Faculty of Economics and Business

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24 June 2019 Master Thesis Page | 1

Mathematical programming-based heuristic for

production inventory systems

with seasonal demand, high product variety, and resource constraints

Research Paper

University of Groningen

Faculty of Economics and Business

Student name: Thijs van de Minkelis

Student mail: G.T.van.de.Minkelis@student.rug.nl

Student number: S3509311

Semester: 2A (2019)

Supervisor: Dr. N.D. van Foreest

Second assessor: Dr. W.M.C. van Wezel

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Abstract

Purpose

The purpose of this research is to propose an efficient planning policy that optimises inventory levels and maximise profit in a multi-product capacitated system, with seasonal demand, modularized products, and production quantities as multiples of a certain batch size.

Methodology

Based on the system characteristics of a case company, the basic product mix planning model of Hopp & Spearman (2011) is extended to a mathematical programming-based heuristic. Simulation is used to test the heuristic and used to investigate possible system improvements.

Findings

A linear model in combination with iterations can be used to find a solution for a complex mixed-integer-program, while keeping a global perspective on the solution space.

It moreover shows that adjustment of parameters can be used to improve system performance.

Research limitation/implications

The heuristic needs accurate forecasting to anticipate early enough on seasonal trends. Moreover scheduling and corresponding set-ups are neglected. Some fixed amount of capacity is reserved to deal with incorrect forecasting and to ensure a feasible schedule is possible. Due to the way the heuristic is built, it is more difficult to apply to systems with other characteristics.

Value

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Preface

Let them gather all the food of these good years that are coming and store up grain under the authority of Pharaoh for food in the cities, and let them keep it. That food shall be a reserve for the land against the seven years of famine that are to occur in the land of Egypt, so that the land may not perish through the famine.”

GENESIS 41:35-36 (ESV; ±1400 BCE)

The citation above shows that the essence of the Dutch proverb “foresight is the essence of

government” has been applied for centuries. Good government has a proactive approach,

instead of reactive damage control. This research paper describes a heuristic that is the result of a journey to find a proactive approach to prevent stock-outs in a particular production environment. The question to answer is the same as the Egyptians faced: how much do we have to store at which time to make sure we are able to satisfy demand in periods available capacity is not sufficient.

The followed journey of half a year contained a lot of iterative steps. From this place, I want to thank all the people who have sharpened my thoughts, gave insights, and helped me to conduct this research. I especially want to thank Dr. N.D. van Foreest for his time and counsel throughout the process. I also want to thank Dr. W.M.C. Wezel, L. Visser, G.M. Loer, K. Scheur, and A. Kasper for the feedback they have given. Finally, I also want to thank my housemates, family, and girlfriend for their support during my master. They made my place to study a place where I am home.

This report is the final stage towards graduation. I hope you enjoy reading. Personally, I have learned a lot about inventory management that I certainly will use in my future career.

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1. Introduction

A consumer electronics manufacturer in the Netherlands faces planning difficulties due to increasing product differentiation and seasonal demand patterns. As a result of mass customisation, the product variety has in the last five years increased from 100 to 366 products, while the total demand has remained the same. The combination with finite production capacity forces the company to adapt its planning methods to be able to satisfy customer demand efficiently. Currently, the lack of adaption results in a high amount of inventory. The challenge is to alter the planning methods to deal with the new mass customisation in place.

In general, the pattern of seasonal demand makes the management of inventory difficult. In periods of high demand, there must be enough inventory to satisfy customer demand, while in periods with less demand less inventory is desired to reduce holding costs and risk of obsolescence. A high variety of products makes this more challenging because products must share the available production capacity, making the inventory important as a buffer against stock-outs and scheduling conflicts caused by not only variation in demand, but also in production or setup times (Winands, Adan and Van Houtum, 2011).

Besides capacity restrictions, other process restrictions can make planning decisions difficult. In practice, manufacturers are for instance constrained by the existent production methods and corresponding material handling in place. Many processes are dedicated to multiples of fixed quantities measured in amount, weight, or volume (Zipkin, 2000). This also holds for the two main successive production processes at the case company.

The system under investigation is consequently a make-to-stock (MTS) production environment of multiple standardized products in a capacitated system and belongs therefore to the category of Capacitated Multi-Item Lot Sizing Problems (CLSPs) (Drexl and Kimms, 1997). Extra additions are the seasonal demand with corresponding anticipation inventory to ensure a high service level. Moreover, the modularisation of products and the production quantities as multiples of a constant batch size must be taken into account. The main aim of solving the CLSP is to determine which products need to be produced in which period and quantity to maximise profit (or minimize overall costs).

The general approach to solve a capacitated single-item lot-sizing problem with seasonal demand is to calculate the required lot-sizes per period. Demand that cannot be produced in a certain period due to limited capacity is assigned to earlier periods to prevent stock-outs. This is, in general, a relatively simple problem that can be solved exact (Salomon, 1991).

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24 June 2019 Master Thesis Page | 7 literature, most effective methods to solve the multi-item problem are heuristics (Karimi, Fatemi Ghomi and Wilson, 2003).

Metters (1998) has for instance proposed five general rules for production planning with seasonal demand in a capacitated system. The rules can be used for prioritising products and decide how early to start with producing anticipation inventory, which products should be produced in advance, and what should be done if capacity is still insufficient. However, despite the insight the rules provide in the prioritisation of products, there is a lack of an efficient way to apply these rules in a high variety system with 366 different products.

Recently a more quantitative approach is investigated by Grewal, Enns, and Rogers (2015). They used simulation in combination with many iterations to find the parameter configuration that optimises the inventory level in a similar production system. Their method shows dynamic adjustment of parameters can be used to increase system performance. However, their conclusion was that the approach was not easy enough to use in practice and more usable methods are required.

An easy to solve model that incorporates many system characteristics is proposed by Hopp & Spearman (2011). However, the model does not take multiple stages and batch restrictions into account, what is needed to represent the system properly.

A model that takes these batch restrictions into account in combination with a multi-item system facing seasonal demand, modularisation, and capacity constraints, is in all honesty not found in literature. However, taking these multiples of constant batch size into account during the planning decision will make the planning more accurate (Jans and Degraeve, 2008). The main research question is therefore formulated as following:

“What is a robust and efficient planning policy that optimise inventory levels and maximise profit in a multi-product, two-stage, capacitated system, with seasonal demand, modularisation, and order quantities as multiples of constant batch size?”

Based on the system characteristics of the case company, the basic product mix planning model of Hopp & Spearman (2011) is extended to a mathematical programming-based heuristic to incorporate all the constraints of the production system. The heuristic is tested with simulation in combination with data of the case company and used to investigate possible system improvements. This paper focusses on the assignment of lot-sizes to particular periods. Visser (2019) considers the complementary scheduling of the case company products within each period.

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2. Theoretical Background

Aim of this research is to come up with an efficient planning policy to minimise inventory, and thereby maximise profit. Therefore, in this section related literature to various aspects of the model considered in this paper are presented to form the context of this research. First used methods to solve similar CLSP-problems will be discussed. Relevant information about the three additions (seasonal demand, modularisation, and batch restrictions) that is not already addressed in the first part will be discussed afterward. A more in-depth survey regarding CLSPs is done by Karimi, Fatemi Ghomi, and Wilson (2003). More general information regarding both deterministic and stochastic lot sizing problems can be found in Drexl and Kimms (1997), Sox

et al. (1999), Jans and Degraeve (2008), and Winands, Adan and Van Houtum (2011).

2.1 Solving methods CLSP problem

In the past, several authors have researched the multi-item CLSP and proved it is NP-Hard (Karimi, Fatemi Ghomi and Wilson, 2003). The more constraints are added, the more complex the problem becomes. This complexity is further increases if it is a multi-product system. This research differs to most CLSPs in the fact that setups are neglected in the model. This should make finding an optimal solution much easier. However, the required order size as multiple of constant batch quantities makes the problem mixed integer (MIP). Consequently, an exact solution is hard to obtain and many solving methods used for CLSPs in general are applicable to this problem as well.

Based on the literature, planning methods for CLSP can be classified into three main categories that try to find a solution that optimises costs. The first is exact methods, the second common-sense or specialised heuristics and the third belongs to mathematical programming-based heuristics (Karimi, Fatemi Ghomi and Wilson, 2003). The found literature that investigates systems similar to ours belong to the two latter categories. In the next section, useful common-sense heuristics will be discussed. Afterwards, mathematical programming-based heuristics will be elaborated. For more information regarding solving practical lot-sizing problems in an exact way we refer to Belvaux and Wolsey (2003).

2.1.1 Common-sense heuristics

A common-sense heuristic is proposed by Metters (1998) in the form of five simple rules applicable to a CLSP in combination with seasonal demand. The aim is to prioritise products to determine which products should be produced in advance if capacity shortages occur. The first rule is to minimize overcommitment risk, or in other words: excess inventory is often more expensive than calculated deterministically. The second rule is to store capacity cheaply. This entails that items with the lowest holding costs per unit of resource used should be produced first as anticipation inventory. The third rule is that production can be seasonal regardless of

demand. In case multiple products are produced, the overall demand pattern should be used

instead of determining the policy product specific. Rule four states to produce the sure-things

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24 June 2019 Master Thesis Page | 9 first, to prevent a mismatch with reality. The last rule can be applied as crisis control and states that when the plan fails, produce the money-makers.

Although the rules seem also applicable for the system under investigation, Metters (1998) does not give a method how these rules can be integrated into an efficient method to determine lot-sizes. However, with 366 different products it is important to use a more coherent approach to ensure overall feasible and as optimal as possible planning decisions are made (Hopp and Spearman, 2011).

A linear model that implicitly takes the trade-off between the second, third, and fifth rule into account is the basic product mix planning model of Hopp & Spearman (2011). This model tries to maximise profit over a certain planning horizon by optimising the trade-off between revenue and holding costs per product. The solution space is constraint by a minimum required service level and maximum available capacity. The linear model can deal with many different products and can be optimised easily by conventional solvers. This model lacks the constraint batches must be multiples of a fixed batch quantity. Adding this constraint will make the problem MIP and conventional linear solvers cannot come up with a solution anymore. Because this model incorporates many of our system characteristics, it is used as starting point for this research as part of a mathematical programming-based heuristic. This is further elaborated in section 4.

2.1.2 Mathematical programming-based heuristic

A drawback of determining an exact solution for a complex situation is the extensive computational time. A common-sense heuristic, on the other hand, entails the risk of resulting in a far-from-optimum solution (Zipkin, 2000; Wu, Shi and Duffie, 2010). Mathematical programming-based heuristic tries to combine the high quality of an exact solution with the computational easiness of common sense heuristics by trying to find an optimum with a mathematical programming methodology. Usually, programming-based heuristics can be more easily made case-specific than common-sense heuristic. Moreover, because commercial solvers can be used, high computational power is easily at hand (Karimi, Fatemi Ghomi and Wilson, 2003).

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24 June 2019 Master Thesis Page | 10 resorting to extensive simulation-optimization (Grewal, Enns and Rogers, 2015). The idea of changing parameters and make iterations to find an optimal solution can however be applied more generally to improve the heuristic that will be tested in this study.

Wu, Shi, and Duffie (2010) propose a Hybrid Nested Partitions and Mathematical Programming (HNP-MP) heuristic with a global perspective on the capacitated multi-item lot sizing problem with setup times. By using Nested Partitions, computational effort is focussed on the most promising region of the solution space while maintaining a global perspective on the problem. The most promising region is determined by giving each region a promising index. The most promising region is used for an iteration and evaluated afterwards until no region with a better lower bound is found (Wu, Shi and Duffie, 2010). They prove that this approach outperforms other common algorithms found in literature. However, their method is quite complex and does not incorporate the requirement of order quantities as multiple of a constant batch size. For the heuristic in this research the idea of solving a problem in several computational steps, starting with the most promising region, is used. Although the idea looks similar, it is applied in a different way and should not be confused with the hybrid nested partitions used by Wu, Shi and Duffie (2010).

2.2 Seasonal Demand and Anticipation Inventory

Most mathematical programming-based heuristics that solve similar CLSPs use backlogging, lost sales, or crashing costs (e.g. outsourcing) to deal with capacity shortage (Karimi, Fatemi Ghomi and Wilson, 2003; Grewal, Enns and Rogers, 2015). Consequently, the service level cannot be guaranteed. As mentioned in the introduction we use anticipation inventory to deal with seasonal demand in a capacitated system to ensure a high service level. Previous mentioned simulation optimisation study of Grewal, Enns, and Rogers (2015) is one of the few found quantitative studies that actually use this approach.

The determination of needed anticipation inventory is based on expected future demand in relation to the bottleneck capacity. The accuracy of this forecast determines if the right quantity of the right product is produced in advance or that there is a misfit between actual and predicted demand. The way anticipation inventory is calculated determines how great the influence of the forecast is on the determined lot-sizes and consequently how great the chance of a misfit (Nahmias, 1989). In most previous mathematical models demand is known in advance for a certain planning horizon. For each period, it is determined what the amount of inventory at the beginning of that period must be to be able to satisfy demand in the period, and consequently what should have been produced in the past (see for example Chen & Chang, 2007; Kimms, 1993). However, in reality, the longer the planning horizon, the more likely the forecast is not completely accurate (Nahmias, 1989).

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24 June 2019 Master Thesis Page | 11 period, stochastic fluctuations in demand can be taken into account more accurately and a closer fit with reality is realised.

In our research, the focus is to come up with a planning methodology suitable for our system and investigate the influence of different production characteristics. By using deterministic demand, the effects of seasonal fluctuations and system characteristics can be analysed more easily. However, to use the proposed heuristic properly in practice, updating the forecast throughout the year is inescapable (Nahmias, 1989). This is therefore marked as further research.

2.3 Modularisation of products

A way to decrease variability caused by product differentiation is the use of mass customisation: building products out of several modules that can be configured in multiple ways (Hoek and Peelen, 2010; Nicholas, 2018). This principle is also used at the case company, to reduce the number of different parts. Demand of products that share the same parts can be aggregated into product families to use economics of scale at processes where the same parts are made (Nicholas, 2018).

A heuristic approach that can be used in systems with modularisation is hierarchical planning, which determines a solution in two successive steps. First, a relatively simple model is solved on aggregated family level. The second step translates the aggregated solution to a product-specific planning, by solving a separate model for each family (Nahmias, 1989; Zipkin, 2000). In this way variety is decreased, resulting in a much easier planning problem. This principle is used in our heuristic to plan the bottleneck machine. The main reason was in the first place that aggregation of same parts result in set-up time reduction and not due to computational complexity. This is further explained in section 3 and 4.

2.4 Multiples of constant batch size

All research above lack the constraint that orders must be multiples of a constant batch size. However, as mentioned in the introduction, many processes use batch production. Thereby the required batch size affects system performance. If batch sizes are large, many products are produced at once. In systems with high product differentiation and consequently smaller customer orders, this will result in large inventories because it takes longer to use up each batch. Therefore, the system has higher holding costs. It moreover restricts the flexibility of the planning process to adapt to changing customer demand (Nicholas, 2018). In the rest of this study, we will refer to the inventory caused by required batch sizes as inventory due to ‘overproduction’. This is done to distinguish it from anticipation inventory due to capacity restrictions.

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3. Methodology

To obtain an efficient planning policy for the system under investigation, it is chosen to use a simulation study. Although the system under investigation is based on a case company, it is important to see the influence of parameter changes. The benefit of using simulation instead of real-life implementation of planning policies is that with simulation many scenarios can be investigated in a short amount of time with little costs (Robinson, 2014; Karlsson, 2016). Another approach could have been to use simulation as part of a design science study. The proposed planning policy could have been the artefact, that was investigated by simulation (Holström, Ketokivi and Hameri, 2009). However, it is chosen to classify this research as a simulation study. The main reason is that the proposed heuristic is not implemented at the case company and only theoretically tested with the simulation of many different scenarios. Consequently, not a complete design cycle has been done.

The three main sub-questions that will be analysed are:

1. How can lot-sizes in a multi-product, two-stage, capacitated system, with seasonal demand, modularisation, and order quantities as multiples of constant batch size be determined efficiently resulting in maximisation of profit and optimal inventory levels? 2. What is the influence of reducing the constant batch-sizes at both stages on the inventory

level and overall costs?

3. What is the influence of increasing capacity on the inventory level and overall costs? The simulation study consist of six basic steps. Every step is covered by one of the subsections below. The steps are consecutive: (1) problem formulation, (2) data collection, (3) mathematical modelling, (4) model verification and validation, (5) heuristic implementation and model experimentation, (6) obtaining of results and drawing conclusions (Winston and Goldberg, 2004). Although the steps are numbered, there exists some overlap and iteration between the steps. This is mostly the case between steps three to six.

3.1 Problem formulation

The consumer electronics producer has grouped their 366 products (k) into 30 different families. Products of the same family (j) use (almost) the same basic module that is produced on the bottleneck machine. Different families can belong to the same type (i). The company produces two types of products: low-end, and high-end. This is done with a make-to-stock policy (MTS). Set-up time between families is between 5-10 minutes. A changeover between types requires 25 minutes. Consequently, demand of products belonging to the same

family is grouped, to take advantage of economies of scale. This demand clustering is also applicable to families. In fact, de sequence dependant set-ups are in this way forced to be minimal.

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24 June 2019 Master Thesis Page | 13 As shown in figure 3.2 the system at the company exists of two important steps: the production of basic elements at the bottleneck machine and the customisation of the products. The lead-time (L) for each stage is assumed as a deterministic one week. An important assumption is that the planning for the bottleneck machine with capacity restriction (𝐶) is also feasible for the other processes. As a result, both stages are jointly optimised (see also Ben-Daya, Darwish and Ertogral, 2008; Grewal, Enns and Rogers, 2015). Consequently, this research focuses on the optimisation of lot-sizes for the bottleneck machine. The lot-sizes at production are determined on family level and must be multiples of a constant batch-size 𝑄1. The lot-sizes at the customisation process are for each product separately and must be multiples of a constant batch-size 𝑄2_{. In the current situation, all process batches are constant multiples of 1440 products} clustered on product level.

This paper tries to determine the lot-sizes per period and does not take the sequencing on the bottleneck machine itself into account. Consequently, it is difficult to predict how many changeovers exist in a period due to load-level policies at the company. However, the number of set-ups have a major influence on the available time and therefore on the available capacity that is an important constraint in this research. Moreover, it determines the way demand will be clustered by different heuristics.

Based on the research of (Mussche, 2018) at the same company, it is assumed that the bottleneck machine has a maximum effectiveness of 0.8. The remaining time of 0.2 gives slack for the scheduling process, maintenance, and R&D activities. Furthermore, the extra time makes sure forecast deviations can be dealt with.

3.2 Data collection

The data necessary to perform this research are related to the product mix, the market, and operational constraints. Data about the product mix, production time per product on the bottleneck machine (𝑎_𝑖𝑗), revenue per product (𝑟_𝑖), and holding costs (ℎ_𝑖) can be found in table 3.1. The revenue per product and holding costs are scaled values.

Due to the new situation, only estimations about the demand exists. Consequently, seasonal demand data is generated by using an adapted version of the method used by Grewal, Enns and Rogers (2015). Figure 3.3 shows the total demand on product type level in amount of production time needed.Because products differ in production time on family level, time is chosen as unit to be able to aggregate demand and show it in combination with the available capacity (134.4 hour/week). A mathematical formula is used to translate the needed production time on product type level 𝑖 into customer order quantities on product level 𝑘 per unit of time (𝐷_𝑖𝑗𝑘(𝑡)). In the current situation, the required minimum order quantity for customers is assumed 100 products a time. Details are further elaborated in Appendix E.

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Table 3.1 - Characteristics product families (product tree, production time per product, revenue, and holding costs)

Type 𝒊 Family 𝒋 # products 𝒌/family 𝒋 𝒂𝒊𝒋 [𝑠𝑒𝑐/𝑘] 𝒓𝒊 [€/𝑘] 𝒉𝒊 [€/𝑘] 1 1 14 6.4 200 2 2 14 6.8 200 2 3 12 7.2 200 2 4 18 7.2 200 2 5 16 7.6 200 2 6 16 8.6 200 2 7 13 9.2 200 2 8 14 9.5 200 2 9 12 9.5 200 2 10 14 9.5 200 2 2 11 8 2.4 100 1 12 10 2.5 100 1 13 12 2.6 100 1 14 12 2.8 100 1 15 12 2.8 100 1 16 10 2.9 100 1 17 10 3.0 100 1 18 10 3.0 100 1 19 10 3.0 100 1 20 12 3.0 100 1 21 12 3.1 100 1 22 12 3.1 100 1 23 14 3.2 100 1 24 12 3.5 100 1 25 14 3.5 100 1 26 10 3.5 100 1 27 10 3.5 100 1 28 10 3.5 100 1 29 10 3.5 100 1 30 13 3.5 100 1

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3.3 Mathematical Model

To obtain an efficient planning policy for the system under investigation, a mathematical programming-based heuristic is built and evaluated with a simulation. It is chosen to use a mathematical programming-based heuristic, because the problem itself is too hard to solve in an exact way. On the other hand, only using simple rules entails the risk of resulting in a far-from-optimum solution in a complex system like ours (Zipkin, 2000; Karimi, Fatemi Ghomi and Wilson, 2003; Hopp and Spearman, 2011). The mathematical programming-based heuristic will combine the benefits of both approaches, and simultaneously reduce their disadvantages. Starting point of the mathematical model in this research is the basic model for product mix planning described by Hopp and Spearman (2011, p.565). Main extensions are the modularisation in the system, the two production stages, and the constraint that order quantities must be multiples of constant batch sizes. The conceptual model is described in section 4.

3.4 Model verification and validation

Both verification and a validation of the simulation are needed to guarantee the model is accurate enough to be able to draw conclusions. Verification tries to verify if the model design has been transferred properly to the simulation. Validation makes sure that it is checked if it is allowed to use the conclusions case specific and for generalizing the findings (Robinson, 2014). To verify if the mathematical model itself is right, it is peer reviewed by an expert. The determined lot-sizes by the heuristic are analysed against the constraints in place. The lot-sizes are also used to calculate the overall costs and check whether it corresponds to the costs determined by the model. The main model is built in Python (see Appendix D). A second model is built in Excel1 to verify if the outcomes are the same. Because using Excel costs much computational time, only one scenario is simulated both in Excel and Python. It turned out the outcome of all the steps are completely similar. In this way, it is guaranteed the behaviour of the simulation is in line with the mathematical model.

To validate the built heuristic, a sensitivity analysis is performed to see how sensitive the heuristic is for parameter and variable differences (section 5.3). If high sensitivity exists, it can be concluded that generalisation of the findings is difficult and the found insights are not reliable. Finally, it is important to benchmark the heuristic to validate the quality of the used method.

3.5 Heuristic implementation and model experimentation

To analyse the influence of changing different parameters on the system, the heuristic is implemented in Python to enable easy simulation of different scenarios. The next two paragraphs describe the two main topics that are investigated: reduction of required batch-size, and increase of capacity.

Required batch-size

1_{The build-in solver of Excel is not suitable for large problems. Consequently, the Add-In of OpenSolver.org is}

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24 June 2019 Master Thesis Page | 16 Many scenarios are tested between the current situation where both stages produce in quantities that are multiples of 1440 pieces, and a theoretical case where products can be produced as single items independently (one-piece-flow). The expectation is that for the overall costs, the batch size does not have a great effect, because holding costs per period are much lower than the product revenue (1:100). If these values lay closer to each other, it is expected the effect is more significant and reduction in batch size will reduce the overall cost more.

From an inventory point of view, it is expected smaller batch steps result in production quantities that have a closer fit with the demand pattern and consequently result in a reduction of inventory level due to overproduction. Because production on the bottleneck (stage 1) is clustered on family level an at customisation (stage 2) on product level, it is expected reductions in required batch size at customisation have initially greater impact on the inventory levels than at the bottleneck. On a certain point, it is possible reducing batch sizes at customisation does not contribute any further and it is also required to reduce the quantities at the bottleneck for a more effective process. The hypothesis are tested by using batch sizes at both stages of 1440, 720, 360, 180, 60, and 1. These numbers are chosen, because in the current situation products are produced in multiples of 1440, resulting in one full pallet per batch. The material handling within the processes are done with trays of 60 pieces each. The chosen numbers between 60 and 1440 are logical quantities within these processes. The last case where batches are multiples of one is used as theoretical one-piece-flow where the batch restrictions are relaxed.

Increasing capacity

It is expected that an increase in capacity reduces the need for anticipation inventory throughout the year. Consequently, inventory levels should drop. Because inventory can be stored cheap, the corresponding cost savings will be small. Several scenarios will be tested where capacity will be increased stepwise from 100% to 200%.

3.6 Obtaining results and drawing conclusions

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4. Conceptual modelling

In this section, the basic mathematical model of the system and the heuristics are presented. To simplify the model, several assumptions have been made. The assumptions that are not mentioned before, will be addressed first.

4.1 Assumptions and notation

In this section, the assumptions in the model are addressed. Because this research considers the determination of lot-sizes per period it is assumed there is an unlimited amount of supply of raw materials. This is justified because the determined lot-sizes can be used as forecasting for the procurement of raw material. Although some variance between actual demand and forecasting will exist, it can be expected they are in the same order of scale.

As mentioned earlier, set-up times and costs are neglected. A maximum process effectiveness of 0.8, ensures there is enough time for all the required set-ups due to load-levelling policies in place.

To determine the required lot-sizes, it is assumed no misproduction occurs. It is justified not to decrease the effectivity of the bottleneck machine because the amount of misproduction at company X is very low. The maximum effectiveness of 0.8 ensures there is enough space in the schedule for the production department to deal with possible scrap.

Although in reality production times and lead-times are variable, it is assumed fixed in this research. This is justified because this research determines the required lot-sizes per period. The production time is used in combination with the capacity constraint per period, and the lead-time for the period the production quantity is finished. Only the average lead-times are therefore important.

There is no restriction on the amount of inventory. This is justified because in line with the previous assumption the main aim is to determine lot-sizes that result in the lowest inventory possible. There is on top of that enough storing space for finished products at company X, so no extra space has to be hired during peak seasons.

It is assumed the system has no production budget constraints per period and product costs are fixed. Regardless of the period produced, the overall production costs will be the same. Consequently, the production costs per product are neglected.

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Table 4.1 - Used notation mathematical model and heuristic

Symbol Meaning

DECISION VARIABLES

𝑚𝑖𝑗1(𝑡) Positive integer to quantify the needed amount of batches 𝑄1 to satisfy demand belonging to family 𝑗 of type 𝑖 in period 𝑡

𝑚_𝑖𝑗𝑘2 (𝑡) _{Positive integer to quantify the needed amount of batches}_𝑄2

to satisfy demand product 𝑘 belonging to family 𝑗 of type 𝑖 in period 𝑡

𝑆𝑖𝑗(𝑡) Amount of sold products 𝑘 belonging to family 𝑗 of type 𝑖 in period 𝑡 𝐼𝑖𝑗1(𝑡) Inventory after stage 1 belonging to family 𝑗 of type 𝑖 in period 𝑡

𝐼𝑖𝑗𝑘2 (𝑡) Inventory after stage 2 of product 𝑘 belonging to family 𝑗 of type 𝑖 in period 𝑡

PARAMETERS

𝐼 Set of product types 𝑖

𝐽_𝑖 Set of product families 𝑗 belonging to type 𝑖 𝐾𝑗 Set of products 𝑘 belonging to family 𝑗 of type 𝑖

𝑄1 Constant batch size stage 1, on family level (same size all families; must be equal or multiple of 𝑄2, 𝑄1= 𝑛𝑄2 𝑤𝑖𝑡ℎ 𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑑 𝑏𝑒𝑓𝑜𝑟𝑒ℎ𝑎𝑛𝑑)

𝑄2 Constant batch size stage 2, on product level (same size all products) 𝐶 Total effective production time, in period t [h/period]

𝐿1 Lead-time stage 1 [# period] 𝐿2 Lead-time stage 2 [# period]

𝑙 Required service level on family level [-] (same for all families) 𝑟_𝑖 _{Revenue per sold product of type 𝑖 [€]}

ℎ_𝑖1 Holding costs belonging to products of type 𝑖 at stage 1 [€ per product/period] ℎ_𝑖2 Holding costs belonging to products of type 𝑖 at stage 2 [€ per product/period] 𝑎𝑖𝑗 Time needed to produce one product belonging to family 𝑗 of type 𝑖

[h/product]

𝐷_𝑖𝑗𝑘(𝑡) Demand product 𝑖 belonging to family 𝑗 of type 𝑖 in period 𝑡 [# products k/period]

𝑇 Planning horizon

VARIABLES

𝑃_𝑖𝑗1(𝑡) Amount produced at stage 1 of family 𝑗 of type 𝑖 in period 𝑡

𝑃_𝑖𝑗𝑘2 (𝑡) Amount produced at stage 2 of product 𝑘 belonging to family 𝑗 of type 𝑖 in period 𝑡

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4.2 Mathematical system description

Objective function

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑝𝑟𝑜𝑓𝑖𝑡: 𝑍 = ∑𝑡∈𝑇∑𝑖,𝑗,𝑘(𝑟_𝑖𝑆_𝑖𝑗𝑘(𝑡) −ℎ𝑖(𝐼𝑖𝑗1(𝑡)+𝐼𝑖𝑗𝑘2 (𝑡))) (1)

The objective function (1) is the summation of all the revenue due to sold products, minus the inventory costs in the system over the entire planning horizon 𝑇. Product value ℎ𝑖 is used to distinguish the holding costs for high and low value products. It is chosen to make the assumption that the inventory points have the same weight. Consequently, from a cost perspective, it does not matter in which inventory point the product is at the end of period t. Normally products have more value after more stages. However, the system uses a make-to-stock (MTS) policy and not a make-to-assembly (MTA) policy. The company has enough storage space for end-products and not for inventory between stages. Moreover, although there is no specified capacity restriction on stage 2, it can be assumed it does not have enough capacity available to produce all required demand during peak seasons. To prevent the accumulation of inventory between the two stages, both inventory points are added to the objective function and it is assumed costs are the same.

Constraints

The constraints beneath are the mathematical representation of the system’s characteristics.

Sold products:

𝑙 ∙ 𝐷_𝑖𝑗𝑘(𝑡) ≤ 𝑆_𝑖𝑗𝑘(𝑡) ≤ 𝐷_𝑖𝑗𝑘(𝑡) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽_𝑖, 𝑘 ∈ 𝐾_𝑗, 𝑡 (2)

The first constraint (2) ensures sales per family 𝑗 is not lower than the required service level and not higher than needed demand. The service level must be taken into account if a high service level must be guaranteed even if it is economically unattractive. This to ensure anticipation inventory is also built if the holding costs are high compared to revenue per product.

Capacity constraint:

∑ 𝑎_𝑖,𝑗 _𝑖𝑗𝑃_𝑖𝑗1(𝑡) ≤ 𝐶 ∀ 𝑡 (3)

The finite capacity 𝐶 of the system has been taken into account by constraint (3). The total required set-up time is not explicitly mentioned, because it is incorporated in the effectiveness of the bottleneck of 0.8.

Quantity balance:

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24 June 2019 Master Thesis Page | 20 Constraints (4) - (5) ensure the amount of products that enter the system is equal to the products that stay in or leave the system.

Multiple of constant batch size:

𝑃_𝑖𝑗1(𝑡) =𝑚𝑖𝑗1(𝑡)𝑄1 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑡 (6)

𝑃_𝑖𝑗𝑘2 (𝑡) =𝑚_𝑖𝑗𝑘2 (𝑡)𝑄2_{∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽}

𝑖, 𝑘 ∈ 𝐾𝑗, 𝑡 (7)

𝑚𝑖𝑗1(𝑡), 𝑚𝑖𝑗𝑘2 (𝑡) ∈ ℤ≥ (8)

Constraint (6)-(8) ensures production quantities at both stages are multiples of constant batch sizes. At stage 1 batches are formed on family level and at stage 2 on product level. These constraints make the model a MIP-problem. Consequently, the solution is very hard to obtain directly with a solver.

Non-negative constraint:

𝑃_𝑖𝑗1(𝑡), 𝑃_𝑖𝑗𝑘2 _{(𝑡), 𝐼}

𝑖𝑗1(𝑡), 𝐼𝑖𝑗𝑘2 (𝑡), 𝑆𝑖𝑗𝑘(𝑡) ≥ 0 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑘 ∈ 𝐾𝑗, 𝑡 (9) Constraint (9) makes sure the production and sales quantities are always equal or greater than zero.

Inventory level at start planning horizon:

𝐼_𝑖𝑗1(𝑡0− 1), 𝐼𝑖𝑗𝑘2 (𝑡0− 1) = 0 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑘 ∈ 𝐾𝑗, 𝑡 (10)

Constraint (10) is used in case no inventory is available at the beginning of the planning horizon. If there already exists inventory in the system, this constraint should be equal to the corresponding inventory level. This is, for instance, the case with a rolling planning horizon.

4.3 Mathematical programming-based heuristic

Constraint (8) makes our model MIP. As described in section 2, our problem is therefore NP-hard and can only be solved efficiently with a heuristic. A mathematical programming-based heuristic is developed to combine computation power of a computer in combination with logic steps that tighten the solution space. In this way, a solution is found efficiently and can be used for analysing the system by running simulations.

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4.3.1 Additional notation

Due to the structure of the heuristic, additional notation is needed. The main cause is the linear part at step 3A where production quantities for stage 1 are determined. Because these variables only have a perspective on stage 1 it is important to distinguish them from the variables in step 4, which have a holistic perspective on the system. This is further explained in the remaining subsections of this section.

Table 4.2 - Additional heuristic specific annotation

4.3.1 Overview heuristic

Figure 4.1 shows the steps of the proposed mathematical programming-based heuristic. Based on customer demand, the production quantities for stage 2 are determined. This is used as input to determine the needed production quantities at stage 1. The aim of the heuristic is to determine production quantities in an efficient way. Thereby the goal is to maximise profit and decrease inventory as much as possible. All steps together must ensure all system requirements as described in the previous section are met. A complete overview can be found in Appendix A. What stands out is that only for stage 1 the capacity constraint is taken into account. Consequently, anticipation inventory is only built at the inventory point after the first stage and a make-to-assembly (MTA) policy is simulated. However, the system under investigation uses a make-to-stock (MTS) policy. Using a capacity constraint at stage 2 as well will shift products from the first inventory point to the second with finished products. This is actually what is needed to represent a MTS-policy properly. Consequently, the capacity constraint of the bottleneck could have been used at customisation as well to ensure the anticipation inventory exists of finished products and not of semi-finished products.

Symbol Meaning

DECISION VARIABLES

𝑃𝑖𝑗1(𝑡) Amount produced at stage 1 of family 𝑗 of type 𝑖 in period 𝑡 (due to linear model now decision variable)

𝑆𝑖𝑗1(𝑡) Amount of produced products belonging to family 𝑗 of type 𝑖 in period 𝑡 to satisfy production need stage 2.

PARAMETERS

𝑏𝑖𝑗 _{Profit per bottleneck time of family 𝑗 of type 𝑖 (}𝑟𝑖 _𝑎 𝑖𝑗 ⁄ )

VARIABLES

𝑉 Subset with families where linear production quantities 𝑃_𝑖𝑗1(𝑡) are transformed to multiples of batch size 𝑄1.

𝑗𝑚𝑎𝑥 Family 𝑗 with highest profit per bottleneck time that is not part of subset 𝑉 yet. 𝑍1 _{Total theoretical profit over the entire planning horizon from the perspective of}

stage 1 in relation to needed production quantities stage 2

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24 June 2019 Master Thesis Page | 23 The best way a capacity constraint at stage 2 could have been incorporated into the heuristic is by replacing step 2, by an altered copy of step 3. However, due to the large number of products, the solver used for the simulation could not compute solutions due to the immense quantity of recursions. Because rewriting the recursions to iterations was too complicated, it was chosen only to use the capacity constraint at the aggregated family level. Consequently, in step 2 only the lead-time and batch sizes as multiples of batch quantity 𝑄2 are taken into account. Step 3 does the same for stage 1 and makes sure the right amount of anticipation inventory is determined per period by including capacity restrictions.

It is important to realise that choosing a MTS or MTA policy should not make any difference for the overall inventory level in the system. It only shifts products from one inventory point to the other. Because holding costs are set the same, the choice for a MTS or MTA policy should also not influence the overall costs. Of course, in practice, it has a lot of side-effects how easily production can anticipate on changing customer demand. Moreover, product value is normally increased after a product is processed. Because also the other KPI’s used for this research do not change, the reallocation of production quantities at stage 2 to other periods, is placed outside the scope of this research. A priority based ‘reallocation-heuristic’ for stage 2 is proposed as further research. In step 4 the KPIs are recalculated to ensure a holistic system perspective.

4.3.2 Mathematical description heuristic

In this section, the most important parts of the heuristic are explained more in dept. The notation used corresponds to the notation at the beginning of this section.

STEP 1. – Initialisation calculation

The first step is to set all the parameters. As shown on the right, the planning horizon in this research equals 52 weeks.

Moreover, it is important to determine if there is inventory in the system at the beginning of the planning horizon. The other parameters are straight forward. Finally, per product family the profit per minute of bottleneck time used, 𝑏𝑖𝑗, should be

calculated. This is used as prioritisation at step 3B.

STEP 2. – Determine production quantities stage 2

The second step determines production quantities at stage 2 without a capacity constraint. Consequently, this step converts the demand to quantities that are multiples of batch size 𝑄2. Therefore only inventory due to overproduction as a result of the batch restrictions is determined and the service level is 100%. It moreover takes the lead-time of stage 2,

STEP 1. Initialisation calculation Set parameters: 𝑡₀= 𝑡, 𝑇 = 𝑡₀+ 52 𝐼𝑖𝑗 1 (𝑡0− 1), 𝐼𝑖𝑗𝑘 2 (𝑡0− 1), 𝑄 1 , 𝑄2, 𝐿1, 𝐿2, 𝑙, C 𝑟𝑖𝑗, ℎ𝑖𝑗, 𝑎𝑖𝑗 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖 𝐷𝑖𝑗𝑘(𝑡) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑘 ∈ 𝐾𝑗, 𝑡 ∈ 𝑡0… 𝑇 Calculate: 𝑏𝑖𝑗 =𝑟𝑖𝑗/ 𝑎𝑖𝑗 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖

STEP 2. Determine 𝑷𝒊𝒋𝒌𝟐 (𝒕) without capacity

restriction Start at 𝑡 = 𝑡0, Do for all 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑘 ∈ 𝐾𝑗: If 𝐷𝑖𝑗𝑘(𝑡)− 𝐼𝑖𝑗𝑘2 (𝑡 − 1)< 0: 𝑃_𝑖𝑗𝑘2 (𝑡 − 𝐿2)= 0 Else, 𝑃𝑖𝑗𝑘2 (𝑡 − 𝐿2)= ቒ 𝐷𝑖𝑗𝑘(𝑡)−𝐼𝑖𝑗𝑘2 (𝑡−1) 𝑄2 ቓ 𝑄 2 End-Do

Increase 𝑡 with one period Repeat step 2 until 𝑡 = 𝑇

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24 June 2019 Master Thesis Page | 24 𝐿2, into account. Of course, inventory due to overproduction should be first used to satisfy demand. Consequently, the calculation starts at the beginning of the planning horizon.

STEP 3. – Determine production quantities stage 1

The third step consists of two parts. First, the linear part of the mathematical system description of stage 1 is solved. Afterwards, the production quantities of the family with the highest priority are made multiples of constant batch size 𝑄1. Step 3 is repeated until all families have production quantities that are multiples of 𝑄1.

In step 3A the capacity constraint at the bottleneck is taken into account and corresponding anticipation inventory is determined. The formulas only contain the formulas related to stage 1. What stands out is that the inventory point of stage 2 is left out of the objective function. As explained previously, only at stage 1 a capacity constraint is taken into account. Consequently, at the second inventory point there is only inevitable inventory due to

overproduction (Step 2). Taken this into account, will not change anything to the obtained solution for stage 1, and is therefore left out of the objective function. For the overall profit it does matter. This is the reason step 4 exists.

A second important notion is that the amount of sales, 𝑆_𝑖𝑗1(𝑡), is determined on family level and not on product level like in the mathematical model in section 4.2. Consequently, the service level represents the percentage of needed products that is produced for stage 2. Therefore, it is possible that the determined production quantities for stage 2 at step 2 must be adjusted downwards after step 3. As a result, also the amount of sold products is determined again at step 4.

Step 3B is very similar to step 2 of the heuristic and converts the production quantities of stage 1 determined at step 3A into multiples of 𝑄1. The difference with stage 2 is that at stage 1 capacity restrictions need to be taken into account. Because making production quantities multiples of 𝑄1 result in overproduction, extra capacity is needed. Therefore, it must be determined which products can be produced at the determined periods, and for which extra anticipation inventory is needed.

STEP 3A. Solve stage 1 linear

Decision variables: 𝑃𝑖𝑗1(𝑡), 𝐼𝑖𝑗1(𝑡), 𝑆𝑖𝑗1(𝑡) ∀𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖,𝑡0… 𝑇 Objective function: 𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑍1 _{= ෍ ෍ (𝑟} 𝑖𝑆𝑖𝑗1(𝑡) − ℎ𝑖1𝐼𝑖𝑗1(𝑡)) 𝑖,𝑗 𝑡∈𝑇 𝑺𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐: SOLD ITEMS: 𝑙 ∙ ∑𝑘∈𝐾𝑗𝑃𝑖𝑗𝑘(𝑡)≤ 𝑆𝑖𝑗 1_{(𝑡) ≤ ∑} _𝑃 𝑖𝑗𝑘(𝑡) 𝑘∈𝐾𝑗 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑡0… 𝑇 CAPACITY: ∑𝑖,𝑗𝑎𝑖𝑗𝑃𝑖𝑗1(𝑡) ≤ 𝐶 ∀ 𝑡0… 𝑇 QUANTITY: 𝐼𝑖𝑗1(𝑡) = 𝐼𝑖𝑗1(𝑡 − 1)+ 𝑃𝑖𝑗1(𝑡 − 𝐿1)− 𝑆𝑖𝑗1(𝑡) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖, 𝑡 NON-NEGATIVE: 𝑃_𝑖𝑗1(𝑡), 𝐼1_𝑖𝑗(𝑡), ≥ 0 ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽_𝑖, 𝑡₀… 𝑇 START INV.: 𝐼𝑖𝑗1(𝑡0− 1) = 𝐼𝑖𝑗1(𝑡0− 1) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖

(solver must use initiated value step 1)

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24 June 2019 Master Thesis Page | 25 As shown in Appendix B, the linear

solver at step 3A assigns anticipation inventory to product families with the lowest profit per minute of bottleneck time. Products of families with the highest profit per minute of bottleneck time are produced as late as possible. This is used at step 3B to prioritise families to determine the sequence in which production quantities of families are made multiples of batch size 𝑄1. The family with the highest value is planned first. This weighted-shortest-process-time-first policy is used in an iterative way. After the production quantities for the family with the highest remaining priority are altered, step 3A is reinitiated to keep a global perspective on the solution. This is done until all families are replanned.

STEP 4. – Calculate the performance of the entire system

Because step 3 only determines quantities from the perspective of stage 1, recalculation of the main process variables is needed to gain insight into the performance of the system as a whole. In this way also the inventory at stage 2, due to batch restrictions, are taken into account. The variables are first recalculated per period (week). To compare different parameter configurations easily, the variables can

be summed over the entire planning horizon (one year).

Because product values are determined on family level, it is not necessary for the KPIs to recalculate the production quantities at stage 2 properly. However, in reality it is important to know which products should be produced and not only the product families. This is marked as further research.

STEP 3B. Make 𝑷𝒊𝒋𝟏(𝒕) with highest 𝒃𝒊𝒋 multiple of 𝑸𝟏

Determine family 𝑗_𝑚𝑎𝑥 with highest profit per bottleneck time that has not yet been transformed to multiples of 𝑄1 (not part

subset 𝑉).

Make 𝑃𝑖𝑗,𝑚𝑎𝑥1 (𝑡) multiples of 𝑄1 over the entire planning

horizon: Start at 𝑡 = 𝑡0, Do for 𝑗𝑚𝑎𝑥: If 𝑃𝑖𝑗1(𝑡 − 𝐿1) − 𝐼𝑖𝑗1(𝑡 − 1) < 0: 𝑃𝑖𝑗1(𝑡 − 𝐿1) = 0 Else, 𝑃𝑖𝑗1(𝑡 − 𝐿1) = ඄ 𝑃_𝑖𝑗1൫𝑡−𝐿1൯−𝐼_𝑖𝑗1(𝑡−1) 𝑄1 ඈ 𝑄 1 End-Do

Increase 𝑡 with one period

Repeat until 𝑡 = 𝑇

Add constraint to linear model: 𝑃𝑖𝑗,𝑚𝑎𝑥 1 ₍

𝑡) = 𝑃𝑖𝑗,𝑚𝑎𝑥 1 ₍

𝑡) (solver must use determined quantities of this step)

Add 𝑗_𝑚𝑎𝑥 to 𝑉.

Go back to step 3A and repeat this sequence until subset V contains all families in the system: ȁ𝑉ȁ=ห∑_𝑖𝜖𝐼𝐽_𝑖ห

Figure 4.4 - Mathematical description step 3B proposed heuristic

STEP 4. Recalculate for the entire system: Start at 𝑡 = 𝑡0, 𝑆𝑖𝑗(𝑡) = 𝑚𝑖𝑛 ቄ∑𝑘∈𝐾𝑗𝐷𝑖𝑗𝑘(𝑡), 𝑃𝑖𝑗 1_{(𝑡 − 𝐿}1_{− 𝐿}2_{) + 𝐼} 𝑖𝑗(𝑡 − 1)ቅ ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖 𝐼𝑖𝑗(𝑡) =𝐼𝑖𝑗(𝑡 − 1)+ 𝑃𝑖𝑗1(𝑡 − 𝐿1− 𝐿2)− 𝑆𝑖𝑗(𝑡) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽𝑖 𝑙𝑖𝑗(𝑡)= 𝑆𝑖𝑗(𝑡) ∑𝑘∈𝐾𝑗𝐷𝑖𝑗𝑘(𝑡) ∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽_𝑖 𝑝𝑟𝑜𝑓𝑖𝑡 𝑍(𝑡) = ∑ (𝑟𝑖,𝑗 𝑖𝑗𝑆𝑖𝑗(𝑡) − ℎ𝑖𝑗𝐼𝑖𝑗(𝑡)) Repeat until 𝑡 = 𝑇

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5. Results and interpretation

More than 500 simulations have been performed to obtain insight into the system. A selection of the most useful outcomes is presented in this section. First, the reduction of constant batch sizes is investigated. Secondly, the consequences of increasing capacity are explored. As last, a sensitivity analysis has been conducted enabling qualification of found insights. It is important to realise that the lines in the graphs are trendlines and it is not possible to predict the behaviour of the system accurately between the different data points without extra simulation due to batch restrictions. The figures belonging to section 5.1 and 5.2 are placed after section 5.2. The numeric results of the experimentations presented can be found in Appendix C.

5.1 Reducing required batch -size

To investigate the influence of altering both 𝑄1 and 𝑄2 simultaneously, different scenarios have been simulated. As mentioned in the assumptions only scenarios where 𝑄1_{is a multiple of or} equal to 𝑄2 are allowed and therefore investigated.

The results are plotted in three different graphs that focus consecutively on service level, profit, and inventory (figure 5.1). Although reducing inventory is important it may not be at the expense of overall profit and service level. Therefore it is necessary to consider them together. In each graph, six different lines are plotted. Each line represents a scenario with another value for 𝑄1_{. In all the scenarios the minimum customer order quantity (min COQ) is hundred.} Capacity is set hundred percent. Because the ratio between the used holding costs and revenue per product is very high (1:100), the solver will try to produce as much as possible products, unless no feasible solution is possible2. This justifies setting the service level at zero during experiments. The found service level will consequently represent the maximum possible service level in our system. Performance is considered acceptable if the service level is above 95%. Important insights of the graphs are (1) the current constant batch size of 1440 at stage 1 result in unacceptable performance in any scenario. (2) Decreasing the constant batch size at stage 2 has a greater impact than changing batch size at stage 1. (3) The best performance occurs when the batch size at stage 2 is smaller or equal to 360.

To investigate more in debt the possible inventory reduction, figure 5.2a presents the scenarios with a service level above or equal to 95% compared with the current situation where products are produced at both stages in multiples of 1440 (1440/1440). As shown (4) both an increase in profit and a reduction in inventory can be realised simultaneously. The increase in profit is mostly caused by the increase of service level, because more products have been sold.

Graph 5.2b presents the possible reduction of only holding costs. This gives a more general perspective on the effect of reducing batch quantities. Again it is shown that the smaller 𝑄2, the higher inventory reduction and corresponding decrease in holding costs. What stands out is that the amount of inventory reduction is higher than the decrease in holding costs. This is not

2_{Mathematically speaking, due to the trade-off between inventory costs and profit per product, the solver will not}

plan an extra production batch if 𝑟𝑖

ℎ_𝑖< 𝑄1

𝑃_𝑖𝑗2(𝑡)(𝑚𝑜𝑑 𝑄1₎. In our case this will result in a decline of less than 1% in

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24 June 2019 Master Thesis Page | 27 logical, because the holding costs per product is 1 or higher. It turned out that the small differences in service level influence the found inventory levels much.

To purely investigate the inventory caused by batch restrictions, without anticipation inventory due to capacity restrictions, an extra simulation for the selected scenarios is performed with an uncapacitated system. The results are shown in figure 5.3. It is found that (5) differences in profit between scenarios are less in an uncapacitated system because all scenarios can satisfy demand completely. However, overall inventory is higher if more products are produced. It stands out that (6) possible inventory reduction is similar for all systems compared to the current state. However, (7) the smaller 𝑄1_{, the higher possible profit increase.}

Finally, for the scenarios with biggest [1440] and smallest [1] batch-sizes, a histogram is drawn to investigate the effect of changing the required constant batch sizes at both stages on the final production quantities (figure 5.4). The figures confirm that decreasing Q2 has a bigger impact than decreasing Q1. It moreover shows (8) the size of production quantities decrease and are more spread.

5.2 Increasing capacity

In figure 5.5 the effect of increasing capacity is shown by plotting different combinations of 𝑄1 and 𝑄2 in relation to a capacity level between 100-250%. It is found that (9) if capacity increases, the need for anticipation inventory declines. The inventory that is still in the system is the result of overproduction due to production batch requirements. This is the point in the graphs where lines become horizontal. Moreover, an (10) increase in capacity results in a decrease of inventory level and both an increase in profit and service level. This is in line with the expectations.

What stands out is the case where 𝑄2 equals 360. To investigate the deviation in behaviour more in detail several other scenarios are simulated (see numerical result appendix C3). It turned out the deviation in behaviour was less significant, but still present in the graphs. Therefore, on period level is searched for explanations for this abnormality. Unfortunately, no reason has been found.

The extra simulated scenarios are also used to compare the influence of capacity increase in combination with constant batch size reduction at both stages. Figures of four scenarios are depicted in figure 5.6. The figure shows that (11) holding costs if 𝑄1 equals 1440 are low compared to scenarios with lower 𝑄1. However, in appendix C3 can be seen the corresponding service levels are also low compared to the other scenarios.

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Figure 5.2 - Inventory reduction versus (a) Profit Increase and (b) Cost reduction of batch configurations Q1/Q2 with service level above 95% in a capacitated system

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5.3 Sensitivity analysis

To investigate the influence of parameter change to the system, a sensitivity analysis is conducted. Table 5.1 shows the different parameter settings per parameter that has been tested. During a simulation one parameter is changed at a time and the rest kept on 100%. An exception is the case where 𝑄1_{is tested. Due to batch restrictions, 𝑄}2_{is set equal to 𝑄}2_{, because otherwise} 𝑄1 became smaller than 𝑄2. Figure 5.6 shows the influence on consecutively the inventory level, profit, and service level. Due to the batch requirements, the system is not linear.

The first finding is that only capacity has a high influence on the overall profit and service level until all demand can be satisfied easily. The current capacity is enough to satisfy all customer demand properly. This is in line with the generated demand with an overall utilisation of 84% (see Appendix E). Because products can be stored cheap, an increase of capacity will only reduce the anticipation inventory and will not result in a large increase in profit or service level. The second finding is that the service level closely follows the profit. This can also be explained by the fact that the revenue per product is hundred times higher than the holding costs. Consequently, the solver tries to satisfy as much demand as possible as long as anticipation inventory can be built.

The third finding is that the profit and service level slightly decrease if required batch sizes increase. The decrease is caused by infeasible production requirements at the beginning of the planning horizon when there is no inventory in the system and the large production batches excess the capacity constraint.

The fourth finding is that decreasing 𝑄2, reduces the inventory in the system.

Test Q1 Q2 C l ri* hi* 300% 2160 2160 3 1.5 300% 300% 200% 1440 1440 2 1 200% 200% 100% 720 720 1 0.5 100% 100% 50% 360 360 0.5 0.25 50% 50% 25% 180 180 0.25 0.125 25% 25%

(34)

(35)

6. Discussion

The aim of this research was to come up with an efficient planning policy that optimises inventory levels and maximises profit in a multi-product capacitated system, with seasonal demand, modularized products, and production quantities as multiples of a certain batch size. In this section, the key findings are discussed and limitations are given.

6.1 Mathematical programming-based heuristic

The proposed heuristic seems to be an efficient way to determine lot-sizes in the system under investigation. By using a linear model it was possible to keep a global perspective on the solution space despite the high variety of products. The iterations enabled the gradual transition from a linear to integer solution. By using prioritisation of products based on their profit of bottleneck time, it was ensured enough capacity is reserved for the moneymakers.

At first sight, looking at the data in our research, the high-end products seem to be the ones with the highest priority, because their revenue is twice as high as the low-end products. However, because the required processing time is three times higher, it turned out their profit per minute of bottleneck time was much lower compared to the low-end products. In other words: the solver evaluates the trade-off between stocking one high-end, or three low-end products in combination with the corresponding profit. Insight is therefore not to quantify products solely on economic values, but also to take the required process characteristics into account during decision making.

Although the proposed heuristic is suitable for the system addressed, the way the heuristic is built limits the generalisation of the heuristic to systems with different characteristics. The main reason is that production quantities at stage 2 are not determined with a capacity restriction in combination with a linear model like the production quantities of stage 1 (step 3 of the heuristic). The reason this could not be incorporated into the model is the excessive amount of recursions needed by the linear solver. Consequently, three constraints cannot be relaxed to make the model more generic.

(1) The first constraint entails that constant batch size 𝑄1_{must be equal or a multiple of}_𝑄2_. This is necessary because currently the service level is determined locally between stage 1 and 2. Thereby the batch constraint at stage 2 is relaxed and should be corrected afterwards. This can only be done properly if the first constraint holds.

(2) The second constraint states that holding costs are similar at each inventory point. This assumption is only acceptable in a MTS-system where the first process is the bottleneck and processes afterwards directly process orders resulting in finished good inventory. The assumption is needed to ensure overall inventory levels can be recalculated at step 4. (3) Processes with high product variety that do not use modularisation, cannot aggregate