Integrated production planning and order acceptance under uncertainty: A robust optimization approach

Abstract

The aim of this paper is to formulate a model that integrates production planning and order acceptance decisions while taking into account demand uncertainty and capturing the effects of congestion. Orders/customers are classified into classes based on their marginal revenue and their level of variability in order quantity (demand variance). The proposed integrated model provides the flexibility to decide on the fraction of demand to be satisfied from each customer class, giving the planner the choice of selecting among the highly profitable yet risky orders or less profitable but possibly more stable orders. Furthermore, when the production stage exceeds a critical utilization level, it suffers the consequences of congestion via elongated lead-times which results in backorders and erodes the firm’s revenue. Through order acceptance decisions, the planner can maintain a reasonable level of utilization and hence avoid increasing delays in production lead times. A robust optimization (RO) approach is adapted to model demand uncertainty and non-linear clearing functions characterize the relationship between throughput and workload to reflect the effects of congestion on production lead times. Illustrative simulation and numerical experiments show characteristics of the integrated model, the effects of congestion and variability, and the value of integrating production planning and order acceptance decisions.

Introduction

Models of production and inventory systems have been developed since the early days of the Operations Research and Management Science field. A major concern in the area has been to formulate models that can be solved efficiently, yet these models should not be based on over simplifying assumptions. Classical production planning models determine the minimum cost or maximum profit production plans in order to meet pre-specified demands. In classical production planning models, three common simplistic assumptions are usually made: (1) The demand is deterministic and known in advance. (2) The production rate or throughput, and consequently the lead time, does not depend on the utilization of the resources. That is, even if the utilization is very high congestion effects that result in increasing lead times is not taken into consideration. (3) The demand is an aggregation of several customer orders without distinction. This means that even if backlogging or shortages are accepted, the customers are not distinguished.

With respect to the first assumption, it is well known that orders are usually subject to great uncertainty in terms of order size and due date which can be critical especially for manufacturers with long production lead times. In the case of semiconductor manufacturing for example, well in advance of the ultimate due date, customers provide an indication, a “demand signal”, of what their orders will ultimately be. As time evolves and after assessment of their needs customers adjust their orders (quantities and due dates) until a “firm order” is obtained. Despite the manner in which orders may change after being signaled, customers still require that orders be met within a short period after their eventual due date, even though this date may only be known with limited advance notice (Kempf, 2004, Higle and Kempf, 2010). The uncertainty inherent in orders should affect both production planning and order acceptance decisions.

One of the most popular frameworks for planning under uncertainty is stochastic programming (Kall and Wallace, 1994, Prékopa, 1995, Birge and Louveaux, 1997). Uncertainty is represented by using a number of discrete scenarios to represent possible future states, which allows stochastic linear programs to be modelled as large linear programming problems. A number of authors have formulated production planning problems as multi-stage stochastic linear programs (M-SLPs) (Peters et al., 1977, Higle and Kempf, 2010), but the approach presents challenges. A significant difficulty of M-SLPs is that the problem size tends to grow exponentially with the number of possible realizations (scenarios) of uncertain parameters, requiring solution methods that exploit their special structure. Chance constrained programming is another possibility to model production planning under uncertainty. It was considered, for example, by Charnes and Cooper, 1959, Charnes and Cooper, 1963, Charnes et al., 1958, Prékopa, 1995. In chance constrained programming, constraints can be violated with a specified probability, which is quite useful to model, for instance, service levels in supply chain problems (Gupta and Maranas, 2003). This approach achieves a substantial decrease in the size of the model, and avoids the problem of defining the penalty function. However, it fails to capture the cost consequences of constraint violations, which can result in anomalous behavior (Blau, 1974). Leung and Wu (2004) develop a robust optimization model to solve the stochastic aggregate production planning problem. Raa and Aghezzaf (2005) use robust optimization to obtain a dynamic planning strategy for the stochastic lot-sizing problem. However, this approach has similar drawbacks to two-stage stochastic programming: (i) the number of scenarios increases with the number of uncertain parameters, leading to an increase in the problem size and (ii) the penalty function describing the relation between constraint violation and achievable profit is difficult to define. The robust optimization (RO) approach followed in this paper is within the framework developed by Ben-Tal and Nemirovski (2000). A recent attempt to model inventory systems using the RO framework was done by Bertsimas and Thiele (2006).

The dependency between resource utilization and lead times (or equivalently available capacity) has been addressed to some degree by some authors. As a result of using queuing models in production planning and scheduling, Hopp and Spearman (2001) show that lead times increase non-linearly as system utilization increases and approaches 100%. Voss and Woodruff (2003) propose a nonlinear model where the function linking lead time to workload is approximated as a piecewise linear function. Ettl et al. (2000) take a similar approach, adding a convex term representing the cost of carrying work-in-process (WIP) as a function of workload to the objective function. Graves, 1986, Karmarkar, 1989, Missbauer, 2002, Asmundsson et al., 2006, Asmundsson et al., 2009 use clearing functions (CFs) to model the dependency between workload and lead times. Several related models are proposed in the recent book by Hackman (2008). Pahl et al., 2005, Missbauer and Uzsoy, 2010 review production planning models with load-dependent lead times. Aouam and Uzsoy (2012) compare the performance of various production planning models with workload-dependent lead times under demand uncertainty. In this paper, the proposed formulations use CFs to model the relationship between throughput and WIP levels. Two production modes are distinguished based on a pre-specified critical utilization level: low utilization mode and high utilization mode. In the latter mode, congestion effects are taken into consideration, i.e., when utilization approaches 100% lead times become increasingly higher.

Grouping orders of customers in a single demand (per time period, for example) is part of aggregation decisions made on data in order to simplify the planning models, or for managerial purposes. However, in practice customer orders need to be distinguished for several reasons. Firstly, even if the finished good is the same, different customers might impose particular conditions on the source of the raw material (this was partially addressed in Brahimi et al. (2006)) or on the quality control tests made during the manufacturing process of their orders. Secondly, there are situations where the planners need to satisfy the demands partially. This happens in case of shortages or for profitability reasons. If shortages happen under the form of backlogs or lost sales, the planners have to decide which order will not be satisfied properly. Furthermore, even if there is enough capacity to avoid shortage, it is not always clear that all orders should be accepted even if the unit price the customer will pay exceeds the variable production cost. There are two possible arguments for this fact. The first reason has to do with economies of scale. In fact, in the case of high fixed or set-up costs it might not be economical to satisfy a single order of a small quantity. The order must be aggregated with additional orders to justify the production setup (Geunes et al., 2006). The second reason has to do with the workload of the production stage. Kefeli et al. (2011) show that the marginal prices of capacitated resources are not necessarily equal to zero when the utilization is less than one. This means that even in the case where capacity is available, the revenue from an additional order should at least offset the variable production cost plus the dual of the capacity constraints that take into account workload. Therefore, models that integrate production planning decisions and order acceptance decisions have a great potential to improve the overall profitability of the firm.

Ivanescu et al. (2002) consider the order acceptance problem in the batch industries where the processing times are uncertain. The authors use regression based models in order to determine whether there is enough capacity to accept a customer order with the due date requested by the customer. Markov decision models are used by Defregger and Kuhn (2005) to decide about the orders to accept or to reject in a planning process over a number of periods. Geunes et al. (2002) consider a production planning problem with order acceptance and call it the “order selection” problem. This problem is a single item lot sizing problem with and without resource capacity constraints. The uncapacitated case is solved using a polynomial time algorithm and propose a Lagrangian relaxation approach for the capacitated case. For a more extensive review of order acceptance literature the reader is referred to Slotnick (2011).

This paper presents a robust model that integrates production planning and order acceptance decisions. To the best of our knowledge, our model (which is a production planning model under uncertainty) is the first model to incorporate: (i) integration of production planning and order acceptance decisions, (ii) a robust optimization approach to model demand uncertainty with demand signals and firm orders, (iii) two production modes based on utilization, reflecting the effects of congestion, (iv) and multiple customer classes. The contributions of the paper are summarized as follows:

• Based on a critical utilization level two production modes are distinguished, low utilization mode in which lead time is assumed to be fixed and equal to an estimated planned lead time, and a high utilization mode in which a non-linear clearing function defines the relationship between throughput and WIP level. That is, in the high utilization mode, congestion effects are taken into consideration and reflect the increasing delays in lead times with increase in utilization. The integration of the order acceptance decision allows the planner to adjust the utilization in order to avoid delays and customer waiting while maximizing profit.

• The presented model incorporates, using a robust optimization approach, demand uncertainty which is inherent in order quantities. A customer order begins as a “demand signal” which represents a demand forecast. As time progresses, the quantity and due date change until the order becomes a “firm order” that must be satisfied. Orders/customers are classified into classes based on their marginal revenue and their level of variability in order quantity (demand variance). The proposed robust model provides the planner with the flexibility to decide among the highly profitable yet risky orders or less profitable but possibly more stable orders. The decision maker can adjust his/her level of conservatism through the budget of uncertainty and shortage penalty and accordingly decide on the fraction of demand from each class that the company commits to satisfy.

• A simulation study was conducted to illustrate the value of integrating production planning and order acceptance decisions and to analyze the effect of variability and congestion. In fact, integrating the two decisions provides the planner with the flexibility to select a reasonable number of orders to be satisfied. This flexibility enables the planner to maintain release quantities and utilization at desirable levels, which leads to high profits and high levels of customer satisfaction.

The rest of the paper is organized as follows. In Section 2, two production planning models based on CFs are presented; one without congestion and the other considers congestion effects. Section 3 summarizes the main concepts of the RO approach and formulate a robust production planning model where demands are aggregated. The integrated production planning and order acceptance models are formulated in Section 4 and its robust counterpart is formulated in Section 5. Section 6 presents numerical experiment and a simulation study to compare the models. The work is concluded in Section 7.