# Determining Due Dates for Rush Orders

Christopher Mundt, Eilis Bernadette Beck, Hermann Lödding

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Christopher Mundt, Eilis Bernadette Beck, Hermann Lödding. Determining Due Dates for Rush Orders. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.293-301, 10.1007/978-3-031-16407-1_35. hal-05386070

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# Determining Due Dates for Rush Orders
## A Simple and Effective Order Acceptance Procedure

Christopher Mundt✉, Eilis Bernadette Beck, and Hermann Lödding

Hamburg University of Technology, Hamburg, Germany
{christopher.mundt,eilis.beck,loedding}@tuhh.de

**Abstract.** Determining realistic delivery dates is difficult for make-to-order manufacturers, especially when customers demand both long and short delivery times (rush orders). In this article, we show a procedure for determining delivery dates for rush orders by extending an approach developed for standard orders. Our procedure only uses early available order information and visualizes it in a throughput diagram. A simulative evaluation proves its effectiveness by achieving both short delivery times and a high delivery compliance.

**Keywords:** Order acceptance · Throughput diagram · Rush orders

## 1 Introduction

Logistical objectives, such as the delivery time, are important decision criteria for customers of make-to-order manufacturers in particular, which is confirmed by recent studies among German manufacturing companies [1, 2].

Many make-to-order manufacturers offer different delivery time classes, such as standard orders or rush orders [2]. This enables them to meet different delivery time requirements. Often, customers are even accepting higher prices for shorter delivery times [3, 4].

Even without rush orders, determining realistic delivery times is difficult, since the incoming order load is uncertain. To begin with, it is unclear whether an offer is accepted by the customer or not. Furthermore, in the early stage of order acceptance management, detailed production schedules with the corresponding routings and work stations are not yet available for planning. Moreover, there is uncertainty about the work content of an order.

Rush orders make planning even more difficult as they are received at short notice. In many cases capacity may already be utilized when rush orders occur. Also, prioritizing rush orders over standard orders increases the throughput times of standard orders [3]. If rush orders are not taken into account in production planning, they will lead to backlogs and schedule deviations, especially of standard orders.

This article introduces a scheduling procedure with throughput diagrams that reserves production capacities for rush orders, so that companies can schedule rush orders at short notice without overloading production. In the following, a brief overview of the
fundamentals and the state of research is given (Sect. 2), the scheduling procedure for rush orders is explained (Sect. 3) and the evaluation results are presented (Sect. 4), before a conclusion is drawn (Sect. 5).

## 2 Fundamentals and Current State of Research

Rush orders are prioritized over standard orders in the workstation queues [5]. Often sequencing by due dates is used to implement this prioritization [6]. Based on the operating curve theory [7] and the funnel model [5], Trzyna employs balance equations to model the effect of rush orders on the throughput times of standard orders [4]. Logistic operating curves illustrate this relation. At low work in progress (WIP) levels, rush orders do not have significantly shorter throughput times than standard orders, since all orders can be processed without long waiting times. Conversely, rush orders have much shorter throughput times than standard orders at high WIP levels, due to their prioritization. However, rush orders thereby also delay standard orders and thus extend their throughput time [3, 4].

Because of the turbulence that rush orders can cause, companies should consciously decide whether and to what extent rush orders offer an advantage [4, 8].

In order acceptance management, rush orders pose an especially great challenge, because of their short term occurrence and their influence on standard orders. Due to the arrival of rush orders on short notice, the available capacities may already be fully utilized. Production may be overloaded, if rush orders will be accepted anyway. Delays, especially for standard orders, would be the consequence, if capacities cannot be adjusted in the short preparation time available. Different approaches from the fields of mathematical optimization, machine learning and procedures based on production models exist to address these challenges.

Procedures based on mathematical optimizations reschedule orders in many cases. They require a comparatively large amount of input data at the work system level, which is usually not available at the early stage of scheduling. The mathematical descriptions are similarly complex, which makes implementation difficult (cf. [9–11]).

Procedures based on machine learning often optimize scheduling at the work system level. Moreover, these methods require comparatively large amounts of detailed input data, such as work plans and capacity requirements at the work system level, which are usually not available at the early scheduling stage. The use of machine learning requires technical expertise, which is not available in many companies (cf. [12, 13]).

The procedure described in this paper is based on models of logistic objectives, such as the funnel model. This approach is also pursued by Kingsman et al., Land and Trzyna (cf. [4, 14, 15]). Especially, in such procedures some authors avoid overloading production by reserving capacity for rush orders (cf. [4, 16]). However, this approach contains the risk of capacity utilization losses, if fewer rush orders are received than expected. Some of these existing scheduling procedures do not consider rush orders separately.

Trzyna, however, takes a different approach to schedule rush orders in his dissertation [4]. His procedure is based on reserving capacities for rush orders. It uses a cumulative input graph in a throughput diagram after a previous throughput scheduling. The procedure checks whether sufficient capacity is available on the planned start date of a rush
order. During the capacity check, Trzyna allows the planned capacity to be exceeded within limits. Since the focus of the dissertation is on rush orders rather than on order acceptance, Trzyna simplifies the order acceptance process. He assumes an existing pool of orders, thereby neglecting the uncertainty about incoming orders, the customer response time and the customers' delivery time requests. Additionally, scheduling takes place at work station level, which requires the existence of detailed production schedules.

A procedure previously presented by the authors, inspired by the method of Trzyna, shows how companies can schedule standard orders using a throughput diagram [17]. The procedure requires comparatively few input parameters and enables effective scheduling on the level of the entire production. Figure 1 shows the respective throughput diagram of the procedure.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Throughput diagram for the basic scheduling procedure [17]</figcaption></figure>

The procedure compares the planned output and the possible output with the planned capacity. The planned output includes both confirmed orders and offers awaiting a decision by the customer. The capacity requirements of the latter are discounted by the expected acceptance rate. The difference between the possible output with planned capacity and the planned output with offers results in the unplanned output, which is used for capacity allocation for incoming requests. In addition to capacity allocation, the procedure also determines the delivery time of an offer on the basis of the customer's requested delivery time and the average throughput time of production.

This approach is associated with the risk of capacity utilisation losses due to capacity reservation under uncertainties. If more offers than assumed are rejected by the customer, the reserved capacities can often not be used due to the short time in advance. However, the procedure can show a high schedule compliance and the capacity utilisation losses can be significantly reduced by releasing orders according to ConWIP [17] or by a short-term capacity reduction. However, rush orders have not been taken into account in this procedure so far.
# 3 Scheduling Rush Orders in a Throughput Diagram

The following sections describe a scheduling procedure for rush orders. Therefore, the construction of the necessary throughput diagram is described (3.1) and the scheduling procedure itself is presented (3.2).

The basic idea is to allocate part of the capacity for rush orders and thus avoid competition with standard orders, as for example suggested by Trzyna (cf. [3]). This approach presents the opportunity to offer short delivery times without sacrificing the schedule of standard orders. Main objectives of the procedure are (1) to secure the logistic objectives of delivery reliability and utilization, (2) a simple implementation with as little input data as possible and (3) a transparent representation of the scheduling situation.

## 3.1 Construction of the Throughput Diagram

The scheduling procedure for rush orders, like the original procedure, is based on the comparison of the planned output with the planned capacity in a throughput diagram. The object of observation is the entire production, as well as the complete offer and order throughput. Possible units to measure the output are standard hours (total work content), the number of orders or other capacity equivalents. The planned output contains orders in different statuses, such as released orders, orders waiting for release, orders in the indirect departments, and open offers with a work content adjusted by the acceptance rate. Figure 2 shows such a throughput diagram.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Throughput diagram for scheduling involving rush orders</figcaption></figure>

The possible output with planned capacity starts with the actual output on the scheduling date and results from the integration of the planned capacity over time. The curve rises linearly at a constant planned capacity and its slope corresponds to the capacity. In
order to avoid the turbulences caused by rush orders, the possible output with maximum planned capacity is split into two parts, the possible output for standard orders and the possible output for rush orders. Therefore, the maximum planned output from capacities is multiplied with the rush order share resulting in a capacity reserved for incoming rush orders (1). The possible output from planned capacities for standard orders is reduced accordingly by the rush order share (2).

$$OUT_{\text{pos,rush}}(t) = OUT_{\text{act,rush}}(T_0) + \sigma_{\text{rush}} \cdot (t - T_0) \cdot \text{CAPA}_{\text{plan}} \quad (1)$$

$$OUT_{\text{pos, std}}(t) = OUT_{\text{pos}}(t) - OUT_{\text{pos, rush}}(t) \quad (2)$$

with

$OUT_{\text{pos,rush}}$ : possible output with planned capacity for rush orders.

$OUT_{\text{pos, std}}$ : possible output with planned capacity for standard orders.

$OUT_{\text{pos}}$ : possible output with planned capacity.

$OUT_{\text{act,rush}}$ : actual output of rush orders.

$\sigma_{\text{rush}}$ : rush order share.

$t$ : time.

$T_0$ : scheduling date.

$\text{CAPA}_{\text{plan}}$ : planned capacity.

The unplanned output is the difference between the possible output and the already planned output. It is essential for the procedure to decide on the schedule of further orders.

## 3.2 Scheduling Rush Orders

Scheduling is carried out separately for rush orders and standard orders. Beforehand planned throughput times for standard and rush orders are determined according to the equations presented by Trzyna [4]. Standard orders are scheduled in the basic scheduling procedure described in Sect. 2 with the planned capacity for standard orders as a reference [17].

The scheduling procedure for rush orders determines the delivery time of the offer, its preliminary planned due date, and after customer feedback the final planned due date of the order.

**Step 1 - Determining a Realistic Delivery Time.** Firstly, the procedure determines the possible delivery time of a rush order based on three conditions.

*Condition 1.* The procedure detects the first date at which the cumulative unplanned capacity for rush orders achieves or exceeds the expected work content of the offer. Scheduling the offer on any earlier date would lead to backlogs and therefore lateness.

*Condition 2.* The procedure checks, if the throughput time of the offer is not shorter than the necessary throughput time for rush orders as determined by the procedure of Trzyna [4] to avoid unrealistic short throughput times.
**Condition 3.** The procedure takes the delivery time, requested by the customer, into account. This ensures that the order is not scheduled earlier than desired by the customer.

The possible delivery time is then calculated by adding a delivery time buffer to the throughput time. The delivery time buffer is intended to provide additional protection against uncertainties in production and associated delays. The company can promise this possible delivery time to the customer in the offer.

**Step 2 – Adding the Offer to the Planned Output.** The offer can now be added to the planned output to update the throughput diagram. The procedure determines the work content that has to be scheduled and calculates the estimated due date of the offer. The work content is discounted with the acceptance rate, thus taking the possible rejection of offers by the customers into account. The procedure calculates the estimated due date for an offer starting from the scheduling date. An estimate of the customer response time is added to the scheduling date since the delivery time does not begin until the customer has placed an order. Moreover, the previously determined delivery time, minus the delivery time buffer, is added, as the delivery time buffer is not taken into account in internal order scheduling. Note that both the acceptance rate and the customer response time may vary between standard and rush orders.

**Step 3 – Adjusting the Planned Output after the Customer Response.** After the customer response, offers are removed from the schedule in the case of acceptance as well as in the case of rejection. For confirmed orders, the procedure determines the planned due date of the order. In contrast to offers, orders are considered with their entire expected work content in the planned output. The planned due date for confirmed orders is based on the actual confirmation date. Again, the previously determined possible delivery time minus the delivery time buffer is added to the actual confirmation date.

# 4 Evaluation by Simulation

The following sections describe the evaluation of the presented scheduling procedure for rush orders. Firstly, the simulation model is described and assumptions are being made. Secondly, the results of the simulation experiments are presented.

## 4.1 Simulation Model

Simulation experiments were carried out to evaluate the procedure in a basic state and to show the potentials of the procedure in combination with production control. The simulation experiments were carried out with a simulation model in the Plant Simulation software. In addition to production, the simulation model also includes areas upstream of production, such as the offer preparation process.

The simulated factory consists of seven work stations, each of which can process one order. These work stations are arranged in a job shop layout. Each work station performs a unique operation and has its own queue. Sequencing and order release can be varied to explore different configurations of the PPC. All workstations prioritize the orders according to the planned order due date to fast-track rush orders. This is a simplification.
Especially if the position of workstations in the routings varies, companies should use an earliest operation due date policy instead. The order release is either Constant Work in Process (ConWIP) or due date based. Capacities are purposely not controlled in the experiments in order to be able to assess the unmodified effect of the scheduling procedure on the logistical objectives.

Orders are divided into rush orders and standard orders. On average, the simulated share of rush orders is 20%. In the short term, the share of rush orders can differ, as each incoming request is randomly assigned. This causes varying inter arrival times between rush orders. Rush and standard orders differ in particular in the average customer response time and the maximum delivery time accepted by the customer. The average customer response time for standard orders is five shop calendar days and for rush orders three shop calendar days. Rush order customers accept a maximum delivery time of six days. Below this value, customers accept an order with an acceptance rate of 50%. Customers of standard orders have no restriction on the maximum delivery time and accept any order with an average acceptance rate of 50%. A delivery time buffer of one day is used for both order types.

Both rush and standard orders are further divided into ten variants. In the long run, the proportion of variants is equally distributed, but for each new order the variant is randomly selected. The average work content across all variants is seven hours. The work content varies between the variants with a coefficient of variance of one.

The simulation experiments were simulated with a balanced ratio of expected load to capacity. In each experimental run, 10,000 customer requests were simulated. The offer request rate fluctuated in the short term, due to varying inter arrival times.

## 4.2 Simulation Results

The aim of the simulation experiments is to evaluate the procedure in terms of delivery compliance, throughput times and utilization. For this purpose, the process was simulated (1) in a basic state and (2) in combination with a ConWIP control. Table 1 shows the results of the simulation experiments.

In the basic state, an order release according to the planned start date and a sequencing according to the planned due date were selected. A high delivery compliance of nearly 99% could be observed for the different order types. The throughput times are short and do not fluctuate excessively. As for the standard procedure, utilisation losses were observed. Especially rush orders are vulnerable to this. These losses result from the fact that capacities are released at short notice due to a higher than assumed share of rejected offers for rush orders. These capacities cannot subsequently be reassigned.

Afterwards, the procedure was combined with order release according to ConWIP with an unlimited advance release window. The sequencing after the planned due date was maintained in order to still prioritize rush orders in production. Compared to the basic state, the following changes occur:

(1) The overall utilization is increasing from 87.7% to 95%. In particular, capacity utilization for rush orders increases from 68.5% to 88.3%.

(2) Delivery reliability increases to 100% due to the combination with ConWIP control. Orders are completed slightly earlier on average due to the advance release window.
(3) Average throughput times increase slightly. This is due to the advanced release window, but it has no negative impact on the delivery times.

(4) The effect of the ConWIP control on the variance of throughput times is very small.

The implementation of the ConWIP control is consequently even more important in the presence of rush orders.

<table><caption>**Table 1.** Results of the simulation experiments</caption><thead><tr><th rowspan="3">Simulation experiment</th><th rowspan="3">Order type</th><th colspan="4">Logistical objectives</th></tr><tr><th rowspan="2">Delivery compliance</th><th colspan="2">Throughput time</th><th rowspan="2">Utilization</th></tr><tr><th>Mean</th><th>SD</th></tr></thead><tbody><tr><td rowspan="3">Basic state</td><td>All Orders</td><td>98.7%</td><td>2.3 SCD</td><td>1.4 SCD</td><td>87.7%</td></tr><tr><td>Rush Orders</td><td>98.6%</td><td>0.3 SCD</td><td>0.6 SCD</td><td>68.5%</td></tr><tr><td>Standard Orders</td><td>98.7%</td><td>2.6 SCD</td><td>1.3 SCD</td><td>92.5%</td></tr><tr><td rowspan="3">With ConWIP</td><td>All Orders</td><td>100%</td><td>2.6 SCD</td><td>1.4 SCD</td><td>95.0%</td></tr><tr><td>Rush Orders</td><td>100%</td><td>0.7 SCD</td><td>0.7 SCD</td><td>88.3%</td></tr><tr><td>Standard Orders</td><td>100%</td><td>3.0 SCD</td><td>1.1 SCD</td><td>96.7%</td></tr></tbody></table>

SD: Standard Deviation SCD: Shop Calendar Day

# 5 Summary and Outlook

The presented paper shows how companies can schedule rush orders using a throughput diagram. This procedure enables companies to schedule rush orders, taking backlogs, previous orders and the expected load of open offers into account. Compared to other procedures, rush orders are taken into consideration from the beginning, so that rescheduling is not necessary. In addition, the procedure works with very little information that is already available at an early stage.

In an evaluation we were able to show the general effectiveness of the procedure, especially if it is combined with a ConWIP control. Without a ConWIP control, a very high delivery reliability and short throughput times can be achieved. However, there were losses in capacity utilisation due to a higher than expected proportion of rejected offers for rush orders at short notice.

Yet the advance release window included in the ConWIP control can avoid more than half of these losses and further improve delivery reliability.

Nevertheless, further simulation experiments are necessary to investigate the process in greater detail. For example, it is not yet clear how robust the procedure reacts to deviations in the planning values. For example, the influence of a systematically higher proportion of rush orders, than taken into account, is unclear. The robustness has to be investigated in further simulation studies.
**Acknowledgement.** The Authors would like to thank Deutsche Forschungsgesellschaft (DFG) for funding the project "Model-based order acceptance and scheduling" (Project No. 405645389).

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