Exact methods for order acceptance and scheduling on unrelated parallel machines

Highlights

• The order acceptance and scheduling (OAS) problem on unrelated parallel machines has been studied from the viewpoint of exact solutions.

• Two MIP formulations have been proposed. One is based on a dummy job while the second is based on linear ordering variables.

• Formulation tightening and valid inequalities have been proposed to improve the efficiency of the two proposed MIP formulations.

• A formulation-based branch-and-bound algorithm has been developed based on the concept of “divide and conquer”.

• Computational results for various instances demonstrate the effective- ness of the proposed branch-and-bound algorithm.

Abstract

This paper studies an order acceptance and scheduling (OAS) problem on unrelated parallel machines to maximize the total net revenue of accepted orders, which is the difference between sum of revenues and total weighted tardiness. Two mixed-integer programming (MIP) models are formulated, which are further improved with various enhancement techniques. A formulation-based branch-and-bound algorithm is developed in an attempt to handle complicated instances following the principle of “divide and conquer”. Extensive computational experiments on various instances are conducted, and the results demonstrate the efficiency of the enhancement techniques for the formulations, as well as the effectiveness and efficiency of the formulation-based branch-and-bound algorithm. The proposed branch-and-bound algorithm can optimally solve instances with up to 50 jobs and different number of machines within the time limit of half an hour.

Introduction

To maintain operational agility and flexibility, many companies from different industries, such as engineering tooling, industrial boilers, construction and contracting, adopt the make-to-order (MTO) operational philosophy, thereby laying more focus on customer satisfaction (Mestry et al., 2011). Additionally, there exists an increase in popularity of the MTO philosophy in the service industry, particularly with regard to E-commerce and O2O takeout & catering services offered within restaurants. For example, the ele.me online platform for O2O takeout & catering service covered more than 200 cities in China and had served more than 260 million customers by June 2017 (source from www.ele.me). In this context, how to coordinate operations and sales for effective use of available resource (or limited capacity) is a big challenge for improving customer satisfaction meanwhile obtaining high profit margins.

The order acceptance and scheduling (OAS) problem arises in different MTO production and/or service systems, wherein limited production and/or service capacity and order-delivery requirements necessitate the use of selective order acceptance to satisfy distinct requirements of customers whilst also maximizing total revenue (profit) (Cesaret, Oğuz, Salman, 2012, Rom, Slotnick, 2009, Silva, Subramanian, Pessoa, 2018, Slotnick, Morton, 2007, Wang, Huang, Hu, Cheng, 2015).

The OAS problem requires one to simultaneously determine which orders should be accepted for processing as well as their corresponding schedule. The complexity of the problem due to their combinatorial nature and intertwined decisions makes optimization extremely difficult, as the problem typically is NP-hard (Ghosh, 1997). Such problems, however, invariably capture the rich and realistic classes of MTO processing, thereby making them easy to explain and tempting to be attempted and solved optimally.

In this paper, a deterministic OAS problem on unrelated parallel machines is addressed, since the prevalence of actual manufacturing environments and service industries are typically equipped with unrelated parallel machines. Typical applications of unrelated parallel machine scheduling include—but are not limited to—semiconductor manufacturing (Şen, Bülbül, 2015, Detienne, Dauzère-Pérès, Yugma, 2011, Shim, Kim, 2007), multiprocessor computer (Fanjul-Peyro and Ruiz, 2010), operating rooms in hospitals (Fanjul-Peyro and Ruiz, 2012), and car factories and food processing plant (Fanjul-Peyro et al., 2017). For the problem under study, there exists a pool of potential orders (jobs) with known processing times, due dates, revenues, and penalty tardiness weights. The objective here is to maximize the total net revenue, which refers to the difference between the sum of revenues obtained from accepted jobs and total weighted tardiness. This paper investigates the problem from an exact solution viewpoint. The contributions of this research are as follows:

• Two MIP formulations are proposed: one is with a dummy job and another is based on linear ordering variables. Formulation tightening and valid inequalities are proposed to improve the efficiency of the two MIP formulations.

• A formulation-based branch-and-bound algorithm is developed based on the idea of “divide and conquer”, in which the branch is to determine how many jobs should be accepted, followed by the subproblem of unrelated parallel machine scheduling.

• The computational results on various instances show the efficiency of the formulations with enhancement techniques, and demonstrate the efficacy of the proposed branch-and-bound algorithm.

The rest of this paper is organized as follows. Section 2 presents literature review relevant to the problem. In Section 3, two MIP models are formulated. In Section 4, some enhancement techniques for the MIP models are presented. The proposed formulation-based branch-and-bound algorithm is described in Section 5. In Section 6, extensive computational experiments are conducted to evaluate the performance of the developed models and the branch-and-bound algorithm. Lastly, Section 7 concludes the paper and suggests some future research directions.