A decision support to assign mould due date at customer enquiry stage in computer-integrated manufacturing (CIM) environments

Achieving better delivery reliability performance is crucial for mould industry. Due date (DD) assignment at customer enquiry stage has a large influence on the kind of performance. This paper focuses on estimating the feasible production lead times of the new arriving mould orders that could be used as a decisional support during DD quoting, given the total workload already in the system at a decision time. An uncertain parameter model firstly addresses the uncertainties in the underlying problem using a discrete time Markov chain, which can obtain the initial probability distribute of the completion date of each new order. Such a model has not been addressed previously in related literature. Then, a mould progress evolution approach with limited capacity checks each realising track contained in the initial probability distribute to meet various capacity limitations. The expectation value of the checked distribute is the estimation value of the completion time of the examining order. The application examples illustrate possible applications of the approach. Further, the simulation experiments-based comparison of the proposed approach with two benchmarks adopted commonly in the real-world mould production is provided, and the results are promising as compared to benchmark decision rules.

Keywords: mould industry; due date assignment; Markov chain; capacity planning; decision support

Mould production is a typical case of Engineering-To-Order (ETO) systems. In mould companies, the arrival of customers' orders is stochastic over time. Each arriving mould is a different product, usually requiring different routings and processing times through production facilities according to the customer's specification (Silva et al.[20], Wang et al.[22]). Among the arriving moulds, due to different constraints from available material and capacity, the company is able to fulfil them only within proper production lead times (PLT).

A poor delivery reliability performance is not necessarily due to bad production scheduling in mould industry. Their origin may well be wrong commitments with reference to too short due date (DD) made in the tendering phase by the sales department (Kingsman [13], Slotnick [21]). On the other hand, if too long DD is quoted to the customer, so as to absorb possible delays, there could be problems of short-term competitiveness, in case company's competitors are able to promise faster deliveries (Alfieri [1]). Under this condition, mould companies have to estimate the feasible PLT for each arriving order at customer enquiry stage. The DD quoted to the customer should be decided on the basis of the PLT estimation (Liu et al.[15]).

The main problem to estimate the PLT in such environments is to determine the actual total workload status of the system. The total workload of the mould production system often consists of three types of workload: the released workload, the unreleased workload and the unconfirmed workload (Liu et al.[15]). The first type of workload derives from the orders called released orders that have been already confirmed and released to the shop floor. The origin of the second type of workload is the orders called unreleased orders that have been confirmed but have not been yet released. The unconfirmed workload derives from the third type of orders called unconfirmed orders which have just entered the system and are still underassessment by the company. The first two types of workload are easily obtainable from the production planning system in computer-integrated manufacturing (CIM) environments, which is common in most of the mould companies (Silva et al.[20]). However, various uncertainties in the mould manufacturing make it difficult to determine the third type of workload. They are: (1) the uncertainties of the bill of materials (BOM) and critical path; (2) the uncertainties of the chosen facilities for each processing stage in the routing; (3) the uncertainties of the number of the unconfirmed orders with different acceptance probabilities.

Thus, the aim of this paper is to present a model in order to estimate the feasible mould PLT that could be used as a decisional support during mould DD quoting at the customer enquiry stage. The analysis is carried out taking into account the total workload at different levels of time aggregation, where determining the unconfirmed workload is the main difficulty in CIM environments. The proposed approach has to simultaneously consider the complex mould BOM structure with random critical path, the uncertain processing facilities for each processing stage and the complex interaction of multiple unconfirmed orders.

The remainder of the paper is organised as follows: A literature review on similar problems is presented in Section 2. Section 3 addresses the uncertainties in the underlying problem using a discrete time Markov chain, which can model correlations among the uncertain parameters for the unconfirmed orders. Based on the Markov model, the mould progress evolution approach with limited capacity is developed in Section 4. In Section 5, the simulation experiments-based comparison analysis of the proposed approaches with two benchmarks adopted commonly in the real-world mould production is provided. Section 6 concludes the paper.

The increasing importance of assigning reliable DDs has spawned various models for addressing this problem (Hopp and Roof Sturgis [11]). Cheng and Gupta ([3]) surveyed DD assignment approaches and divided them into: (1) exogenous; and (2) endogenous methods. The former does not account for any information regarding the arriving order, orders already in the manufacturing system, future order arrivals or shop structure when assigning DDs of the new arriving orders. On the other hand, the latter accounts for information such as the characteristics of both accepted orders and potential orders. Most modelling approaches address setting endogenous DDs (Hopp and Roof Sturgis [11]).

When speaking of DD, it often distinguishes internal DD from external one in endogenous methods (Alfieri [1]). Internal DD is the date when managers expect the order to be completed, which usually equals to the summation of the arrival time and the PLT of the examining order. While external DD is the delivery date firm promises to customer, which equals to the summation of the internal DD and a safety time to account for the transportation and the uncertainty of PLT, especially in cases when there are huge penalty for late delivery. The safety time can be treated as a static parameter or dynamic decision variable (Moses et al.[16]). The two main research streams on PLT estimation are based upon static and dynamic problems.

In static problems, the information of all orders in manufacturing system is available at a given time, such as the routing, the resource required by a project task and the critical path. Two common static approaches are total work content (TWK) (Conway [5]) and processing plus waiting (PPW) (Kanet and Christy [12]). More information concerning the current work in progress, available resources, and activity precedence relationships provided a better DD estimation for a new order than the TWK (Dumond and Mabert [7]). Fry et al. ([9]) used a DD setting method based on the total work on the critical path (TWKCP). Park et al. ([17]) developed a heuristic algorithm (called HDDDA) to find reliable DD for each order by considering the actual residual capacity of the bottleneck resource. In Kingsman's ([13]) method, a heuristic algorithm (called WORKCON) focuses on capacity test rather than on lead time verification. Corti et al. ([6]) integrates the HDDDA and the WORKCON to develop a capacity-driven approach, which can verify the feasibility of the DD required by a new order by comparing the capacity requirements of the order with the actual available capacity of the system. The application-related researches are quite limited in complex production systems (Zorzini et al.[25]). Such a gap between theory and practice can be partially explained by the tendency of such methods to rely on provisional data that are not often available in real environment, because in practice most of the information is dynamic and uncertain on future arrivals.

To make the PLT estimation closer to the real-world situations, the dynamic environments should be considered. The information of orders available for processing varies over time. Currently, most of the related research adopts the simulation-based methods to deal with the dynamic problem, such as Weeks ([23]), Roman and Del Valle ([18]), Grant et al. ([10]), Alfieri ([1]), Weigert and Henlich ([24]). Such a method needs to maintain the complex simulation model, and the simulation lead time is enormous.

Not much research has been performed on the capacity planning methods on PLT estimation in dynamic environments. In Moses et al. ([16]), the DD assignment is tackled by considering the time phased availability of resources. When a new order arrives, considering the dynamic availability of resources, a starting time, and then the PLT is estimated. Sawik ([19]) proposes a simple integer programming approach for multi-objective DD setting over a rolling planning horizon in a dynamic Make-To-Order (MTO) environment. In Liu et al. ([15]), a multi-agent-based DD setting approach is proposed by considering the uncertainties of processing facilities for the project tasks in mould manufacturing.

Motivated by the literature, the main disadvantage of previous research is the lack of a capacity planning method which is able to assign the DD by considering the actual workload of the system in dynamic environments. Hence, in this paper, a comprehensive capacity-planning method is proposed to estimate the PLT of a new arriving order at the customer enquiry stage of mould manufacturing by taking into account the actual workload situations. In order to consider various uncertainties in dynamic mould manufacturing, the proposed method will adopt an original probabilistic model by means of Markov chains. To the best knowledge of the authors, this is a new and significant contribution from the theoretical perspective.

The mould production system under study is a flexible flowshop that consists of m processing stages in series, where each stage is made up of non-identical parallel work centres. In the system, various types of mould products are manufactured according to customer orders, where each product type usually consists of multiple types of mould component. Similarly, each component type consists of multiple mould parts. Each part requires processing in various stages, however some parts may bypass some stages. The customer orders are single product type orders.

The PLT estimating decisions over a rolling planning horizon are usually made periodically upon the arrival of a number of orders in a specific time interval (the batching interval, e.g. a day). Let be the set of the unconfirmed orders collected over a batching interval. At the tendering phase, the mould BOM structure and processing routing of an order cannot be determined immediately because of their complexities and uncertainties. Nevertheless, it can be determined which mould product types the order roughly belongs to. In CIM environments, various mould product types that had once been produced in the company are easily classified by means of the statistical analysis of historical data. Each product has a fixed BOM structure, i.e. the components contained in the mould product are fixed. Moreover, among the parts contained in these components, the key parts usually have much longer processing time than other ones. The key part actually decides the manufacturing lead times of the corresponding components. Order i can always roughly match with a product type, i.e. it has the approximate BOM structure and routing with the product type. Let be the set of the components of the matching product type. The PLT of order i depends on the manufacturing lead times of the key parts contained in each component .

Each key part requires processing in multiple stages. Each stage is made up of multiple non-identical parallel work centres, and it is assumed that the key part only requires one of the work centres for processing in a stage. In actual, which work centres are chosen is uncertain. It appears that such uncertain parameters in related literature have not been addressed previously. In this paper, the probabilistic correlation among the uncertain parameters is based on the premise that the work centre, workload (i.e. capacity requirement) and duration of adjacent processing stages in the routing of a key part are correlated. For example, if a current stage is completed by the work centre with higher precision, then the precision of the work centre chosen in the next stage also tends to be higher. The assumption is appropriate for mould manufacturing because the parts requiring higher processing precision are more likely to choose the work centres with higher precision than other common parts from the point of view of manufacturing cost. In general, the correlation can exist between any two stages in the routing of a key part and can be modelled by introducing corresponding transition probability. However, this paper assumes the probabilistic correlation between two adjacent stages only for simplification. The probabilistic correlation among uncertain parameters can be modelled with discrete time Markov chains. The jth stage of a key part has mj realisations and each realisation consists of the values of the chosen work centre, the required workload, and the duration of the stage from a discrete set as shown in Figure 1. For example, 'WC11, WL11, D11' (the first realisation set of the stage 1 for the key part 1 in Figure 1) shows that the stage 1 of the key part 1 chooses WC11 for processing with WL11 workload and D11 duration. The possible discrete values for the parameters may represent the actual values or the mean values for the parameters (Choi et al.[4]). An explicit representation of the probabilistic correlation of the uncertain parameters in a key part with five processing stages is shown in Figure 1.

Graph: Figure 1. Uncertain parameter modelling for a key part with five processing stages.

Here, there are 3, 2, 3, 2 and 2 possible realisations for processing stage 1, 2, 3, 4 and 5 of the key part 1, respectively. A Markov model for the key part is defined with five probability matrices (vector), PI1, PM11, PM21, PM31 and PM41. The realisation of the stage 1 is governed by 'initial probability vector' (PI1), which consists of three probabilities and for different potential realisation of this stage. The realisation of the second stage are conditioned by the realisation result of the previous stage and governed by PI1 by PM11 transition matrices. The occurrence probabilities for the two realisations in the second stage can be determined by the following computation.

Graph

The summation of each column of the matrix is equal to 1 and the kth column of the matrix represents a conditional probability vector when the previous stage is completed with the kth realisation. The realisation linked with red dashed lines in Figure 1 represents the realising track of '1 (work centre WC11, workload WL11, duration D11)-2 (work centre WC22, workload WL22, duration D22)-1 (work centre WC31, workload WL31, duration D31)-1 (work centre WC41, workload WL41, duration D41)-2 (work centre WC52, workload WL52, duration D52)': each number represents realisation index of the key part 1. For the key part 1, 2 and 3, there can be 72, 24 and 18 possible tracks, respectively. With the uncertain parameter model representation, the illustrative example with 31,104 possible realising tracks are represented with the parameters for order O1 shown in Figure 1. In a word, the proposed representation compactly represents quite complex interactions between stage outcomes.

Let Ei be the set of the realising tracks of the key parts of various components for the mould product type that matches with an order . For a track , the completion date and the PLT of this track are denoted by ECDie and PLTie, respectively. The value of ECDie can be computed by Equation (1).

Graph

where ERDi is the earliest release date of order i, which can be estimated based on the lead time of customer response stage and the lead time of material procurement stage. Dei,j is the duration of the jth processing stage of the track e, which can be obtained based on the Markov model. In addition, the occurrence probability Pie of the track e can be computed by Equation (2).

Graph

where Pei,j is the occurrence probability for the realisation of the jth stage in track e, which can be obtained by the Markov model. Once all of tracks in the set Ei are considered, an initial probability distribute for the completion dates of the order i is able to be determined.

The obtained initial probability distribute is based on the mean durations of various processing stages. In fact, it is a mould progress evolution approach with infinite capacity. In the remainder of this paper, the approach is called IC-MPE approach. The IC-MPE allows the setting of two different dates with reference to each processing stage of a track e: stage starting date (SSD) and stage completion date (SCD). Hence, for the jth stage in the track, the period-of-time (SSD, SCD) available for completing the stage can be determined from the scheduling. In addition, the acceptance probability (AP) of the unconfirmed order has to be considered at the customer enquiry stage (Kingsman and Mercer [14]). Thus, the unconfirmed workload UWei,j of the jth stage for the track e of order i can be computed by Equation (3).

Graph

where APi denotes the acceptance probability of order i. WLei,j denotes the workload required by the jth stage in the track e of order i, which can be obtained from the Markov model. It is supposed that the workload required by a stage in a track is considered to be equally split among time units requested for its processing (SSD, SCD).

Each work centre can only provide limited available capacity. Therefore, the completion date of a track in the initial probability distribute is possibly infeasible due to not enough capacity. In this case, a limited-capacity mould progress evolution called LC-MPE approach in the remainder of this paper has to be implemented in order to get the feasible completion date of each track. The overall flowchart is shown in Figure 2.

Graph: Figure 2. The overall flowchart of mould progress evolution with limited capacity.

In this paper, the priorities of the unconfirmed orders are dispatched based on the same approach proposed by Ebadian et al. ([8]). The interested reader is referred to the reference. Then the unconfirmed orders contained in O are sorted in the non-increasing order of their priorities: {O1, O2, ... ,Oi, ... , On}.

Different components may compete for the same resources amongst themselves. Thus, each of them has to be prioritised as well. Considering the real-world situations of mould production, the mould components are prioritised based on the complexities of the component BOM structure, i.e. these components with more complex structure have the higher priorities. The components contained by Oi are sorted in the non-increasing order of their priorities: Ci = {Ci1, Ci2, ... , Cil, ... , Cili}. For the work centres chosen by various processing stages in a realising track of Cil, their unconfirmed workload in the current total workload derives from the other components with higher priorities than Cil. Let CHP be the set of these components with higher priorities.

It can form a group of tracks by stochastically selecting one track from the set of tracks of each component contained in CHP. Let E(Cil) be the number of tracks of Cil in CHP. Then, the number of the above different groups of track can be determined by Equation (4).

Graph

The unconfirmed workload of each group of tracks in Ng can be computed based on the workload required by various processing stages in each track contained in this group. Thus, the current total workload (which includes the released workload, the unreleased workload and the unconfirmed workload of the group under considering) on the work centres for a track e of Cil, i.e. the available capacity of these work centres in the period-of-time (SSD, SCD) can be determined. Comparing the available capacity to the workload required by the various processing stages in the track under considering, the feasibility of the completion date of the track in the initial probability distribute can be checked. If it is feasible, the completion date of the track is saved, otherwise, it has to add the durations of corresponding stages in the track until the capacity is enough, the modified completion date of the track is then saved. Hence, there are Ng completion dates for the track e of Cil, and each completion date corresponds to an occurrence probability which is the product of the occurrence probability of each track in the corresponding group of tracks (including the occurrence probability of the track e of itself). When all of the tracks of Cil are considered, the probability distribution of its completion date can be determined. Further, doing this component by component, the probability distribution of the completion date of the order can be determined, and its mathematical expectation can also be computed.

At customer enquiry stage, the customers only consider the suppliers that provide the bid before a deadline. The deadline is often very short, even requested to answer immediately sometimes. Hence, the order-winning fundamental for mould companies is to set the DD rapidly.

With the quantity increase of the unconfirmed orders, the calculation times for estimating the completion dates of all of these orders are enormous so that it cannot be allowed in a real bid process (it suffers from what Bellman ([2]) referred to as 'the curse of dimensionality', meaning that its computational requirements grow exponentially with the number of orders). An approximate approach has to be proposed in order to get the faster calculation speed. For this purpose, this paper reduces the state space via eliminating the states with low probabilities in the Markov model. A probability threshold value (PTV) is set first. The probabilities in probability matrices will be set at 0 if they are lower than PTV. The other probabilities which are greater than PTV will be set again based on their proportion of original probabilities. An example is illustrated in Figure 3. In the proposed approximate approach, the realising tracks with low occurrence probabilities will not be taken into account as well.

Graph: Figure 3. An example for modifying probability matrix based on the PTV.

In this section, the simple computational examples are presented to illustrate possible applications of the proposed approach. The examples are modelled after a real-world shop floor for tyre-mould products. The shop floor is a flexible flowshop consisting of five processing stages with parallel work centres. The first two stages design the mould. This stage is carried out in the design department. The team of mould designers can similarly be considered as a type of work centre. Then the designed mould is decomposed into the corresponding components, and they will be respectively produced in the next two stages which consist of computer numerical control (CNC)/electro-erosion operations. Finally, various components are assembled together in the fifth stage.

Heretofore, the authors have developed a production planning system (PPS) for the shop floor, using the following tools: Microsoft visual studio 2005 (C#), SQL server 2005. The PLT estimation module for newly arrived mould orders in PPS has embedded the LC-MPE approach. Under this condition, some real data from the module will be utilised to conduct the examples, shown in Table 1. In the examples, two unconfirmed orders were selected. Factors 2, 3 and 6 in Table 1 present their related parameters. Table 2 shows the workload and duration required by each processing stage of various components for the two product types, and the various transition probability matrices for adjacent processing stages are shown in Table 3.

Table 1. Configuration of the computational examples.

Table 2. Workload (h)/duration (d) required by each processing stage of various components for the two mould product types.

Table 3. Transition probability matrices for adjacent processing stages.

The computational outcomes were obtained by means of running the PLT estimation module in PPS, shown in Figure 4 (where P is the occurrence probability of a track; D is the PLT of a track; E is the expectation value of the PLT for the order). For O1, 576 realising tracks can be achieved and thus, probability distribution and expectation value of the PLT of this order can be determined, shown in Figure 4(a). Similarly, 147,456 tracks for O2 can be achieved, for which probability distribution and expectation values are shown in Figure 4(b). Hence, the estimation values of the completion dates for O1 and O2 are 20 July 2011 and 23 July 2011, respectively.

Graph: Figure 4. The computational outcomes. (a) Probability distribution of the completion dates for O1. (b) Probability distribution of the completion dates for O2. (c) Probability distribution of the completion dates for O2 based on the approximate approach with PTV = 0.35.

Based on the same parameters in above computation, the proposed approximate approach with PTV = 0.35 is carried out as follows. For O2, only 544 realising tracks are obtained. The new probability distribution and expectation value for this order are shown in Figure 4(c). The estimation values of the completion dates for O2 are 20 July 2011. Hence, the number of tracks can be greatly decreased in the approximate approach (from 147,456 to 544), consequently improving the calculating speed. Moreover, the difference between two estimation values is little relative to the whole PLT. Thus, the approximate approach may be more appropriate for the real production. Nevertheless, the outcomes of the approach strongly depend on the PTV. It is often difficult to be properly set at the real-world mould production.

In this section, the experiments-based comparison of the proposed IC-MPE and LC-MPE approaches with two benchmarks adopted commonly in the real-world mould production is provided. In the remainder of this paper, the two benchmarks are called: AvPLT-based-I and AvPLT-based-II, respectively. In AvPLT-based-I, PLT estimation of a new arriving order is based on the average PLT of these already completed orders that belong to the same product type with the new order. In AvPLT-based-II, PLT estimation of a new arriving order is based on the average PLT of these already completed orders that belong to the same product type and have the same importance rating with the new order. According to the approach proposed by Ebadian et al. ([8]), the importance of a new arriving order includes three ratings: high, medium and low.

The simulation model is developed using the software Tecnomatix Plant Simulation 8.2, shown in Figure 5. The characteristics of the shop are summarised in Table 4. The shop consists of six processing stages. Each stage contains multiple non-identical work centres, where each contains a single machine and can process only one job at a time. No pre-emptions are allowed. The shop is able to produce three types of mould product. The Markov model related to the uncertain parameters of each product type has been known and embedded into the simulation model. Considering the real situation of mould production, a hybrid dispatching rule is used on the shop floor.

Graph: Figure 5. Simulation model.

Table 4. Configuration of the simulation experiments.

The batching interval for making PLT estimation decisions is set at 5 days. Order arrivals follow Poisson process. Moreover, in order to analyse the impact of the order arrival intensity, two different arrival rates (i.e. mean 2 and 3 orders per batching interval) are used. The index of product type matching with a new arriving order is uniformly distributed between one and three. In addition, each arriving order is randomly dispatched an importance rating. Two different sets of the proportion relevant to the number of the new arriving orders with high, medium and low importance (i.e. 1:7:2 and 3:5:2) are used in order to analyse the impact of the important order arrival intensity. In mould manufacturing, the orders with lower AP have more possibility to be rejected. Hence, whether a new order will be released into the shop floor finally is randomly determined based on its AP in the simulation model. The LC-MPE approach takes into account this uncertainty by Equation (3). A distribution (Uniform [0.8, 1]) is used to randomly produce the AP for each arriving order. The characteristics of arriving orders are summarised in Table 4.

In order to assess the impact of four decision rules, specific performance criteria must be selected. Due date-related performance measures of flowshop performance were primarily considered, which are indicative of customer satisfaction and deliverability under different PLT estimation decisions: percentage of early orders, mean earliness, percentage of tardy orders and mean tardiness.

Simulation data were collected with reference to the steady state of the system. To remove the effects of the warm-up period, all statistics were set to zero and restarted after a warm-up period of 1000 working (simulated) days. Statistics were then collected for 2000 days, where the data of first 1000 days was used to obtain various average PLTs and durations (such as average PLTs of various product types, average PLTs of various product types with different importance ratings). Based on the above statistics, the four decision rules were carried out after 1000 days to compute the estimation values of the completion date of each new arriving order, and the actual values of the completion times were obtained from the collected simulation data. Comparing the two values, the relevant performance measures were determined. Ten replications were performed for each set of experimental conditions and the average is reported.

The preceding section discusses the design of the experiments, from which the outcomes are obtained by means of running the developed simulation model. This section will analyse the results of the simulation for each of the decision rules considered. Main attention is paid, for each performance measure, to analyse the impact of the order arrival intensity (i.e. order arrival intensity + proportion of the orders with high, medium and low importance ratings: mean 2 order arrivals per 5 days + 1:7:2, mean 2 order arrivals per 5 days + 3:5:2, mean 3 order arrivals per 5 days + 1:7:2 and mean 3 order arrivals per 5 days + 3:5:2).

The results concerning the percentage of tardy orders under different order arrival intensity are presented in Figure 6. It can be seen that, under all order arrival intensities, the percentages of tardy orders for LC-MPE rule are far lower than the ones for the other three rules. Increasing the order arrival intensity or the proportion of the orders with high rating enables the percentage of tardy orders for LC-MPE rule to add slightly. Moreover, the LC-MPE seems to be more sensitive to the parameter related to the proportion of the orders with high rating. In addition, the percentages of tardy orders for the AvPLT-based-I, AvPLT-based-II and IC-MPE rules under various order arrival intensities are highly approximate.

Graph: Figure 6. Percentage of tardy orders under different order arrival intensity. Experimental settings are shown in Table 4.

Figure 7 compares the performance of the four decision rules on two sets of conflicting objectives (Mean earliness vs. Mean tardiness) under various order arrival intensities. This figure shows that four points of the LC-MPE is in the left of all points for the other three rules, which means the tardiness-related performance measure of the LC-MPE is better than the other three rules under all order arrival intensities. But on the other hand, the earliness-related performance measure for the LC-MPE is approximate with the one for the IC-MPE. This is because the LC-MPE only increases the completion dates of the realising tracks with infeasible capacity in the initial probability distribute obtained by the IC-MPE, and the completion dates of the realising tracks with feasible capacity are not modified. In addition, the LC-MPE has slightly larger mean tardiness under higher proportion of the orders with high rating. For same order arrival intensity, both earliness- and tardiness-related performance measures for the IC-MPE are better than the ones of the AvPLT-based-I and AvPLT-based-II. But the AvPLT-based-II under lower proportion of the orders with high rating is better than the IC-MPE with higher proportion of the orders with high rating. The AvPLT-based-I is the worst in the four decision rules. In the simulation process, the running time of LC-MPE is much longer than the ones of the other rules, even in the case of few order arrivals in the batching interval. Once the number of the arriving orders increases, the LC-MPE seems impossible to be carried out within a proper lead time. In this case, the presented approximate approach has to be adopted.

Graph: Figure 7. Performance of the four decision rules on two sets of conflicting objective (Mean earliness vs. Mean tardiness). Experimental settings are shown in Table 4.

With more and more competition in mould industry, the more reliable delivery performance is emphasised by the customers. The effective approach to assign the due dates of the new arriving orders at customer enquiry stage is strategically important for this performance. Most of the existing researches dealing with the issues are still rarely suitable for practical mould production.

The approaches proposed in this paper serve this purpose, being a decision support tool that can help tendering preparation at the customer enquiry stage. The reference context is dynamic environments. At first, this study addresses the uncertainties in the underlying problem using a discrete time Markov chain, which is able to model correlations among the uncertain parameters for the unconfirmed orders. This is a new contribution from the theoretical perspective. Then, the mould progress evolution approach with limited capacity is developed based on the Markov model, which can obtain the probability distribution of the feasible completion date of each new arriving order. Moreover, the application examples illustrate possible applications of the approach. Finally, the simulation experiments-based comparison of the proposed approaches with two benchmarks adopted commonly in the real-world mould production is provided, and the results are promising as compared to benchmark decision rules.

A further work is to assess the performances of the proposed approaches in the real-world mould production. For this purpose, the authors will further apply the PPS in the cooperative mould company, consequently obtaining the experiences for setting various parameters required by the approaches, etc. The internal DD is only assigned based on the PLT estimation so that the research on external one assignment has to be carried out in the future. A method for considering mould DD and price assignment simultaneously should be an extension of the proposed approach.

The authors wish to thank the anonymous referees for providing helpful suggestions. This research was jointly supported by the National Science and Technology Ministry (2012BAF12B10), the National Natural Science Foundation of China (51175094, 51205068) and the Guangdong Natural Science Foundation (S2012040007784).