Title:Risk-Averse Supply Chain for Modular Construction Projects

Pei-Yuan Hsu* Imperial College LondonPhD Student, Dyson School of Design EngineeringImperial College London, South Kensington Campus, London SW7 2AZPhone: 07761770155, Email: p.hsu15@imperial.ac.uk

Dr Marco Aurisicchio Imperial College LondonSenior Lecturer, Dyson School of Design EngineeringImperial College London, South Kensington Campus, London SW7 2AZPhone: (0)20 7594 7095, Email: m.aurisicchio@imperial.ac.uk

Dr Panagiotis Angeloudis Imperial College LondonSenior Lecturer, Department of Civil and Environmental EngineeringImperial College London, South Kensington Campus, London SW7 2AZPhone: (0)20 7594 5986, Email: p.angeloudis@imperial.ac.uk

*Corresponding author

Declarations of interest: none

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AbstractThe traditional in-situ construction method is currently being replaced by modular building systems, that take advantage of modern manufacturing, transportation, and assembly methods. This transformation poses a challenge to construction supply chains, which have, thus far, been concentrated on raw material transportation only. A mathematical model is conceived in this study for the design and optimisation of risk-averse logistics configurations for modular construction projects under operational uncertainty. The model considers the manufacturing, storage, and assembly stages, along with the selection of optimal warehouse locations. Using robust optimisation, the model accounts for common causes of schedule deviations in construction sites, including inclement weather, late deliveries, labour productivity fluctuations and crane malfunctions. A school dormitory construction project is used as a case study, demonstrating that the proposed model outperforms existing techniques in settings with multiple sources of uncertainty.

Keywords: logistics, supply chain management, robust optimisation, modular construction, risk aversion

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1. Introduction

Modular construction has in recent years emerged as a transformative movement within the construction industry. The benefits of modular construction stem from the use of factory-based production environments, where bespoke building components are manufactured before being shipped to construction sites. Such techniques allow for higher levels of operational efficiency and product quality while shortening project timelines as fabrication of building components can take place in parallel to site and foundation works [1,2].

For most in-situ construction projects, there usually exist multiple material suppliers who make direct shipments to construction sites. Within this framework, various raw building materials are conveyed to construction sites at a specific time following requests made by main contractors or sub-contractors [3]. Raw building materials are dispatched to manufacturing factories, where they are transformed into modular components that are ultimately shipped to construction sites. As such, there is an urgent and persistent need to maintain close coordination among the manufacturing, transportation, inventory management and assembly sequences. Due to such interdependencies, operational deviations are expected to reverberate across the entire supply chain, resulting in significant variations in construction schedules [2,4].

As a result of large component sizes and the urban location of modular construction sites, it is often necessary to utilise warehouses to overcome storage limitations [5]. Site selection has the potential to affect operational capacities, running costs [4], and delivery punctuality [2]. Furthermore, given the just-in-time nature of deliveries, any transportation disruptions are bound to affect assembly schedules, further exacerbating the timeliness of project execution. As such, site selection represents a key aspect of supply chain design in such projects.

Compared to conventional construction, projects that employ modular components have more predictable material demands. However, timings over their course of execution remain volatile and are likely to change as production and assembly progress [6]. This is because the assembly process is typically affected by various delay factors such as extreme weather conditions, human errors, and equipment failures [7]. Thus, any plans for manufacturing, transportation, and inventory holding activities in subsequent periods must acknowledge the potential for variations, and the risk of insufficient component delivery rates or excessive inventory levels across the supply chain.

The bespoke nature of modular construction components necessitates a close alignment of production and consumption schedules, with inventories being entirely consumed by the end of a project [2]. This contrasts common practices in retail supply chains, where it is expected that a certain proportion of the inventory will be preserved as safety stock. Moreover, the requirement of further assembly operations in construction sites causes additional costs upon product delivery, with penalties that also need to be considered in case of stagnating supply. In this context, existing in-situ construction supply chain models are not directly transferable to modular construction. New techniques are therefore needed in order to ensure that the sector can benefit from modularisation.

Hsu et al. [8] previously proposed a two-stage stochastic programming model (2SP) for the design of modular construction supply chains, which seeks to generate optimal production schedules while considering the potential variations in product demand scenarios within sites. That approach, however, did not provide a means to accommodate decision makers with varying levels of risk-aversion, who aspire to reduce the overall hazard exposure significantly.

To address this gap in literature, the study presented in this paper seeks to achieve a risk-averse and highly coordinated logistics system using a robust optimisation (RO) approach, focusing on the design of a resilient prefabricated construction supply chain configuration. The proposed model accommodates the presence of stochastic site demands for modular components and recommends optimal distribution schemes that specify manufacturing quantities and timings, and transportation plans between factories, warehouses and sites. The optimal location for a warehouse that could significantly contribute to the mitigation of stochastic effects is also determined. The outputs of the model are expected to provide a useful guide to managers responsible for issuing

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supply chain configurations under stochastic site demands.

2. Background

Previous work on supply chain networks focused on the formulation of dynamic, time-dependent mathematical models that capture the interactions among the various parties involved. Chandra and Fisher [9] point out that activities across various supply chain echelons should be considered simultaneously to achieve prominent levels of operational efficiency. Accordingly, coordination efforts should be extended to operations taking place across organisational boundaries, for the creation of seamless, value-added processes that can fulfil customer needs [10].

More recently, Coelho and Laporte [11] developed integrated mathematical models which coordinate the manufacturing process planning, delivery scheduling and stocking management for the transportation of time-sensitive (perishable) goods. Similar models have also been used for tactical decision making in several complex supply chains [12-14]. A review of this topic by Díaz-Madroñero et al. [16] indicates that in recent years most practitioners and researchers have adopted mixed integer linear programming (MILP) models for the design of integrated supply chains, commonly solved using branch and bound/cut algorithms, meta-heuristics or Bender’s decomposition [6,8,16].

Ierapetritou and Pistikopoulos [17] were among the first to use stochastic programming for the determination of optimal production rates in multi-echelon supply chains with uncertain customer demands. The resulting model sought to minimise the total operating cost and expected penalties to be incurred by unfulfilled customer demands, with a large set of scenarios used to represent operational uncertainty and the effects of supply chain disruptions. Similar models were later developed for the optimal design of multi-merchant, multi-time horizon and multi-customer supply chain networks under various forms of demand uncertainty [18-22].

On a separate thread, Feng et al. [23] performed a probabilistic analysis of historical demand variations in construction sites. Later works [24,25] led to the development of stochastic delay models focusing on the underlying causes of demand variation with consideration of expected impact severity levels. As these are known to affect upstream activities [26] directly, their consideration during initial design planning is deemed essential.

Even though material demands and shipment schemes are enacted at the design stage, the modern construction sector remains particularly exposed to delays [27], with schedule deviations practically regarded as unavoidable. Gündüz et al. [7] list 83 distinct factors known as the reasons for resulting schedule deviations in large building projects, and more than 90% of the identified reasons can be traced to the adverse incidents that happen within construction sites. The construction schedule delays are often linked with the extension of overall project timespans, and unforeseen project costs [28].

As it currently stands, most of the commercial tools for construction supply chain scheduling and resource allocation are based on the discrete event simulation (DES) models. Mohsen et al. [29] adopted a software toolkit called Simphony.NET for executing DES to determine the construction durations under various perturbations due to uncertainties in worker and construction equipment efficiencies. More recently, Taghaddos et al. [30] developed a simulation-based protocol for optimising the construction tasks scheduling and performing the resource levelling at the same time for construction projects using prefabrication method. Bu Hamdan et al. [31] used building information models (BIM) to extract the geometrical dimensions and the lifting sequence of prefabricated building modules and then employed DES to identify the most economical strategy for managing onsite inventories.

Although DES models are extensively used in practice, they commonly adopt a deterministic approach, resulting in schedules that do not consider situations that deviate from a small set of previously identified scenarios. In contrast, stochastic programming techniques can determine logistics arrangements that simultaneously consider the entire spectrum of potential variations. However, only a few studies have adopted such methods in the context of construction supply chains. Hsu et al. [8, 54] proposed a series of such models that seek to optimise the modular construction supply chains with uncertain site demands and fluctuating component productivities. Nevertheless, such models cannot be used in settings where strongly risk-averse solutions are

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preferable - while the aggregate exposure to uncertainty effects is minimised, individual solutions do not perform well concerning the particular scenario that is eventually realised [31,32].

Robust optimisation models have the potential to overcome this limitation [33] through the generation of solutions that are progressively less sensitive to the perturbations in the scenario set. One of the first studies on robust optimisation with direct applications to supply chains was carried out by Gutiérrez et al. [34]. Later works by Leung and Wu [35] and Leung et al. [36] developed robust optimisation models for the solution of multi-site production planning problems with uncertain demands. The overall objective was to minimise the costs of production, inventory and hiring or layoff workforce. Furthermore, the trade-off between solution robustness and model effectiveness was demonstrated by adjusting the weight of penalty parameters. Pan and Nagi [37] proposed a cost-minimising robust optimisation model, which considered expected total costs, total cost variance due to demand uncertainties, and expected penalties for unmet demand. The model was solved using a heuristic approach, obtaining near-optimal solutions.

While robust optimisation has been used extensively in other disciplines, it has been rarely used in the past to solve civil engineering problems. Specific applications of the technique can be found on the allocation of water resources under uncertain demands [38,39] and the design of construction policies under uncertain construction costs, schedules, and product qualities [40,41]. As modular construction projects are characterised by stochastic onsite demands that can only be determined accurately once projects are underway, conventional DES tools cannot be relied upon for dynamic process models characterised by high levels of uncertainty. At the same time, the previously developed two-stage stochastic programming methods cannot be used to determine a holistic risk-averse strategy that ensures cost optimality against any realisation of a future site demand.

This study contributes to the current state of the art with a novel, robust optimisation model for prefabricated construction supply chains. The resulting model not only considers the various sources of uncertainty and volatility that are associated with construction logistics processes but also allows for customisable levels of risk aversion following from the adoption of the robust optimisation approach.

3. Methodology

The principal argument provided by Mulvey et al. [33] for the use of robust optimisation (RO) was the need to account for varying levels of risk aversion on behalf of decision makers. A range of RO applications has been studied since, including power capacity expansion, matrix balancing, airline scheduling, financial planning, and minimum weight structural design. Such models commonly adopt a goal-oriented programming formulation, with input data drawn from large scenario sets. The overall objective is to generate a series of solutions that are progressively less sensitive to realisations of these scenarios [35].

An RO model has two distinct constraints, namely the structural constraint and the control constraint. Structural constraints target input data sets that are free from any fluctuation (noise). Control constraints are taken as auxiliary constraints which can be affected by the noise of input data. Two sets of variables - design and control - are defined, with the former being those that cannot be modified once a specific realisation of the data has been perceived. Conversely, control variables are subject to adjustment when uncertain parameters are observed [33,36].

The mathematical formulation of the RO model is provided below, with x∈ Rn1being the vector of design variables and y∈Rn2the vector of control variables.

Min cT x+¿dT y ¿ (1)

Subject to: Ax=b (2)

Bx+Cy=e (3)

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x , y ≥ 0 (4)

In the above, Eq. (2) is the structural constraint whose coefficients are fixed and free from noise; while Eq. (3) is the control constraint whose coefficients are noise-sensitive. Eq. (4) ensures that the two vectors are non-negative. The problems to be solved by robust optimisation normally contain a set of scenarios Ω={1, 2 , 3 ,. . . , S }.For each scenario s∈Ω, the coefficients associated with the control constraints or variables are denoted as {d s , Bs , C s , es }

with a fixed probability of occurrence ps, and ∑s=1

S

ps=1. The optimal solution of this model is robust with respect

to optimality if it remains “close” to the optimal after the realisation of any scenario s∈Ω (solution robustness). With respect to feasibility, solutions are regarded to be robust if they remain “almost” feasible for any scenario realisation (model robustness). The notions of “close” and “almost” are defined through the choice of norms [33,36].

There are conditions under which obtaining a solution that is both feasible and optimal for all scenarios is impossible. Hence, the trade-off between solution robustness and model robustness is determined using the concept of multiple-criteria decision-making. Robust optimisation models that can manage such trade-offs are defined as follows [36]. Let ys be a control variable and δ s be an error vector, introduced for each scenario s∈Ω. The latter is used to measure the degree of infeasibility to be allowed in control constraints (i.e. δ s=es−Bs x−C s y s, ∀ s∈Ω). The earlier formulation is Eqs. (1) – (4) is thus transformed as follows:

Min σ ( x , y1 , y2 , …, ys )+ωp ( δ1 , δ2 ,…, δ s ) (5)

Subject to:Ax=b (6)

Bs x+C s ys=es ∀ s∈Ω (7)

xs , ys≥ 0 ∀ s∈Ω (8)

Since the RO model considers multiple scenarios, the value of the objective function in Eq. (1): ξ=cT x+dT y ,

becomes a random variable taking the value ξ s=cT x+dT ys with probability ps under scenario s∈Ω. Hence,

there is no single choice for an aggregate objective. Instead, a mean value σ ( ∙ )=∑s∈Ω

psξ s could be used in the

first term of Eq. (5). This is also employed in the formulation for stochastic linear programming. As in this research, we focus on the application of stochastic non-linear RO models, we introduce a higher distribution moment of ξ s to account for variance, therefore reducing sensitivity to noise present within input data.

The second term of the objective function in Eq. (5), p (δ 1 , δ 2 , …, δ s ) , is a feasibility penalty function, which is used to penalise violations of the control constraints under some of the scenarios. The violation of control constraints means that an infeasible solution to a problem has been chosen. By using the weight ω, the trade-off between solution robustness evaluated from the first term σ ( ∙ ) and model robustness measured from the penalty term p ( ∙ ) can be modelled under the multi-criteria decision-making process. For instance, in cases where ω = 0, the objective becomes to only minimise the term σ ( ∙ ). Even though outputs will be solution-robust, they are highly likely to be infeasible under certain scenarios. If, instead, ω is assigned with a sufficiently large value, the term p ( ∙ ) dominates the objective and the solution would tilt toward model robustness, resulting in higher costs, and a more conservative solution. The choice of appropriate forms of σ ( ∙ ) and p ( ∙ ) has been discussed in several studies [32,33,36]. The term σ ( x , y1 , y2 ,…, ys ) was proposed by Mulvey et al. [33], and represents the mean value σ ( ∙ ) plus a constant λ, multiplied by the variance as follows:

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σ ( x , y1 , y2 ,…, ys )=∑s∈ S

ps ξ s+λ∑s∈ S

ps(ξ s−∑s' ∈S

ps' ξs ')2

(9)

Increasing values of λ render the solution less sensitive to scenario variations, as the emphasis in the objective shifts to variance minimisation. However, as the second term in Eq. (9) is quadratic, the determination of exact solutions would have significant computational time requirements. To streamline the computation effort, the following formulation has been proposed to replace Eq. (9) [32]:

σ ( x , y1 , y2 ,…, ys )=∑s∈ S

ps ξ s+λ∑s∈ S

ps|ξ s−∑s'∈ S

ps ' ξs'| (10)

Several alternatives have been proposed to address the non-linearity of Eq. (10). Li [42] and Yu and Li [32] proposed an approximation based on goal-programming, outlined as follows:

Z=min∑s∈S

ps ξ s+ λ∑s∈S

ps[(ξ s−∑s' ∈S

ps' ξ s')+2 θs] (11)

Subject to:ξ s−∑

s'∈ S

ps ' ξs'+θs ≥0 ∀ s∈Ω (12)

θs≥ 0 ∀ s∈Ω (13)

with θ s being a non-negative slack variable. It can be verified that in either case, ξ s−∑

s'∈ S

ps' ξ s' ≥ 0∨ξ s−∑s'∈S

ps' ξ s' <0 ,Eq. (11) is equivalent to Eq. (10). Thus, the quadratic function in Eq. (9)

can be transformed into a mean absolute deviation minimisation problem in Eq. (10) and then into a linear programming model as given in Eqs. (11) – (13).

3.1 Model formulation

The RO framework described in the previous section is used to define a model for selecting an optimal warehouse location and identifying a supply chain configuration which can facilitate the execution of the project at the minimum operational cost under any adverse site assembly perturbations. A table describing all indices, parameters and decision variables of the model is provided in the Appendix.

In modular construction, most of the building components are produced in manufacturing facilities (factories). This is reflected in the model formulation which dictates that the factory can produce multiple types of modular products j (where j∈ J , J is the set of all product types). There exists a weekly fixed cost (MF) for the factory which is independent of production quantities. The fixed production cost (FPC) is defined as the product of MF with the “manufacturing timespan of the entire project”, which encompasses the initial inventory

preparation period (∑p∈P

y p), and the production period (∑a∈ A

ya) that proceeds in parallel with the site construction

process, where y is a binary decision variable defining whether the factory carries out manufacturing on a certain week and the notation “❑” refers to a state before the site construction commences. The time index p is defined for representing the week in the initial inventory preparation period, whereas index a is for the week during the site assembly period.

The factory-based production has a basal unitary manufacturing cost ( MV j) and proceeds in accordance

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with a maximum weekly manufacturing rate (MMR j) that cannot be exceeded. An adjustment term is introduced in our model to reflect the rise of MV jas the weekly production rate approaches MMR j ,due to requirements for more resources, workers, and longer shifts.

The variable production cost (VPC) includes two parts. The first part focuses on the production cost for the initial inventory, defined as the product of MV j and the weekly production rate m pj. The MV j value needs to be adjusted accounting for further labour and resource utilisation. Here, we assume that the usual weekly production rate is half of MMR j, and any production rate higher than this value will increase MV j linearly.

Therefore, VPC is calculated as mpj ∙ MV j+max(m pj−MMR j

2, 0) ∙ MV j. The second component of the VPC

formula refers to production activities after the site construction has commenced, which are determined in the same manner, except for mpj which has been replaced with a weekly production quantity maj.

Our model anticipates and accounts for assembly delays that affect the weekly demand for modular products on the site. In our case study presented later in this paper, we focus on the following disruptions that perturb the site demand: bad weather conditions, lateness in module delivery, worker efficiency variations and crane reliability problems or breakdowns. After taking into account all the disruptions mentioned above simultaneously, a set of demand scenarios can be generated.Dsaj indicates the weekly demand for modular product j in week a (a∈ A where A= 1, 2 ... LW), in scenario s (s∈S, where S is the set of demand scenarios) and each demand scenario has a probability of occurrencePs. The time required to reach the project assembly target is defined as DDs and varies among scenarios in accordance to disruption expectations. The longest duration among all scenarios is denoted as LW.

The site fixed cost SFCs in each scenario is defined as the product of working durations DDs and weekly site overheads SF. The extension cost (EX s) refers to the “extra operational cost on site”, and it occurs in a scenario when the weekly site demands for modules are not met momentarily. Under this circumstance, the construction site must carry out works beyond the initially planned completion date for finishing the modules that arrive late.

Aspects of inventory management are also considered, including the choice among several candidate warehouse locations that provide intermediate storage between the site and the factory. Warehouses have different storage capacities, distances to site/factory and establishment costs. Our study assumes that a single, most beneficial location will be selected. The binary decision variable ez represents whether a certain warehouse z is established or not (z∈Z ,where Z is a set of warehouses). Based on the case study, we have assumed that items manufactured during the initial inventory preparation period are solely deposited in the warehouse, because the storage spaces at the factory are limited and must be shared with other ongoing projects, and the inventory protections are not available on the site during this period. As such, the initial inventory quantities in the factory and the sites are zero, while the quantity at the chosen warehouse z (i¿¿ pzjW)¿ will be increasing weekly. The superscript notation W refers to warehouse and F refers to the factory while C refers to a construction site in the following paragraph for specifying the location of the decision variables and parameters.

The inventory cost ( IC) for the initial inventory preparation period is defined as the sum of weekly storage costs for each product, in turn expressed as: i pzj

W ∙V j ∙ SCW , where V j is the volume of the modular

products j and SCW is the weekly unitary storage cost in the warehouse. Weekly inventory levels after

construction work starts in the factory, warehouse and site are denoted as: iajF ,iazj

W and isajC , respectively. Site

inventory levels are scenario-specific owing to demand variation in each scenario. The corresponding inventory costs are defined in the exact same way as the product of inventory volumes with respective weekly unit storage costs, i.e. SCF , SCW and SCC for the factory, warehouse, and site, respectively. The aggregate inventory cost across all locations and product types for each scenario is defined as ICs. Given the bespoke nature of modular

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products as mentioned earlier, the model binds the values of iajF , iazj

W and isajC to zero at the end of the project, as we

expect all inventories to have been consumed. With regards to transportation, all finished modular products are shipped from the chosen warehouse to

the construction site following the enacted transportation scheme for accommodating the weekly site demand in all scenarios. The number of truck needed for transporting each type of product is calculated by dividing the weekly transportation quantity for product j from warehouse z to construction site¿¿) by the quantity of product j

that can be loaded onto a single truck running on the route (L jWC ). As such, the weekly transportation cost between

the warehouse and the site (TRCWC) is defined as: TRCWC=(t ¿¿azjWC/ L jWC) ∙(D¿¿ zWC ∙ TCWC+LP z)¿¿, where

D zWC is the distance between site and warehouse z, and TCWC is the unitary transportation cost, while LP z refers

to the lateness penalty owing to the distance and traffic condition between the site and warehouse z, as detailed in the following section.

Conversely, transportation costs for component movements between the factory and warehouse during the preparation and construction period (TRC FWand TRC FW , respectively) are calculated using the weekly

transportation quantities during preparation (t ¿¿ pzj FW )¿. and construction ¿¿), truck loading quantity L jFW ,

distance D zFW and unit transportation cost TC FW .

We define the objective of the model as the minimisation of the costs incurred by manufacturing, transportation, stock-keeping (inventory) and warehouse establishment under the influence of various site demand variation factors. It is formulated as follows:

Minimise: ETC+λ∑

s∈SP s(TC¿¿ s−ETC+2θs)+ω∑

s∈SPs δ s¿ (14)

where:

FPC=MF (∑a∈ A

ya+∑p∈ P

y p) (14.1)

VPC=∑a∈ A

∑j∈ J

maj ∙ MV j+max(maj−MMR j

2, 0) ∙ MV j+∑

p∈ P∑j∈ J

m pj ∙ MV j+max(m pj−MMR j

2, 0)∙ MV j(14.2)

IC=∑p∈ P

∑z∈ Z

∑j∈ J

i pzjW ∙V j ∙ SCW

(14.3)

ICs=∑a∈ A

∑j∈ J

iajF ∙V j ∙ SCF+∑

a∈ A∑z∈Z

∑j∈J

iazjW ∙ V j ∙ SCW+∑

a∈ A∑j∈J

max ( isajC , 0) ∙V j ∙ SCC

(14.4)

TRC FW=∑p∈ P

∑z∈ Z

∑j∈ J

(t ¿¿ pzjFW / L jFW) ∙ D z

FW ∙TC FW ¿ (14.5)

TRC FW=∑a∈ A

∑z∈ Z

∑j∈J

(t ¿¿azj FW / L jFW )∙ Dz

FW ∙ TC FW ¿ (14.6)

TRCWC=∑a∈ A

∑z∈Z

∑j∈ J

(t¿¿ azjWC /L jWC)∙ (D¿¿ zWC ∙ TCWC¿¿¿+LP z)¿¿¿¿¿ (14.7)

SFCs=SF ∙ DDs (14.8)

EX s=max ( x−DDs ,0 )∙ SF (14.9)

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WO=∑z∈ Z

ez ∙WEC z (14.10)

δ s=−∑a∈ A

∑j∈J

min (i sajC , 0)+∑

a∈ Amax [∑j∈ J

max (isajC ,0 )−N ,0 ] (14.11)

TC s=FPC+VPC+ IC+ ICs+TRC FW+TRCFW +TRC WC+SFCs+EX s+WO (14.12)

ETC=FPC+VPC+ IC+TRC FW +TRC FW +TRCWC+WO+∑s∈ S

Ps ( ICs +SFC s+EX s ) (14.13)

Eq. (14) is the aggregate objective function, defined as the sum of three terms: total expected cost, cost variance (solution robustness) and feasibility penalty (model robustness). Eqs. (14.1) and (14.2) calculate the fixed and variable production costs (FPC and VPC) for producing all types of modular products, respectively, in the period of both preparation and construction phases.

Eq. (14.3) determines the initial warehouse inventory cost IC before construction work starts at the site, while Eq. (14.4) computes the total inventory cost for each scenario after construction work starts which covers the factory, the warehouse, and the construction site. It should be noted that in Eq. (14.4) only non-negative quantities of inventory on the construction site are included in the calculation. In some demand scenarios, the demands on the site are not fulfilled for a certain week momentarily, and the calculated inventories on the construction site are negative in these time periods. These negative values stand for the temporarily unmet demands for modular products, but do not represent the real inventory quantities on the construction site, the costs of which are assigned to zero under these circumstances. Eq. (14.5) captures the transportation cost between the factory and warehouse for the initial inventory preparation period (TRC FW). Eqs. (14.6) and (14.7) are used to model the transportation costs between the factory and warehouse (TRC FW) and between the warehouse and construction sites (TRCWC ¿, respectively after the beginning of construction works.

Eq. (14.8) calculates the fixed cost on the construction site in each scenario ( SFCs). Eq. (14.9) evaluates the extension cost for each scenario, incurred when weekly demands are not met, and the site is forced to extend operations beyond the initially planned period. Here x represents the total duration of the optimal transportation scheme recommended by the model measured in weeks and the difference between x and DDs indicates the length of site extensions in each scenario. Eq. (14.10) calculates the warehouse establishment cost for choosing one of the candidate locations. Eq. (14.11) computes the proxy measure of model infeasibility which is composed by two parts. The first part focuses on calculating the total quantity of temporarily unfulfilled demands for all product types across the entire construction period in each scenario. The second part calculates the numbers of all types of products that exceed the site inventory capacity in each week in each scenario. N is the maximum number of products that can be stored on the site without stacking on one another.

Eq. (14.12) determines the total cost for each scenario (TC s¿, defined as the sum of all the above cost terms. Eq. (14.13) defines the expected value of the total costs (ETC). In this study, we set λ=ω=1 , therefore assigning equal priority to solution and model robustness.

The constraints of the model are implemented as follows:

Subject to:

0 ≤ mpj ≤ MMR j∀ p∈P ,∀ j∈ J (15)

0 ≤ maj ≤ MMR j∀a∈ A ,∀ j∈ J (16)

∑a∈ A

maj+∑p∈ P

m pj=∑a∈ A

Dsaj∀ s∈S , j∈J (17)

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iajF =i (a−1 ) , j

F +maj−∑z∈Z

tazjFW ∀ a∈ A , j∈ J (18)

iazjW =i (a−1 ), zj

W +t azjFW−t azj

WC∀ a∈ A , z∈Z , j∈ J (19)

i pzjW =i ( p−1) , zj

W + t pzjFW ∀ p∈P , z∈Z , j∈ J (20)

isajC =is , (a−1) , j

C +∑z∈ Z

t azjWC−Dsaj∀ s∈S , a∈ A , j∈ J (21)

isajC =0 ∀ s∈S , a=LW , j∈ J (22)

∑j∈ J

i pzjW ∙V j∧∑

j∈Jiazj

W ∙V j ≤WCAP z ∙e z∀ p∈P , a∈ A , z∈Z (23)

∑j∈ J

iajF ∙ V j ≤ FCAP ∀a∈ A (24)

t pzjFW ¿ t azj

FW ≤ WCAPz ∙ ez∀ p∈P , a∈ A , z∈Z , j∈J (25)

∑z∈Z

∑j∈ J

t azjWC ≤ ca ∙ M ∀ a∈ A (26)

x=max (c¿¿a ∙a)a=1. . LW ¿ (27)

∑j∈ J

maj≤ ya ∙M∧∑j∈J

mpj ≤ y p ∙M ∀a∈ A , p∈ P (28)

TC s−ETC+θs ≥0∧θ s≥ 0∀ s∈S (29)

iajF ,iazj

W , i pzjW , t pzj

FW ,t azjFW , t azj

WC ≥ 0 ∀a∈ A , p∈ P ,z∈Z , j∈ J (30)

i0 , jF , is , 0 , j

S =0∧i0 , zjW =i pzj

W ∀ s∈S , p=PW , j∈J (31)

Constraints (15) and (16) bind the weekly manufacturing rate for each product type within a pre-defined range during the initial inventory preparation and the construction periods, respectively. Constraint (17) makes sure that the total production quantity for each type of modular product follows the total demand, a condition that applies to every scenario.

Constraint (18) represents the inventory balances in the factory, while constraints (19) and (20) relate to balances in the warehouse during the construction and initial inventory preparation periods, respectively. Constraint (21) handles inventory balance at the construction site. Constraint (22) assures that for every scenario the site demand for each type of modular product must be fulfilled when the construction project is finished. Constraint (23) guarantees that the warehouse capacity can never be exceeded. Constraints (24) imposes the same restriction on the inventory at the factory. Constraint (25) ensures that finished modules cannot be directed towards warehouses that have not been established.

Constraints (26) and (27) ensure that the duration of an optimal transportation scheme is determined correctly, where ca is a binary decision variable representing whether any product has been sent to the site or not on week a, and M is a big positive number. Constraint (28) guarantees that the lengths of the production periods for the initial inventory preparation and site assembly are rightly calculated. Constraints (29) is introduced to transform the objective function from a mean absolute deviation minimisation problem into a linear programming problem as discussed in the previous section. Constraint (30) asserts the non-negativity of the inventory levels at the factory and warehouse, as well as of transportation quantities between all places at all time. Finally, the first part of constraint (31) ensures that the initial inventory will not be diposited in the manufacturing factory and the construction site, while the second part represents the continuity of the warehouse inventory between the initial preparation phase and the construction phase, where PW refers to the last week of initial inventory preparation period.

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4. Case Study

The model was applied to a case study for constructing a 4-story-high school dormitory using prefabricated room modules (excluding stairs, elevators and foundations). The floor plan of the dormitory is shown in Figure 1. Two types of modules have been adopted in this project. The first type (module A) is the accommodation room, and there are 14 module A on each floor. The second type (module B) is the toilet and bathroom room, and each floor has 2 modules B. Thus the total demand is 56 module A and 8 module B. There exists a strict installation sequence.

The assembly of the higher floor room modules cannot commence until all the lower floor room modules are installed, while B modules must be assembled after all A modules have been assembled. Construction is expected to take place during the winter school break, therefore limiting assembly periods to a maximum of 4 weeks. Module designs are expected to be finalised ten weeks before the site operations begin, therefore establishing a maximum duration for the initial inventory preparation period.

Figure 1 The floor plan for the case study.

The manufacturing factory is located 60 km away from the construction site (which lies in the centre of London). There exist six candidate warehouse locations (Z1 to Z6) that could be used to store modular products, each of them having a different storage capacity and establishment cost, as shown in Table 1. The other parameters applied in this case study are given in Table 2.

Table 1 The parameters for the warehouse located in 6 candidate locations.Candidate No. Z1 Z2 Z3 Z4 Z5 Z6Capacity (m3) 3,150 3,550 3,938 4,326 4,714 5,100Establishment Cost (£) 14,436 13,917 12,028 11,359 10,043 9,349Distance Factory to Warehouse (km) 51 42 31 24 18 10Distance Warehouse to Site (km) 9 18 29 36 42 50

Table 2 The parameters for the case study project.Data Names Values

Type of modular product Room module A, BMaximum number of assembled modules 36 modules/weekConstruction site fixed cost (overhead) £11000/weekMaximum storage capacity on site 360 m3/ 4 modulesMaximum storage capacity at factory 650 m3/ 8 modulesInventory management cost on site £0.2 /m3/ weekInventory management cost at warehouse £0.3 /m3/ weekInventory management cost at factory £0.8 /m3/ weekVolume of module A 78 m3/module A

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Volume of module B 84 m3/module BTransportation cost from factory to warehouse £4.57/truck/kmTransportation cost from warehouse to site £3.06/truck/kmTruck capacity from factory to warehouse 2 modules/truckTruck capacity from warehouse to site 1 module/truckManufacturing fixed cost at factory (overhead) £13200/weekMaximum manufacturing rate for module A 12 modules/weekMaximum manufacturing rate for module B 4 modules/weekVariable manufacturing cost for module A £3264/moduleVariable manufacturing cost for module B £3497/module

During project execution, there exist four significant sources of uncertainty that would cause the assembly process to deviate from the original construction schedule. These include lateness in module deliveries, adverse weather conditions (e.g. strong wind and heavy rain), efficiency perturbations among site workers and equipment breakdowns. This study has quantified these unwanted disruptions for generating all the possible module demand scenarios that may happen during the assembly period. The procedures followed for the quantification of the uncertain site demands and generation of scenarios are introduced in the following sections.

4.1 Transportation-related disruptions

Previous studies have suggested that the transportation time on a specific route within urban areas can be approximated as a Burr distribution [43,44,45,46]. The probability density functions (PDF) of the transportation time between the construction site and each of the candidate warehouse location are presented in Figure 2. Note that the spread becomes wider with increasing distance in between. Thus, if the warehouse is located farther from the site, the higher the probability of being late. The late delivery of modular products could disrupt the assembly process and is penalised for causing idleness of equipment and labour force, and a waste of site utilities. The probabilities of lateness in delivery in different ranges of severity derived from the PDF shown in Figure 2 are given in Table 3. The most severe delivery delay ever happened in this case study is three hours. The site idleness penalty in this case study is £405.4 per hour. The expected lateness penalty (LP z in Equation 14.7) for a single shipment from each candidate warehouse locations is calculated and presented in Table 3.

Figure 2 The PDF of the transportation time between the site and the six candidate warehouse locations.

Table 3 The probabilities of distinct levels of late delivery when the warehouse sets in various locations. The expected lateness penalty for a single shipment from each warehouse location is also calculated.

 Candidate No.

Probability of lateness in delivery (%) Expectedlateness penalty (£)On time 1 hour 2 hours 3 hours

Z1 98.86 1.14 0 0 5.68

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Z2 94.77 5.23 0 0 21.20Z3 64.02 31.77 3.32 0.89 166.53Z4 60.59 33.83 4.32 1.25 187.41Z5 57.17 35.90 5.33 1.60 208.17Z6 53.74 37.96 6.33 1.97 229.17

4.2 Weather-related disruptions

Inclement weather conditions have been generally recognised as one of the most common reasons for disrupting the tasks on construction sites by previous studies [7,47,48]. Assembly of modular products is also very susceptible to adverse weather conditions. Historical data is commonly used to predict future weather conditions, such as the probabilities of wind speed and rainfall amount in times of the year. For this study, we obtained meteorological data for Greater London through the UK Met Office, such as daily precipitation records (in mm) over 85 years (1931-2015) and wind speed data (in m/s) over 21 years (1995-2015). Hsu et al. [8] previously carried out a statistical analysis of occurrence probabilities for various levels of rainfall amounts and wind speeds in January. A semi-trapezoidal function model [8,49] was used to develop a relationship between module assembly rates and weather conditions, with values for a range of intervals illustrated in Table 4.

Table 4 The probabilities of rainfall amount (a) and wind speed (b) lie in the various intervals, and the corresponding disrupted weekly assembly rate.

(a) RainfallRainfall

Interval (mm/h)Probability

(%)Assembly rate

0 mm <= X < 5 mm 85.26 365 mm <= X < 6 mm 4.73 326 mm <= X < 7 mm 4.01 257 mm <= X < 8 mm 2.40 188 mm <= X < 9 mm 1.91 11

9 mm <= X < 10 mm 1.16 4X >= 10 mm 0.53 0

(b) Wind speedWind speed

Interval (Km/h)Probability

(%)Assembly rate

0 Km/h<= X <36 Km/h 83.73 3636 Km/h<= X <38 Km/h 3.74 3338 Km/h<= X <40 Km/h 3.58 2840 Km/h<= X <42 Km/h 2.42 2342 Km/h<= X <44 Km/h 1.77 1844 Km/h<= X <46 Km/h 1.94 1346 Km/h<= X <48 Km/h 1.45 848 Km/h<= X <50 Km/h 0.81 3

X>=50Km/h 0.56 0

4.3 Worker efficiency and crane-related disruptions

Although the implementation of modular construction practices has significantly simplified the onsite construction procedures and reduced the number of needed workers [1,2], manual operations are still required during the module assembly process. When the modules arrived at the site, a number of tasks are carried out by workers such as unloading, inspection, unwrapping, and facilitating crane lifts. Precise module alignment is necessary before they are permanently fixed into place, while tasks such as unhooking, bolting and welding require manual intervention. Therefore, the efficiency of construction workers on-site is crucial for the achievement of weekly assembly targets. Previous surveys mentioned fatigue, overtime work, overdue payments, absenteeism or negative working attitude as the factors that may affect workers’ performance [50,51].

In this case study, the discount rates of the worker efficiency and the corresponding probabilities of occurrence as given in the upper panel of Table 5, were derived from both the case study data and the studies which focused on construction workers’ productivity perturbations [50,51]. It is worth mentioning that there are a large number of factors that can affect construction workers’ efficiency. Thus, these figures may be particularly diverse across different locations and construction project instances. Similarly, module assembly can be disrupted by crane failure due to mechanical malfunctions or breakdowns. By consulting the research literature [52,53] regarding crane operation stability and the site event log of our case study, we established a 20% reduction of weekly productivity when such incidents occur, with an approximately 1.5% probability of occurrence (see Table 6).

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Table 5 Discount on the assembly rate incurred by labour productivity fluctuation Delay factor Influence Level Discount rate Probability (%)

Labourproductivity

Normal 1 91.00Medium low 0.8 7.50

Low 0.5 1.50

Table 6 Discount on the assembly rate incurred by crane failure.Delay factor Influence Level Discount rate Probability (%)

Crane Normal 1 98.50Down 0.8 1.5

4.4 Generation of assembly scenarios

The complete range of possible weekly assembly rates was determined by considering all combinations of onsite delay factors (i.e. precipitation, wind, workers, and cranes), as mentioned in sections 4.2 and 4.3. These rates would in the first instance be derived using the relationships described in Table 4, and subsequently penalised by the worst performance reduction penalties induced by labour and crane (described by Tables 5 and 6). While heavy rain and strong winds may occur individually or simultaneously, we consider that construction site assembly rates are governed by the highest delay factor among the two processes. A total of 14 possible weekly assembly rates are calculated, which are listed in Table 7 alongside their respective occurrence probabilities.

Table 7 Distinct Assembly Rate Scenarios and Occurrence Probabilities Assembly rate No. 01 02 03 04 05 06 07 08 09 10 11 12 13 14

Module/Week 36 33 32 28 25 23 21 18 13 11 8 4 3 0Probability (%) 63.98 2.86 3.71 9.14 3.92 2.04 1.42 4.18 2.18 1.67 1.67 1.23 0.82 1.19

Given the above range of identified assembly rate scenarios, there will exist a total of 144=38,416 possible demand scenario combinations for four consecutive weeks. The probability of having a total assembly capacity that is equal or larger than 64 modules (with 56 A modules and 8 B modules) is 99.63%, i.e. installation of 64 room modules in 4 weeks can be achieved with 99.63% confidence. There exist 25,162 demand scenarios that are expected to meet the assembly target of 64 modules. The remaining scenarios (with occurrence probability of less than 0.37%) are regarded to be invalid (for timely project execution) and are therefore excluded.

Among the valid scenarios that were considered, there exist instances with assembly rates that exceed targets and are therefore expected to complete assembly tasks in less than four weeks (as there exist fewer onsite disruptions). Therefore, a corresponding adjustment has been conducted to ensure that, in every valid scenario, only the exact target number of room modules (64 modules in total) are assembled on the site within the designated construction period. This is followed by combining scenarios with the identical pattern of weekly assembly rates, which ends up identifying 1,680 distinct demand profiles

4.5 Model implementation and execution

The model is built and solved using the IBM CPLEX solver (version 12.6) on a mid-range workstation with an Intel Core i7-4790 CPU and 8GB RAM, using linear programming relaxations to reduce computational time. The solution time for obtaining an exact solution to the current problem instance is 576 seconds. The value of the objective function (Eq. 14) is minimised, which simultaneously considers all the 1,680 demand profiles.

Model outputs indicate that the warehouse should be established at Z1, and the most economical production duration is six weeks for initial inventory preparations and a subsequent manufacturing period of 3 weeks once site assembly commences. The best production plan and transportation schemes between all places as

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well as the change of inventory levels at the warehouse for the entire project timespan are shown in Table 8. Owing to the site demand being scenario specific, the change of site inventory level is also scenario dependent. For demonstration purpose, a demand profile is chosen and presented in the 2nd row from the bottom of Table 8, along with the corresponding site inventory level changes (the last row). The model can disclose the inventory information for all demand profiles.

Table 8 The optimal risk averse supply chain configuration recommended by the model, and the information about the site inventory for a chosen demand profile.

Location Events Period for initial inventory predation (week) Period for modules assembly (week)

1 2 3 4 5 6 1 2 3 4

Factory

Module type A B A B A B A B A B A B A B A B A B A BProduction quantity 6 0 6 0 6 0 6 0 6 0 6 2 6 2 6 2 8 2 0 0Transp. Fac to Whse 6 0 6 0 6 0 6 0 6 0 6 2 6 2 6 2 8 2 0 0Factory inventory - - - - - - - - - - - - 0 0 0 0 0 0 0 0

Warehouse

Transp. Fac to Whse 6 0 6 0 6 0 6 0 6 0 6 2 6 2 6 2 8 2 0 0Transp. Whse to Site - - - - - - - - - - - - 17 2 25 2 14 4 0 0Warehouse inventory 6 0 12 0 18 0 24 0 30 0 36 2 25 2 6 2 0 0 0 0

SiteTransp. Whse to Site - - - - - - - - - - - - 17 2 25 2 14 4 0 0Demand - - - - - - - - - - - - 16 2 16 2 16 2 8 2Site inventory - - - - - - - - - - - - 1 0 10 0 8 2 0 0

5. Discussion

This research proposed an RO model to determine optimal warehouse locations, production plans, transportation rates and inventory schemes for modular construction projects. The established model is explicitly designed to deal with demand uncertainties at the construction site incurred by inclement weather, lateness in module delivery, workers’ efficiency perturbation and crane mechanical failures.

To ensure solution robustness, our model includes a higher-order distribution moment of variance which is a critical risk-aversion manner absent in the two-stage stochastic approach. When the variance of the total operational cost across all scenarios is minimised, the operational cost in each scenario is expected to converge to the lowest expected value given by the first term of the objective function in Eq. (14). As such, the total operational cost will remain within the desired value range, therefore guaranteeing that the budget limit would never be exceeded regardless of which demand scenario unfolds. Our formulation also adopts a penalty term, which is included in the objective function. This seeks to minimise the relative infeasibility of solutions attributed to excessive site inventory and insufficient delivery of room modules. Therefore, the proposed RO model not only depicts the fundamental parameters and decision variables for optimising the logistics process in a modular construction project under demand uncertainties but also embodies the risk-aversion of the logistics system.

As the module manufacturing activities commence much earlier than onsite construction, it is essential for component designs to be finalised much earlier than in conventional projects. Should these timelines be not achievable, the production timespans would inevitably be shortened, making it essential to ensure that further delays will be avoided as much as possible. In this context, an RO-based approach would be particularly valuable, given its ability to observe rigid delivery and inventory constraints. To demonstrate the benefits of this approach, we carried out a comparison against the 2SP model that was developed by Hsu et al. [8]. By applying the optimal solutions recommended by the two models (RO and 2SP) respectively to each demand profile, we obtain the minimised total operational costs for all 1,680 profiles. The result is shown in Figure 3 where:

(a) Profiles 1 to 4 (occurrence probability: 61.46%) can complete assembly in 2 weeks.(b) Profiles 5 to 131 (occurrence probability: 33.74%) can complete assembly in 3 weeks.(c) Profiles 132 to 1,680 (occurrence probability: 4.43%) require 4 weeks to complete assembly.

Except for the first four demand profiles, the solution of the RO model can generate lower total operational costs than that of the 2SP model for the remaining 1,676 profiles. During the optimisation process, the 2SP model allocates a higher priority to the minimisation of total operating costs in the first four profiles, given their higher occurrence probability. This results in higher total operating costs for the remaining demand profiles

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when compared to the RO model, considering the entire set of demand profiles. It is clear that the approach dictated by the two models varies significantly.

Figure 3 The total operational costs of all 1,680 profiles derived from the RO and 2SP solutions, respectively.

Differences between the two models can also be discerned in the variance and standard deviation of total operational costs among all demand profiles, which are found to be 58.53% and 25.91% higher the 2SP model compared to RO, respectively (see Table 9).

Table 9 A comparison between the solutions given by the RO and 2SP models.RO 2SP Difference (£) Improvement (%)

Total Costs (Expected) 396,130 394,629 -1,501 -0.38%Total Costs (Variance) 22,233,500 35,246,618 13,013,118 58.53%Total Costs (Standard Deviation) 4,715 5,937 1,222 25.91%Total Costs (Worst Case) 412,983 419,363 6,380 1.54%

When worst-case costs are considered (i.e. the demand profile with the highest operational cost), we observe that the figure produced by the 2SP model is £6,380 higher compared to RO. This is mainly because the RO objective function includes a term that minimises total cost variance and by extend reduces the chance of deviation from the minimum expected total cost. This holds regardless of which demand profile is ultimately realised and despite the number of temporarily unfulfilled demands and excess site inventories. The expected total operational costs determined by the 2SP model are 0.38% lower than the RO model (see Table 9). This can be attributed to 2SP’s prioritisation of dominant demand profiles, but we can argue that an 0.38% cost premium is a small price to pay for RO’s risk aversion.

These results indicate that a strategy developed using the RO model would be more advantageous compared to 2SP for decision-makers who seek to optimise supply chain configurations while ensuring that total costs cannot exceed a certain budget ceiling. It is worth noting that, based on the risk aversion attitude of the decision maker, the values of λ and ω in Eq.14 can be adjusted to make the model solution tilts towards cost variance minimisation or avoidance of certain undesirable site conditions.

With regards to the warehouse location, Table 10 provides the objective function value determined by the RO model for each distinct locations. Z1 was chosen as it produces the arrangement with the lowest cost value. At this point, it must also be mentioned that there is a trade-off associated with the selection of warehouse locations. As shown in Table 1, Z1 is closest to the site and therefore is associated with the least probability of delivery delays (Table 3). On the other hand, it also has the least capacity and highest establishment cost. Nevertheless, our analysis indicates that a distance reduction between the construction site and the warehouse can reduce the probability of delivery delays, in turn reducing idleness in the assembly process and higher overall construction efficiency.

Table 10 The minimised objective function values at different candidate warehouse locations. Warehouse locations Z1 Z2 Z3 Z4 Z5 Z6

Objective function values 397,101 398,021 405,983 406,589 406,898 407,945

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Using the RO model we can also recommend an optimal manufacturing duration. As shown in Table 11, the manufacturing duration yielding the lowest objective value involves six weeks for initial inventory preparation and continues production for a further three weeks beyond the commencement of on-site assembly. At this point, we can observe that there exists a trade-off relating to the selection of optimal durations. In shorter total production timespans, the factory will be forced to operate at higher production rates, therefore increasing variable production costs. In longer production timespans, in addition to increasing inventory costs, lower production rates may result in unfulfilled demands, which in turn incur project extension costs.

Table 11 The objective function values under different production duration.Module assembly period (week)

    0 1 2 3 4

Initi

al in

vent

ory

prep

arat

ion

peri

od (w

eek)

1 - - - - 430,0782 - - - 420,890 423,8493 - - 420,484 413,832 418,0354 - - 414,256 408,623 411,6755 - - 408,215 402,768 406,091

6 - - 402,362397,10

1 410,3017 - - 415,552 404,063 417,2548 - - 428,752 417,261 430,4549 - - 441,952 430,655 443,65910 - - 455,479 443,654 456,854

Note: “-“ indicates that there insufficient time for module production

6. Conclusions

Modular building systems offer the opportunity to achieve higher efficiency and quality. However, successful implementation of the practice requires an understanding of how to design and optimise time-critical logistic systems. In this study, a robust optimisation model is proposed to identify the optimal supply chain configuration for modular construction projects, under demand uncertainties from the site. The model can recommend the most favourable production and transportation schemes, disclose the variations of inventory by considering all site demand profiles generated based on significant schedule deviation factors, and help select the best warehouse location. A customisable range of modular products, warehouse locations, and manufacturing periods can be considered, therefore ensuring that our formulation remains versatile and can be implemented across a range of setting without requiring major changes in the model structure.

By incorporating solution robustness, model outputs can remain optimal regardless of which demand scenario is realised in the future. Furthermore, using this approach logistics practitioners can hedge between cost optimality and risk-averseness, with particular reference to the effects of unfulfilled demand and excess site inventories. Therefore, the mathematical model proposed in this research can serve as a basis to support decision making by managers involved in the design of logistic processes for modular construction projects who possess

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an attitude toward risk aversion. The model can account for any number of delay factors that are quantifiable with occurrence

probabilities. In this study, such probabilities were derived for inclement weather and transport delays through a statistical analysis of historic records, while the impact of reduced labour productivity and crane operation reliability was determined using event logs from real-world construction sites. Uncertain processes were modelled using continuous distributions and approximated using probability mass functions and discretisation methods (e.g. n-points estimation, Gaussian quadrature). This results in a finite number of demand scenarios that can be analysed using a reasonable amount of computational resources, and are therefore appropriate for most real-world modular construction projects.

Finally, the approach described in this study can provide insights on inventory variations over time, and across the entire supply chain of the project. This level of information clarity is invaluable for effective management of production buffers, as well as the optimisation of resource and capital allocations across their entire operations portfolio.

AcknowledgementsThe authors acknowledge the support Top University Strategic Alliance (TUSA) PhD scholarships from Taiwan.

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Appendix Indices

j Product types; j∈ J , J is the set of module product typess Demand scenario; s∈S, S is the set of demand scenariosz Warehouse location; z∈Z , Z is the set of warehousesa Working weeks of the construction project; a∈ A , A = {1, 2, …., LW}

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p Weeks for preparing initial inventory; p∈P, P = {1, 2, …., PW}Parameters: Construction sites

Ps The probability that scenarios occursDDs The construction duration of scenarios in weeksLW The longest working duration among all scenarios in weeksDsa j The demand of product j at the site in the week a in scenariosSF The fixed overhead at the construction site per week N The maximum number of products can be stored on the site without stacking

SCC Storage cost per m3 per week at the construction site

Parameters: WarehouseVj The volume of product j in m3 WCAPz Maximum storage capacity for warehouse z in m3

SCW Storage cost per m3 per week in warehouses

WEC z The cost for establishing warehouse zParameters: Factory

MMR j The maximum weekly manufacturing rate of product j

MF Fixed cost per week if the factory is operatingMV j Basal unitary (variable) cost for manufacturing one product j

FCAP Maximum inventory capacity in the factory in m3

PW The last week of the initial inventory preparation period

SCF Storage cost per m3 per week in the factory

M M is a very big positive real numberParameters: Transportation

D zFW The distance between factory and warehouse z in km

D zWC The distance between warehouse z and site in km

TC FW Transportation cost from factory to warehouse per truck per km

TCWC Transportation cost from warehouse to all sites per truck per km

L jFW The quantity of product j that can be loaded onto a single truck running on the route FW

L jWC The quantity of product j that can be loaded onto a single truck running on the route WC

LP z The lateness penalty between the site and warehouse zDecision variables

maj Manufacturing quantity of product j per week at the factory in week a

m pj Manufacturing quantity of product j per week at the factory in week p

ez Whether a certain warehouse z is established or not

i pzjW The quantity of initial inventory of product j in warehouse z in week p

iajF The quantity of inventory of product j in the factory in week a

iazjW The quantity of inventory of product j in warehouse z in week a

isajC The quantity of inventory of product j at the construction site in week a in scenario s

t pzjFW Transportation quantity for initial inventory product j from factory to warehouse z on week p

t azjFW Transportation quantity of product j from factory to warehouse z in week a

t azjWC Transportation quantity of product j from warehouse z to site in week a

y p Whether the factory carries out manufacturing on a specific week during the preparation period

ya Whether the factory carries out manufacturing on a specific week during the construction period

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x The total duration of the optimal transportation scheme in weeksca Whether any product has been sent to the site or not in week a

θ Dummy decision variable for linearising the second term of the objective function

λ λ =1, the parameter controls the trade-off between solution and model robustnessω ω =1,the parameter controls the trade-off between solution and model robustness

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