Journal of the Operational Research Society (2007) 58, 413–422
r 2007 Operational Research Society Ltd. All rights reserved. 0160-5682/07 $30.00 www.palgrave-journals.com/jors
A robust optimization model for production planning of perishable products SCH Leung1*, KK Lai1,2, W-L Ng1 and Y Wu3 1
City University of Hong Kong, Hong Kong; 2Hunan University, Hunan, China; and 3University of Southampton, Southampton, UK In this study, a robust optimization model is developed to solve production planning problems for perishable products in an uncertain environment in which the setup costs, production costs, labour costs, inventory costs, and workforce changing costs are minimized. Using the concept of postponement, the production process for perishable products is differentiated into two phases to better utilize the resources. By adjusting penalty parameters, decision-makers can determine an optimal production loading plan and better utilize resources while considering different economic growth scenarios. A case from a Hong Kong plush toy company is studied and the characteristics of perishable products are discussed. Numerical results demonstrate the robustness and effectiveness of the proposed model. An analysis of the trade-off between solution robustness and model robustness is also presented. Journal of the Operational Research Society (2007) 58, 413–422. doi:10.1057/palgrave.jors.2602159 Published online 22 February 2006 Keywords: production management; stochastic programming; robustness; perishable
Introduction This study is particularly motivated by the problem faced by a company manufacturing two types of plush toy products— animal-type plush toy products with light music, and Christmas-theme plush toy products with Christmas songs. From the sales report, it is known that animal-type plush toy product can be sold throughout the year, while the Christmas-theme plush toy products can only be sold in November and December as Christmas gifts. Furthermore, demand for the Christmas-theme plush toy products in these two months comprises over 50% of the company’s total sales volume for the entire year. In addition, it is noted that the demand for Christmas-theme plush toy products is time-sensitive, and dramatically increases as 25th December approaches. It is also recognized that the Christmas-theme plush toy product is perishable; in this aspect, it is similar to fashion and high-technology products in that they all have a short lifecycle. The product is comprised of two components, that is, a musical box mounted on a printed circuit board (PCB) and an outside body with plush fabric finish. The final assembly is a spliced fabric body with musical box. From here on, we refer to the musical box as the semi-finished product and to the outer body with musical box as the finished product. Based on reports and from past experience, the current annual production planning strategy is to produce animal*Correspondence: SCH Leung, Department of Management Sciences, City University of Hong Kong, Kowloon, Hong Kong. E-mail:
[email protected]
type products from January to October and Christmastheme products for the remaining two months. In order to fulfil the substantial increase in demand in December, currently more Christmas-theme plush toy products are produced and stored in November in order to compensate for the limited resources available in December. Moreover, it may not be practical for the company to subcontract production to other manufacturers owing to quality control, special technologies, plant specifications and client contracts. In addition, adding more machines leads to higher capital and maintenance costs, and the machines may be underutilized during the non-peak season. A backorder strategy is not appropriate here since this will not only lower customer satisfaction, damaging the company’s goodwill and customer loyalty, but also does not deal with the fact that demand falls once the special event is over. No customer is willing to send Christmas gifts at any time other than Christmas. In this study, we employ the postponement concept to produce semi-finished products in November and transfer semi-finished products to finished products in December so that the resources can be better utilized to meet the dramatic growth in demand. In production planning, postponement refers to a common intermediate product being manufactured in a first phase, and, according to the differentiating options—such as colour, sizes and types, production line activities such as dyeing, compounding, final assembling, packaging and so on—being postponed to a second phase until customer orders are received (van Hoek, 2001; Aviv and Federgruen, 2001a, b). The most recent review of postponement can be found in van Hoek (2001). A
414 Journal of the Operational Research Society Vol. 58, No. 4
well-known real-life postponement example is the redesign of the European DeskJet Printer line by Hewlett Packard, as illustrated by Lee and Billington (1994). Van Hoek (2001) reviewed the literature on postponement and identified postponement opportunities in operations. Pagh and Cooper (1998) stated that the advantage of postponement is the reduction or full elimination of risk and uncertainty in manufacturing and logistics operations. Garg and Tang (1997) investigated two points of postponement in the manufacturing stage—early postponement and late postponement—and investigated the importance of demand variabilities, correlations and the relative magnitudes of the lead times in determining the appropriate points of differentiation. Aviv and Federgruen (2001b) studied the benefit of postponement with unknown demand distributions. Many real-world planning problems involve noisy, incomplete or erroneous data. In the last four decades, many studies have addressed the formulation of risk-averse decision-making in stochastic programming models. Høyland and Wallace (2001) proposed a mean-variance model, in which risk aversion can be formulated in stochastic linear programming using a piecewise linear utility function. A robust optimization model (a special type of stochastic nonlinear programming model) was developed, in which a concave risk aversion function can be incorporated in the specification of the objectives (Bai et al, 1997). Robust optimization, presented by Mulvey and Ruszczynski (1995) and Mulvey et al (1995), is able to tackle decision-makers’ favoured risk aversion or service-level function, and has yielded a series of solutions that are progressively less sensitive to realizations of the data in a scenario set. The optimal solution provided by a robust optimization model is called robust if it remains ‘close’ to the optimal if input data change: this is called solution robustness. The solution is also called robust if it is ‘almost’ feasible for small changes in the input data: this is termed model robustness. Robust optimization integrates a goal programming formulation with a scenario-based description of input data. The model generates a series of solutions that are progressively less sensitive to realizations of input data from a set of scenarios. The purpose of this study is to develop a robust optimization model to optimize the production planning problem for perishable products under an uncertain environment, from which we can determine (1) how many finished products should be produced from raw materials directly (direct production), (2) how many semi-finished products should be produced from raw materials (master production), and (3) how many finished products should be produced from semi-finished products (final assembly) so that resources can be better utilized to meet the dramatic growth in demand and total costs (consisting of the setup costs, production costs, labour costs, inventory costs, hiring costs and lay-off costs) can be minimized. This paper has been organized as follows. After this introduction, the background to robust optimization model is described. Then
a robust optimization model is formulated to solve the production planning problem for perishable products using a postponement strategy, and a set of data from the toy company is used to test the effectiveness and efficiency of the proposed model. Conclusions are given in the final section.
Framework of robust optimization model Robust optimization incorporates a goal programming structure with a set of scenarios involving stochastic inputs. The aim of robust optimization is to obtain a set of solutions that are robust with respect to changes in the realization of the input data from a scenario set. Goal programming consists of system and goal constraints. Robust optimization includes two distinct constraints: a structural constraint and a control constraint. Structural constraints are formulated following the concept of linear programming and its input data are free of any noise, while control constraints are taken as auxiliary constraints influenced by noisy data. Moreover, two sets of variables—design and control—are defined. Unlike a design variable, which cannot be adjusted once a specific realization of the data has been observed, a control variable is subject to adjustment once uncertain parameters are observed. In the following, the framework of robust optimization is briefly described. First, let x 2
