GAP1(%) CPU(s) SOF-SA DASH SOF-SA DASH 10 0.1 0.2 13.7 16.2 31.63 3.15 10 0.1 0.4 7.1 9.5 20.32 1.21 10 0.1 0.6 14.4 15.8 13.87 0.89 10 0.2 0.2 16.6 19.1 43.57 0.75 10 0.2 0.4 21.4 23.6 24.33 0.67 10 0.2 0.6 15.5 22.7 15.21 0.62 10 0.3 0.2 18.2 19.1 50.92 0.56 10 0.3 0.4 7.2 15.8 28.69 0.53 10 0.3 0.6 20.6 23.4 24.36 0.27

Table 3.  The algorithms' performance for $n = 20$

 $n$} $\tau$ $R$ GAP2(%) CPU(s) SOF-SA DASH SOF-SA DASH 20 0.1 0.2 5.8 11.0 97.04 3.76 20 0.1 0.4 11.7 19.1 45.73 1.61 20 0.1 0.6 9.2 14.4 39.97 1.45 20 0.2 0.2 11.9 12.3 120.38 1.32 20 0.2 0.4 5.0 8.8 52.67 1.19 20 0.2 0.6 9.4 12.1 46.89 0.96 20 0.3 0.2 7.4 13.0 168.42 0.88 20 0.3 0.4 12.2 15.6 38.54 0.73 20 0.3 0.6 9.1 13.5 37.45 0.65

Table 4.  The algorithms' performance for $n = 50$

 $n$ $\tau$ $R$ GAP2(%) CPU(s) SOF-SA DASH SOF-SA DASH 50 0.1 0.2 9.5 15.8 735.14 10.68 50 0.1 0.4 17.0 20.4 530.87 5.49 50 0.1 0.6 20.9 21.3 521.22 5.31 50 0.2 0.2 12.6 15.0 822.36 5.16 50 0.2 0.4 16.2 17.7 579.54 4.88 50 0.2 0.6 17.1 19.0 566.97 4.79 50 0.3 0.2 18.4 22.6 899.53 4.45 50 0.3 0.4 22.7 26.5 620.28 4.15 50 0.3 0.6 15.3 15.8 619.99 3.65

Table 5.  The algorithms' performance for $n = 100$

 $n$ $\tau$ $R$ GAP2(%) CPU(s) SOF-SA DASH SOF-SA DASH 100 0.1 0.2 25.8 19.2 1365.49 32.54 100 0.1 0.4 30.3 17.7 954.21 25.57 100 0.1 0.6 19.7 10.4 996.73 24.66 100 0.2 0.2 25.1 18.8 1681.32 23.09 100 0.2 0.4 27.9 26.1 1307.76 21.85 100 0.2 0.6 29.4 27.0 1178.54 21.73 100 0.3 0.2 32.0 19.9 1873.55 20.42 100 0.3 0.4 26.5 21.2 1054.27 19.81 100 0.3 0.6 35.2 25.5 1175.92 17.65
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