Parallel machine replacement with shipping decisions

International Journal of Production Economics 218 (2019) 62–71

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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe

Parallel machine replacement with shipping decisions a

b

Brett A. Shields , Javad Seif , Andrew Junfang Yu a b

T

b,∗

Department of Physics and Engineering Francis Marion University Florence, SC, 29502, USA Department of Industrial and Systems Engineering University of Tennessee at Knoxville Knoxville, TN, 37996, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Parallel machine replacement Physical asset management Logistics Project management Fleet replacement

In this research, the Parallel Machine Replacement Problem is adapted to include shipping decisions between demand sites. The formulation arises from an application in the construction industry, where the shipping of assets is vital. The benefit of including shipping decisions with replacement analysis is presented in a numerical example where heavy machinery is considered. The formulation is shown to significantly increase in computational complexity as the number of demand sites increases, and therefore an efficient heuristic algorithm is presented to solve large-size problems. Managerial implications are derived from sensitivity analysis on the shipping cost-per-mile, and economic impact is discussed. It is shown that the optimal solution does not change even when shipping cost inflates significantly.

1. Introduction In industry, companies must make decisions for the physical assets they use to satisfy demands. Replacement decisions are often studied in fleet management and economic analysis, see (Hartman and Tan, 2014). Shipping decisions are also studied vastly, yet never in conjunction with replacement analysis to date, see (Hartman and Tan, 2014) and (Montoya-Torres et al., 2015). Some industries such as transportation, trucking and construction can benefit from considering the decisions simultaneously. Specifically, the construction industry is known to ship assets from project to project to satisfy machinery needs. The machines may spend some time at a project and then are shipped to satisfy demand elsewhere. With a well-defined demand, there can be significant cost reduction, given an optimal replacement and shipping schedule. Simultaneous decision making for replacement and shipping sheds light on opportunities that minimizes the cost of satisfying demand across the network, through exchange of machinery between demand sites. These alternatives would be missed if the decisions were to be made separately. Projects can be located within one city or may exist across the country or world. Depending on the location of the project, machine salvage values, and purchase prices, shipping rates may change, and it is often that construction companies need to make annual, monthly, and even weekly decisions on where to send an asset to satisfy demand. Because the shipping of some assets is difficult, it is imperative that an accurate movement of machines is in place that also tracks age and utilization. Consider a 50,000lb excavator; this machine can take

∗

multiple days to transport, gain permits, schedule escorts, and may require alternative routes. Therefore, having an optimal schedule can mitigate delays in arrival time and potentially reduce run-overs of projects. Replacing the heavy equipment must happen at some point in time. Therefore, it is desired to know when and where the machines are salvaged and purchased within the network. A significant cost in the management of large assets is that of shipping. Shane, Molenaar (Shane et al., 2009) state that poor estimation and project schedule changes lead to escalation of project costs. Large construction equipment is frequently transported from project to project as either new demand is realized, or an asset needs to return to a holding station or previous project. Without an optimal shipping and replacement schedule, it may be the case that excess costs will be incurred. The reason that these specific considerations have not been addressed before in the literature is that few have considered multiple demand sites, as seen in the construction industry. To solve this, a new modelling structure is proposed that allows for the assets to be used at various demand sites. This includes a cost matrix for the transportation of machines in various time-periods. To represent this problem, the Parallel Machine Replacement Problem (PMRP) is adapted to include a shipping network. The PMRP is the problem of finding an optimal schedule for replacing a set of assets that are economically interdependent and satisfy demand in parallel. Classically, this problem decides when to keep or replace assets, yet more recently many other decisions have been included such as: renting, holding, operating, and now shipping. PRMPs have appeared in the literature since 1955 according to (Hartman and Tan, 2014). The

Corresponding author. E-mail addresses: [email protected] (B.A. Shields), [email protected] (A.J. Yu).

https://doi.org/10.1016/j.ijpe.2019.04.032 Received 22 August 2018; Received in revised form 23 April 2019; Accepted 25 April 2019 Available online 04 May 2019 0925-5273/ © 2019 Published by Elsevier B.V.

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B.A. Shields, et al.

term “parallel” replacement was coined by (Vander Veen, 1985; VanderVeen, 1985). Multiple asset replacement is concerned with replacing multiple assets utilized in an integrated system, therefore the replacement decisions of all assets in that system must be considered simultaneously (Leung and Tanchoco, 1990).

in the UK (Ansaripoor and Oliveira, 2018). Des-Bordes more recently considered the PMRP under technology change as well as a case study on MRI machines in 2017 with Büyüktahtakin (des-Bordes and Büyüktahtakın, 2017). In this work the authors allowed for machines to satisfy demand that was service dependent by allowing for two sets of machines: a set whose demand can only be satisfied by their own asset style, and a set that can be satisfied by its own style as well as the first set. Most recently in 2018, Seif et al. studied parallel machine replacement considering the uncertainty of planning horizons in construction projects (Seif et al., 2018). A significant literature gap is that of multiple demand sites and transportation decisions. Only one paper to date has considered multiple demand sites mathematically, namely (Hartman and Ban, 2002). In this work, the machines are serial-parallel and have various operations. Yet, the way in which Hartman and Ban formulated their mathematical model, the structure is equivalent to multiple demand sites, without a distance matrix or transportation decisions. This was modelled in such a way to account for the locations of the machines in various sites within a warehouse. As far as we know, no research to date has considered shipping decisions integrated with PMRP. In this paper, we present a novel mixed-integer program that integrates the replacement and shipping decisions. Because the new formulation makes the already-NP-hard PMRP more difficult to solve, we also introduce a heuristic algorithm and through computational experiments demonstrate its effectiveness in solving large-size problems. The rest of the paper is organized as follows: in Section 2, the methodology is discussed in which a MILP model and the heuristic algorithm are presented. In the following section, a numerical example is presented using data from (Seif et al., 2018) that provides a realistic look at how the replacement of assets is scheduled alongside of shipping decisions. Here, sensitivity analysis is discussed on shipping cost-per-mile. Next, computational results are presented that provide the algorithm efficiency in terms of optimality and computation time. Lastly, the concluding remarks and future work is presented.

1.1. Literature review The earliest work in the replacement of a set of machines was (Leung and Tanchoco, 1986), where the authors provided the initial framework for such a problem. Early works in multiple asset replacement such as (Jones et al., 1991) simplified the solution space using two rules: 1. The no-splitting-rule and 2. The older cluster replacement rule. The no-splitting-rule states that all same aged assets are kept or replaced at the same time, while the older cluster replacement rule states that a machine of older age is always replaced before a newer one. The rules require assumptions of non-decreasing operations and maintenance costs, non-increasing salvage values, and their sum to be nondecreasing (Hartman and Tan, 2014). Stinson and Khumawala (1987) studied the multiple machine replacement problem considering lost production costs and machine downtime costs. The authors also present a heuristic algorithm for solving large problem size and provides multiple ranked alternative solutions. As problems became more realistic, more advanced models and solution procedures arose (Lotfi and Suresh, 1994). considered a nonlinear mixed-integer linear programming (MILP) formulation of the replacement of computer numerical control machines, solved with a two-level exact solution method via Dynamic Programming. In (Chand et al., 2000) Chand extended the standard PRMR to include capacity expansion. That is, demand is assumed to increase over time, Chand also considered economies of scale (McClurg and Chand, 2002). developed a forward-time algorithm that determines the optimal replacement plan for the generalized PMRP. Proofs of the no-splitting rule and the older cluster replacement rule were presented by (Tang and Tang, 1993), and extensions of these proofs are found in (Hopp et al., 1993). Buy, lease, and rebuild decisions are studied by (Hartman and Lohmann, 1997), where the authors use Integer Programming to solve computational experiments. In (Rajagopalan, 1998; Rajagopalan et al., 1998) capacity expansion is introduced into the equipment replacement literature. Here, future demand changes and economies of scale are considered in a general model that solves efficiently. Keles and Hartman (2004) considered a mixed integer programming formulation for the replacement of a bus fleet. They determined implications of the model via sensitivity analysis and solved the problem with a commercial solver. Büyüktahtakin and Hartman considered the PMRP under technology change and deterioration and gave an integer programming formulation and provide analysis on the effects the considerations had on the solution (Büyüktahtakin and Hartman, 2009). A famous work by Hartman was that of replacement under stochastic horizon in 2004 (Hartman, 2004). Here, Hartman provided optimal utilization of machines given various assumptions of the input data (Parthanadee et al., 2012). studied the multiple machine replacement problem with assumptions that newer machines would be utilized more and studied: purchase-new-vehicles-only, no-splitting-in-selling, onepurchase-choice, older-vehicles-selling, and all-or-none rules. In 2014 Büyüktahtakin studied the parallel replacement problem under economies of scale (PRES) (Esra Büyüktahtakin et al., 2014). Here the author proved the NP-hardness of the problem formulation by transforming a 3SAT problem into the proposed formulation. Also in 2014, Seif and Rabbani studied the PMRP considering component based life cycle costing and used a Genetic Algorithm as a solution method for a real industry problem (Seif and Rabbani, 2014). More recently (Büyüktahtakın and Hartman, 2016) studied the PMRP under technology changes, considering capital gains where newer machines have a larger capacity. In 2016, Ansaripoor et al. studied a replacement problem from a risk management standpoint and provide a case study

2. Methodology In this section, a Mixed Integer Linear Programming (MILP) model is presented that adapts the PMRP to include a network for shipping decisions. The model generalizes previous works by changing the problem structure to include network-flow type constraints. The flow of assets in and out of each construction project is preserved, while keeping an account on the age and cumulative utilization of the assets over time. Subsequently, a Clustering Decomposition Algorithm (CDA) is presented to solve for a large number of projects (construction or mining work zones), or when monthly decisions need to be made (both of which significantly increase the problem complexity). 2.1. PMRP with shipping (PMRP-S) Here, the considerations of replacement are integrated with transportation to and from various demand sites, denoted as projects, throughout the planning horizon. For the first time, network flow constraints that allow for shipping decisions to be made are included into the PMRP model. Replacement, holding, renting, and operating decisions are also considered for practicality. The problem formulation stems from applications in the construction industry where shipping is often part of the demand satisfaction. Although decisions are made at the end of the time-period, shipping occurs in between time periods, presented as beginning of time-period decisions in terms of notation, as seen in the results. The presented formulation is a Parallel Machine Replacement Problem with Shipping, denoted PMRP-S. Sets and Parameters.

U Threshold for a machine's cumulative utilization L Threshold for a machine's cumulative age 63

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B.A. Shields, et al.

yijtp , z tp, Iijtp, aijtp ∈ +,

P Number of demand sites in varying location T Length of the planning horizon (may be annual, monthly, etc.) cijtp Cost of purchasing an asset with age and cumulative utilization (i, j ), in time-period t at project location p qtp Cost of renting an asset for one time-period t at project location p htp Cost of holding an asset for one time-period t at project location p otp Operating costs of an asset for one time-period t at project location p mijtp Cost of maintaining an asset with age and cumulative utilization (i, j ) for one time-period t at project location p sijtp Salvage/Selling profit for an asset with age and cumulative utilization (i, j ) in time-period t at project location p rtp Number of required assets in time-period t for project p tc (nm) Cost of shipping an asset from n to m , where (n, m) ∈ P

yijtp Number of assets with age and cumulative utilization (i, j ) to buy in time-period t for demand site p z tp Number of assets to be rented in time-period t for demand site p Sijtp Number of assets with age and cumulative utilization (i, j ) to sell (salvage) in time-period t for demand site p Iijtp Number of assets with age and cumulative utilization (i, j ) that are idle in time-period t for demand site p aijtp Number of assets with age and cumulative utilization (i, j ) in time-period t for demand site p that are operating TCijt(nm) Number of assets with age and cumulative utilization (i, j ) in time-period t shipped from n to m

2.2. Clustering Decomposition Algorithm (CDA) Because of the large number of possible routes available in each

(

Minimize ζ= ∑t = 1 ∑p = 1 ⎡qtp z tp + ∑i = 1 ∑ j = 1 (cijtp yijtp + htp Iijtp + (otp + mijtp) aijtp − sijtp ⎢ ⎣ L

U

(1) The objective is to minimize the total cost of satisfying machine requirements over the duration of multiple projects. The objective is given by the sum of three components: 1. The cost of satisfying demand by renting, 2. The life-cycle cost of purchased machines (cost of purchasing plus the cost of the operating, holding, maintaining, and subtracting end of life salvaging value), and 3. The cost of satisfying demand with shipping an asset that is currently in the fleet. All costs are assumed to be in United States dollars and are discounted to time zero. The objective is subject to the following constraint sets: M

⎛

L

U

⎞

k=1

⎝

i=1 j=1

(2)

P−1

aijtp + Iijtp + Sijtp = yijtp +

Shipping is observed to be demand and cost dependent, therefore the frequency matrix will give more insight into which projects send and receive machines than simply considering the closest or cheapest shipping routes. As an example, a shipping may occur when the demand of one project decreases while another project has a surplus of machines for a particular time-period, even if a cheaper shipping route exists to a third project. It should be noted that if demand is stationary, shipping may not occur. This provides an efficient solution time, while allowing for a minimal trade-off in optimality. To illustrate the process, a small general graphical representation is presented, as well as the pseudocode below.

∀ p ∈ P , ∀ t ∈ [2, T ]

⎠

P−1

∑ TCijt(np) − ∑ n≠p

+ I(i − 1) j (t − 1) p ,

TCijt(pm) + a(i − 1)(j − 1)(t − 1) p

m≠p

∀ t ∈ [2, T ], i ∈ [2, L], j

∈ [2, U ], ∀ p ∈ P

)

1. Breakdown the planning horizon into a number of smaller timewindows. 2. Solve the problem for each time-window iteratively. 3. After each problem is solved, we record the number of times a shipping occurs between any pair of projects. 4. Add these values to a matrix that show the frequency of shipping between any two projects. 5. Cluster the projects based on the frequency of shipping between them. 6. Now that the problem is decomposed into problems with fewer number of projects, solve the small problems individually. 7. Aggregate the solutions.

T L U P P Sijtp) ⎤ + ⎡∑t = 1 ∑i = 1 ∑ j = 1 ∑m = 1 ∑n = 1 tc (nm) TCijt(nm) ⎤ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦

∑ ⎜z tp + ∑ ∑ aijtp⎟ ≥ rtp,

P (P − 1)

, where P is the number of projects , in order to time-period 2 satisfy demand while considering the optimal replacement schedule, a clustering of the projects is used to efficiently solve large problem size. The premise is to decompose a large size problem into smaller instances, with much fewer number of decisions and then solve the presented model for each smaller test problem and aggregate the solutions. The process involves the following steps:

The mixed-integer model is formulated as follows:

P

(3)

ai (U − 1) tp = 0

∀ t ∈ [2, T ], i ∈ [2, U ], ∀ p ∈ P

(4)

a(L − 1) jtp = 0

∀ t ∈ [2, T ], j ∈ [2, U ], ∀ p ∈ P

(5)

Clustering Decomposition Algorithm (CDA)

(6)

Inputs: T Length of the planning horizon s Maximum cluster size Length of the planning horizon in the sub-problems d P Set of all the nodes (projects) Define F = 0 P × P //The matrix of frequencies

S(L − 1) jtp = y(L − 1) jtp − a(L − 2)(j − 1)(t − 1) p + I(L − 2) j (t − 1) p, ∀ t ∈ [2, T ], j ∈ [2, U ], ∀ p ∈ P Si (U − 1) tp = yi (U − 1) tp − ai (U − 2)(t − 1) p ,

(8)

Constraint (2) is the demand constraint, ensuring at least the number of required machines are placed in each project for each timeperiod. Constraint (3) ensures the flow of machines entering and leaving a project is balanced. This constraint ensures that no shortage or excessive number of machines are in the system. The remaining constraints enforce the machines to be salvaged or sold when they reach either of their thresholds (U or L ). It can be seen in Constraint (3) that shipping decisions are made at the beginning of the time-period, while other decisions are made at the end. This is to attempt to model the decision in between the time-period, which is a more accurate depiction of reality. Constraints (4) and (5) ensure that any machine that has reached its threshold for (U or L ) does not operate. Likewise, constrains (6) and (7) ensure an at threshold machine must be salvaged. Lastly, constraints (8) require integer solutions. Note that because it is not possible to operate a machine more than the available hours in a year (365 × 24 = 8,760 hours per year), the utilization level cannot be more than 4 (1 ≤ j≤ 4 ) when its age is one-year old (i= 1), and the same logic is used for all time-periods. For a more details see (Seif et al., 2018).

Decision Variables.

T

∀ t ∈ [1, T ], i, j = 1, …, U , ∀ p ∈ P

∀ t ∈ [2, T ], i ∈ [2, L], ∀ p ∈ P (7) 64

International Journal of Production Economics 218 (2019) 62–71

B.A. Shields, et al. Define C = ∅ //Set of clusters Define θ a (0,1) -matrix of size P by P //θi, j = 1 if shipping has happened between nodes i and j , and 0 otherwise. Define P ′ = P Define t = 1 Solve the master problem for the planning horizon [t , t + d] Get the value of θi, j , ∀ i&j ∈ P Set Fi, j = Fi, j + θi, j , ∀ i&j ∈ P Set t = t + 1 While t + d ≤ T do Solve the master problem for the planning horizon [t , t + d] Get the value of θi, j , ∀ i&j ∈ P

Fig. 2. Frequent shipping routes of a network (bold).

Set Fi, j = Fi, j + θi, j , ∀ i&j ∈ P Set t = t + 1 End While Find(i, j ) ∈ (P , P ), i ≠ j such that Fi, j ≥ Fk, l, ∀ (k , l) ∈ (P , P ) //Find the largest element of F Add the set {i, j} to C SetFi, j = 0 Set P ′ = P ′\ {i, j} While P ′ ≥ 1 do Find(m, n) ∈ (P , P ), m ≠ n, m or n ∉ C such that Fm, n ≥ Fk, l, ∀ (k , l) ∈ P For all subset ∈ C do If subset < s AND ∃ (a, b) ∈ ({m, n}, subset ) such that Fa, b > 0 do Add the qualified node to subset \break the ties by largest connection Else do Add the set {m, n} to C End If End For SetFm, n = 0 Set P ′ = P ′\ {m, n} End While For all subset ∈ C do Solve the master problem for the planning horizon [1, T ] End For Aggregate the solutions

Fig. 3. Clusters determined by the shipping frequency.

two special cases. The problem that considers all possible shipping routes between all projects is denoted as the Master Problem. This is the desired problem to solve and should be used if computational complexity is not of concern. If no significant frequencies are observed and every project is in its own cluster, then this case is equivalent to solving the special case of P number of Master Problems without considerations of shipping, denoted P − Base . The optimal Objective Function Value (OFV) of the Clustering Decomposition Algorithm falls between the OFVs of the Master Problem and P − Base Problem. Solving without shipping would yield a faster, yet sub-optimal solution, while including shipping would result in a computationally more expensive problem that yields a global optimal solution. This provides a valid upper and lower bound for the algorithm's optimal OFV. If we let Z ∗, H ∗, and Y ∗ be the optimal OVFs of the Master Problem, Clustering Decomposition Algorithm, and the P − Base Problem respectively, the bounds on optimal OFVs for the minimization problem defined by Equations (1)–(8) would be:

To understand the general clustering, consider a small network of four nodes and all possible edges, denoted a K 4 , or a complete graph with four vertices, shown in Fig. 1. In general, as defined in (Chartrand et al., 2010), a Kn is a complete graph of n number of nodes or vertices and we will use this notation to discuss the algorithm. Suppose that a number of smaller time periods is solved, and shipping only occurs between nodes (1 and 2) and (2 and 3), as seen in Fig. 2 denoted with bold edges. In this case, we would aggregate these three nodes into one cluster, and node 4 would be set into a separate cluster, Fig. 3. In Fig. 2, the bold edges represent the most frequent shipping routes. These frequencies determine that projects 1,2,3 should be clustered together (K3) , while project 4 is clustered separately (K1) . Subsequently, each cluster would be solved independently, and the objective function values and solutions aggregated to form the final solution. It should be noted that if the frequencies yield a connected graph between all projects, this structure would be equivalent to solving the problem defined by equations (1)–(8), KP with P number of projects, and the reason for the limit on cluster size, s . We use this notation to define three problem types: the presented formulation and

Z∗ ≤ H∗ ≤ Y ∗ Because the solution space of the Master Problem is a superset of both the P − Base and CDA Problems, the optimal OFV of the Master Problem will always be less than or equal the OFV of those two problems. Additionally, because in an extreme scenario of the CDA, each project is considered as a cluster, the OFV of the P − Base Problem is always greater than or equal to the OFV of the CDA. The proofs are presented in detail below. Proposition 1. For the defined minimization problem, the optimal OFV of the Master Problem is less than or equal to the optimal OFV of the P − Base Problem. Proof Let Equation (1) define the OFV of the Master and P − Base Problems. Let Z ∗ and Y ∗ be the optimal OFVs and x ∗ and y∗ the corresponding optimal solutions of the minimization problem for the Master and P − Base Problems defined by Equations (2)–(8), respectively. Without shipping, the Master Problem is equal to the P − Base Problem. Similarly, solving a problem instance of Master Problem in which the shipping costs are sufficiently large, is practically the same as solving the P − base Problem. Decreasing the shipping costs will allow new solutions in which shipping happens. That is, any feasible solution of the P − base formulation is a feasible solution of the Master Problem. Therefore, the solution space of the Master Problem

Fig. 1. A starting graph with four projects and all possible shipping routes for one time-period (K 4 ) . 65

International Journal of Production Economics 218 (2019) 62–71

B.A. Shields, et al.

3. Numerical example

contains all points of the P − base problem, and we can say that the solution space of the master problem is a superset of the P − Base problem’s solution space. We can say that Z ∗ is never worse than Y ∗.

In this section, we demonstrate the application of the model on a realistic dataset. The data for the input parameters is derived from two main sources: a recent publication that includes a case study in construction industry (Seif et al., 2018), and various online sources (see (Keles and Hartman, 2004), (Transport, 2017) and (MachineryTrader.com., 2017)). Here the replacement of excavators is considered in multiple construction projects across the United States over a ten-year period. The specific parameter values are assumed for a CAT 320 EL (Caterpillar. 320Hydra, 2013), which is a popular asset in the construction industry as defined in the mentioned case study by (Seif et al., 2018; Caterpillar. 320Hydra, 2013). Here the authors generalized functions first appearing in (Hartman, 2004), and fit real data to the functions to derive purchase, salvage, and maintenance costs accurately. The maximum age of a machine (L ) is assumed to be 6 years. Excavators usually operate around 2000 h per year, up to 12,000 h in a lifetime, which leads to a maximum cumulative utiliza12000 tion level (U ) of 2000 = 6 . Note that if it is desired to make monthly decisions, U and L have to be multiplied by 12, giving U = L = 72 . This greatly increases the complexity of the problem, as discussed in detail in Computational Results. We consider an initial purchase price of $220,000 and annual maintenance cost of $10,000. Salvage values are assumed to be 57% of the initial purchase price, see (Keles and Hartman, 2004), in each time period and equivalent for each project. We also assume a holding cost of $5000 and an operating cost of $50,000, annually. We consider six cities as demand sites: San Antonio, TX, Bozeman, MT, Los Angeles, CA, Orlando, FL, Chicago, IL, and San Francisco, CA. In general, the demand sites can be geographically close (within one state) to vastly distant (internationally). The cities where chosen to represent a case in between the two extremes (nationally). The number of excavators demanded in each city's construction site can be seen in. Table 1. The length of the project in each city varies from 5 years, being the shortest duration in San Francisco, CA, to 8 years in Orlando, FL. The distribution of demand was chosen to represent various scenarios that may occur: stationary demand periods, a monotonically increasing or decreasing demand, and a demand which increases to the middle of a project and decreases until the end of the planning period. The assumed demand structures and values are consistent with the case study in (Seif et al., 2018). The distances between each pair of cities were determined online, taking only the routes in which a large excavator could be sent (interstates), shown in Table 2. It should be noted that shipping cost is direction-independent in the numerical example yet the model can easily handle various costs in each direction. The assumed cost-per-mile for shipping is $4.00 over a 10-year period (Transport, 2017). Fig. 4 and Fig. 5 from (Pdf Printable Us States M, 2018) gives a visual representation of the city locations and all possible combinations of

Furthermore, by the properties of infimum, if a set A is a subset of a set B , any function defined on the appropriate domain mapping to R gives:

min(f (B )) ≤ min(f (A)) Thus, with the objective function defined for the minimization problem,

= > ζ (x ∗) ≤ ζ (y∗ ) = > Z∗ ≤ Y ∗ ■ Proposition 2. For the defined minimization problem, the optimal OFV of the Clustering Decomposition Algorithm (CDA) is greater than or equal to the optimal OFV of the Master Problem, and less than or equal to the optimal OFV of the P − base problem. In short,

Z∗ ≤ H∗ ≤ Y ∗ Proof We have already shown the Master Problem has an optimal OFV at most that of the P − Base Problem. Here we have three cases as the output of the CDA: Case a. The number of clusters is equal to the number of projects Case b. The number of clusters is exactly one Case c. The number of clusters is strictly greater than one and strictly less than the number of projects. For Case a: the number of clusters is equal to the number of projects, CDA is equal to the P − Base problem, and we are done.

Z∗ ≤ H∗ = Y ∗ For Case b: the number of clusters returned is one, then all projects are included into the problem, and CDA is equal to the Master Problem. In this case we are done.

Z∗ = H∗ ≤ Y ∗ For Case c: Because all clusters have to be considered and the CDA utilizes the Master Problem and the P − Base formulations only, then the least number of clusters is 2 and the most number of clusters is P − 1 (one cluster with two nodes and the rest are isolated nodes), with integer number of clusters required. P − 1 clusters is the closest case to the P − base . If shipping happens between the two nodes, the optimal OFV (minimization) will be strictly less than that of the P − base and otherwise it will be equal. As the number of clusters decreases the possibility of shipping, and hence a lower OFV, increases due to a greater number of potential shipping candidates. A two-cluster scenario is the closest scenario to the Master Problem case. If the optimal solution of the Master Problem includes an edge between any two nodes from two different clusters, we have to remove that edge in the CDA method because otherwise we will go back to Case b. Therefore, the optimal OFV of CDA method cannot be less than that of the Master Problem formulation. Therefore, the solution space is a superset of the P − base problem formulation and a subset for the Master Problem. Then, using the properties of infima as before, the minimum of the CDA is at most that of the P − Base , and at least that of the Master Problem formulation. ■ These results are shown to hold true in the Computational Results Section.

(

)

6(6 − 1)

= 15edges . Even shipping routes available for each time-period 2 for a small network, the number of possible solutions is quite large for shipping alone. Here P = 6, I = J = 6, and T = 10 and the decisions being made include buying, holding, operating, selling, and shipping Table 1 Machine requirements at each city in each year. Project

San Antonio, TX Bozeman, MT Los Angeles, CA Orlando, FL Chicago, IL San Francisco, CA

66

Time-Period (Year)

Total

1

2

3

4

5

6

7

1 3 5 1 2 4

1 4 4 1 4 5

3 4 4 4 4 5

5 5 3 4 4 6

5 4 2 5 4 7

3 4 1 4 1

2 2 3

8

2

9

10 20 26 19 24 19 27

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B.A. Shields, et al.

Table 2 Distance matrix between cities. Project Location

San Antonio, TX Bozeman, MT Los Angeles, CA Orlando, FL Chicago, IL San Francisco, CA

Project Location San Antonio, TX

Bozeman, MT

Los Angeles, CA

Orlando, FL

Chicago, IL

San Francisco, CA

0 1613 1350 1160 1200 1730

1613 0 1100 2397 1388 1000

1350 1100 0 2500 2000 382

1160 2397 2500 0 1124 2813

1200 1388 2000 1124 0 2132

1730 1000 382 2813 2132 0

happens in time-period six, in which two machines near the end of their life are sent from San Francisco to Bozeman, MT to finish out their useful lives. Lastly, one machine is sent from Chicago to Orlando in year seven, also at the end of its life. Considering the first shipping instance, it is easy to see that demand in San Francisco dropped, while the demand in Los Angles increased, allowing for the opportunity for transportation to satisfy demand. It can also be observed that demand increased in Chicago as well, yet the cheaper option was to ship to the closer city. In the next shipping schedule, demand increased in San Francisco, yet there were already excess machines in holding in said city from the previous time-period, and the model took advantage of the idling ability to send the assets to Los Angles in the next time-period. This demand increase was less than the number of machines available in San Francisco; therefore, the model would need to decide to hold for another year, sell the machine, or ship it. This instance would not be possible with the current modelling techniques that exist. The third shipping happens in the sixth time-period, when demand in San Francisco drops to zero (i.e. the project ends). In Bozeman, two machines are at the end of their lives and need to be salvaged. The model globally realizes the advantage and ships two machines to Montana. The last shipping occurrence is like that of the shipping in time-period six. The Chicago project is coming to an end and although demand in Orlando is decreasing, it is advantageous to ship an excavator there and finish out the machines life. The shipping routes can be seen visually in Fig. 5, where orange nodes are cities that have been transported to or from. The benefit of considering the integration of shipping decisions into PMRPs can be quantified, even in this small example, as significant. The

machines. Note: renting is not considered for ease of understanding yet can be included application. The results in Table 3 show the optimal schedule for purchasing, holding, salvaging and shipping the excavators to satisfy demand. The solutions are represented as follows: the first letter indicates the type of decision (P for purchase, I for inventory/idle, S for salvage/sell, and O for operating) followed by the (i, j ) values (age and cumulative utilization level) and the number of machines, except for shipping, which is represented as an age and cumulative utilization pair T (i, j ) , then the number of machines shipped, followed by a pair of cities shipped from, (g → h) , for some g , h ∈ P . Consider the first time-period schedule for San Antonio, TX. The demand is one machine and the model simply obtains a new machine to satisfy the requirement. The decision also shows that the purchased machine is operated in that same time-period. In the subsequent time-period, demand stays uniform at one, yet another excavator is purchased to take advantage of the increase in demand for the next time-period, which jumps to three. This means that the decision to obtain and hold the asset (un-utilized) is a cheaper option than to buy two new in the next time-period. The schedule continues in this standard fashion until time-period six, when an asset has reached the end of its useful life and must be salvaged. Notice that the sole machine required in the last time-period is not salvaged until the next time-period after it has operated to satisfy demand. There are four shipping instances in this example. In time-period two, one excavator is shipped from Los Angeles to San Francisco of age one-year-old and that has worked 2000 h. Next, in year 5, one excavator is transported back to Los Angeles from San Francisco of age four years old and has worked around 6000 h. Note: this is a different asset than was sent to that project originally. The next shipping occurrence

Fig. 4. All possible shipping routes between the 6 cities. 67

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Fig. 5. Visual representation of the shipping solutions (decisions). Table 3 Results for the numerical example of six cities. Time-Period (years) 1

2

3

4

5

6

7

8

Demand Schedule for San Antonio, TX

1 P(1,1)1 O(1,1)1

1 P(2,1)1 O(2,1)1 I(2,2)1

3 P(2,1)1 O(2,1)1 O(3,2)2

5 O(3,2)2 O(4,3)1 O(5,4)2

3 O(4,3)2 O(5,4)1 S(6,5)2

2 O(5,4)2 S(6,5)1

S(6,5)2

Demand Schedule for Bozeman, MT

3 P(1,1)1 O(1,1)1 P(2,1)2 O(2,1)2

4 P(2,1)1 O(2,1)1 O(2,2)1 O(3,2)2

4 O(3,2)1 O(3,3)1 O(4,3)2

4 O(3,2)2 O(5,4)2 S(6,5)2

4 O(4,3)2 O(5,4)2 S(6,5)2

2 O(5,4)2 S(6,5)2

S(6,5) 2

Demand Schedule for Los Angeles, CA

5 P(1,1)2 O(1,1)2 P(2,1)3 O(2,1)3 1 P(2,1)1 O(2,1)1

4 O(2,2)1 O(3,2)3

4 O(3,3)1 O(4,3)3

5 P(2,1)2 O(2,1)2 O(3,2)1 O(4,3)2 5 P(2,1)2 O(2,1)2 O(4,3)1 I(4,4)1 O(5,4)2 3 I(4,4)1 O(5,4)3

2 O(4,3)1 O(5,4)1 S(6,5) 3

1 O(5,4)1 S(6,5)1

S(6,5)1

1 P(2,1)1 O(2,1)1 I(3,2)1

4 P(2,1)2 O(2,1)2 O(3,2)1 O(4,2)1

4 O(3,2)2 O(4,3)1 O(5,3)1

4 O(3,2)2 O(5,4)2 S(6,5)1

3 O(4,3)2 O(5,4)1 S(6,5)2

Demand Schedule for Chicago, IL

2 P(1,1)1 O(1,1)1 P(2,1)1 O(2,1)1

4 P(2,1)2 O(2,1)2 O(2,2)1 O(3,2)1

4 O(3,2)2 O(3,3)1 O(4,3)1

Demand Schedule for San Francisco, CA

4 P(1,1)4 O(1,1)4

5 O(2,2)5

5 P(2,1)3 O(2,1)3 I(3,3)3 O(3,3)2

4 P(2,1)1 O(2,1)1 O(4,3)2 I(4,4)1 O(5,4)1 6 O(3,2)3 O(4,3)3 I(4,4)2

5 P(2,1)2 O(2,1)2 O(4,3)2 O(5,4)1 S(6,4)1 4 O(3,2)1 O(5,4)3 S(6,5)1

Demand Schedule for Orlando, FL

Shipping Routes

T(2,2) 1 from (3 → 6)

7 O(4,3)2 O(5,4)5

T(4,3)1 from(6 → 3)

2 O(5,4)2 S(6,5)1

9

10

S(6,5)2

1 O(4,3)1 S(6,5)3

S(6,5)5

T(5,4)2 from(6 → 2)

T(5,4)1 from(5 → 4)

Computational Results. Likewise, comparing this solution to a current suboptimal solution followed in practice (and not another PMRP solution) will provide significant cost savings of much more than $109,010.52 , as discussed in the Economic Impact section.

objective function value (OFV) of the Master Problem considering all shipping routes is $12,176,448.24 ; while, solving the problem without considerations of shipping and has an objective function value of $12,285,458.76, a $109,010.52 difference. This benefit will obviously amplify with a larger problem size or demand, as discussed in the 68

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(

$109,010.52

)

procurement $2,361,171.60 . The authors believe that the numerical example is a good representation of a standard construction project, and depending on the number of locations, planning horizon length, and asset type, the benefits may deviate. It should be noted that the savings mentioned is for shipping alone, considers one asset type, and is compared to another replacement system from the PMRP literature (without shipping). PMRP already provides significant economic benefits when implemented in construction industry (see, for example (Seif et al., 2018),). Using the 4.6% value can lead us to an approximate estimation of the model's impact in a larger scale. According to a 2018 market research report by Grand View Research, Inc., the global market for construction equipment is expected to reach a total of USD $112.25 billion by 2025, increasing annually from the 2017 market total of USD 76.87 billion. Of the 2017 market value, 70% was attributed to excavators alone. The US was responsible for over 36% of the total market for construction equipment in 2017 and the report predicts significant growth for the US excavator market. In general, the market is expected to increase for many construction equipment types (Construction Equipment Ma, 2018). From shipping cost savings alone, the impact can be significant. Combining this benefit with implementation of the model on all construction asset types and within multiple companies, and the expected economic impact is fundamentally noteworthy. Yet, although we will not attempt to extrapolate an exact predicted economic benefit in this research, the impact can be quite large in this case. For instance, if we generalize the results of the numerical example, and the methodology is implemented for each relevant asset type, for each large construction company, while assuming the same problem size, the total savings from the company perspectives could be estimated up to billions of dollars. While PMRP itself provides much cost benefit by optimizing the procurement and other strategic decisions, adding other assets intensive industries to the picture (e.g., commercial aerospace, mining, trucking) makes the impact even larger, exhibiting the value of the proposed model. The benefit of implementing the presented methodology in companies that currently do not have optimal replacement policies would be much greater than the presented benefit in the numerical example. For instance, consider a company that has a policy that only new purchases can be made and has not considered shipping optimization. Then, the OFV for the data presented in the numerical example would be $13,311,026.49, a difference of $1,134,578.25. This vast difference is because, in the example, we compared the presented methodology with the most recent PMRP method in the literature, which does not necessarily represent all or average application instances (because of novelty). It is estimated that a typical construction company can improve their procurement schedule with PMRP alone, yet shipping optimization provides further added value.

Fig. 6. The sensitivity of the OFV to a percentage increase in the shipping cost per mile.

3.1. Sensitivity of shipping cost Shipping costs are known to fluctuate due to changes in global or local economies. Here it is desired to determine managerial implications of an inflation of shipping cost-per-mile on an optimal solution to the defined problem. Other researchers have performed in depth sensitivity analysis on varying parameters, see for instance (Keles and Hartman, 2004) and (Seif et al., 2018), yet the focus here is to provide insight specifically to shipping. As described in the numerical example, shipping is both demand and cost dependent. Because demand is especially sensitive, and any fluctuation of demand may yield an infeasible solution, the sensitivity analysis is performed on the distance matrix, changing the assumptions of the cost-per-mile of the shipping cost. This is performed starting with a base cost of $1/mile and increasing to over $20/mile. As seen in Fig. 6, the increase in cost-permile has a linear effect on the value of the objective function. In fact, a large increase in shipping cost has a minimal effect on the percent objective increase. Note that each time the assumptions are changed, a MILP is solved for the new input and an optimal solution is determined. This means that observing a linear relationship in Fig. 6 implies that the decision variables associated with the shipping cost-per-mile are not changing, or an equivalent solution is available. Any nonlinear observation would imply variation in the decision variable from the previous optimal solution. And in fact, although not shown in the graph, the optimal solution for the particular numerical example did not change until the shipping cost-per-mile exceeded $30/mile, which we believe to be an unrealistic inflation of shipping price. For instance, even if the cost-per-mile of shipping an excavator increased to $20 a mile, the overall cost of satisfying demand only increases by 0.06%. This is due to the value of having an optimal shipping schedule incorporated into the replacement and demand satisfaction decisions. It is arguable that implementing this model formulation is imperative to any multiple demand site instance. That is, with the optimal shipping and replacement decision in place, outside influences on costs that would normally affect shipping are mitigated.

4. Computational results In this section, several large-size problems are solved for the instance where shipping is of concern. For the models that only consider one demand site, the solution time for any realistic problem size is instantaneous (solved with Gurobi, implemented in Python). For instance, solving a single project for over 100 years was less than a 60 s computation time. Even making monthly decisions, for any realistic case (monthly decisions for say three years) the solution time is also instant. Any monthly or annual decisions beyond these thresholds lose a realistic scheduling in that demand realization may change greatly over time, and when this is the case, Stochastic Programming may be the best solution method. For the shipping instances where the problem size and solution time grows much faster as more demand sites are solved for, an efficient solution methodology is possibly required. Nineteen test problems of various medium and large size are presented to evaluate the performance in terms of optimality and solution time for this case. The experiments were performed on a computer

3.2. Economic impact In this section we discuss the potential economic impact of the proposed model, specifically for shipping considerations. In doing so we compare the savings from shipping considerations with total expenditures on equipment in the numerical example. We use this comparison as a basis for estimating an approximate economic impact of the model on a larger scale. We showed in the numerical example that the amount of saving from shipping is $109,010.52 . We calculate the portion of the objective function value that excludes operations, maintenance, and holding costs, which is $2,361,171.60 . Therefore, the savings from shipping considerations is 4.6% of the total expenditure on asset 69

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complexity would require significant changes in the model and is not in the scope of this work. The computational results can be seen in Table 5. The objective function value (OFV) and the solution time in seconds are presented for each problem type and size. The gap between the Master Problem and the CDA is presented as Gap 1, and the gap between the CDA and the P − Base Problem is presented as Gap 2. The problem size changes in number of projects every five instances; consider the first five. The number of projects is five, and the Master Problem solves the model efficiently for the largest size in that realm in under 500 s. The performance of the CDA for small problem size is much less efficient, although the gap is always within 1% of the optimal solution. As the problem sizes increase, it is observed that the gap of the CDA stays significantly closer to the Master Problem, as the advantage of shipping is still in place. The average of Gap 1 is 0.16% while the average Gap 2 is 1.06% , both of which are considered good. Yet, the CDA performs much better in terms of optimality as the problems become much more complex, in retrospect to not considering shipping. The CDA provides the possibility of obtaining the optimal solution, as seen in Problem 7. For Problems 10,14,15,18,19, and 20, the Master Problem and the No Shipping was not able to find a solution, while the CDA provided a solution relatively fast. In one case, Problem 17, the Master Problem could not find a solution, yet the No Shipping formulation could. This shows the advantage of a less complex model on solution time, yet the CDA algorithm provided a solution that was 2.58% better. In general, the fastest of the three methods is without shipping considerations, yet when the problem size is sufficiently large, even this formulation cannot find a solution. This is due to the CDA taking advantage of the structure of the problem formulation and shipping frequencies, allowing the problem to be broken up into clusters of maximum defined size, here six.

Table 4 The problem sizes solved for each presented parameter. Problem Size Number

P

T

U

L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

5 5 5 5 5 10 10 10 10 15 15 15 15 15 25 25 25 25 25

6 8 12 24 36 6 8 12 36 6 8 12 24 36 6 8 12 24 36

72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72

72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72

cluster made up of 11 Dell DSS 1500 servers, each containing 2 Intel Xeon E5-2630 8-core processors running at 2.4 GHz. Each server has 32 GB of RAM. There are a total of 176 processor cores and 352 GB of RAM. Table 4 gives the sizes of the sets for each problem size solved. Up to 25 projects are solved because it was observed that the largest construction company in the world, Bechtel, has presented around this number of projects in the US that are currently under construction (although they have 100s of projects worldwide). It is desired to determine at what point the CDA is required for solution. Because the complexity grows significantly with U and L , it was decided to make monthly decisions, versus annual. If daily decisions were desired, these parameters would need to take on the values of 365x 6 = 2190 ; yet, this is not practical in this case, as most construction projects seem to leave assets in place for at least one month. It was also determined that the most practical planning horizon for making monthly decisions would most likely not exceed that of 3 years, and 36 months is the largest considered here. It should be noted that it is not desired to determine how large of a problem the CDA can solve, yet more importantly the threshold in which the solution method is necessary. Furthermore, it may be the case that weekly decisions be made, yet to solve this level of

5. Conclusion We presented a new Parallel Machine Replacement Problem with Shipping, denoted PMRP-S, that allows for replacement decisions to be integrated with shipping decisions between demand sites. The problem has vast application in the construction industry specifically. As commonly seen in construction, the shipping of machines is an expensive, yet necessary endeavor. The integration of shipping and PMRP allows for important asset management decisions to be made simultaneously. Having an optimal replacement and shipping schedule in place can reduce costs significantly, as well as potentially reduce delays. With the

Table 5 Computational results of three solution methods for each defined problem size. Problem Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Master Problem

CDA

No Shipping

Solution Time (s)

OFV

Solution Time (s)

OFV

Solution Time (s)

OFV

23.0 28.0 60.0 133.0 413 68.0 89.5 228.0 N/A 150.0 214.22 349.0 N/A N/A 375.0 N/A N/A N/A N/A

11487914.7 10761291.0 15717444.0 30161638.3 38915377.0 16897557.8 16646129.0 23135014.9 N/A 25527551.1 26204783.2 36719271.6 N/A N/A 44153495.2 N/A N/A N/A N/A

58.0 67.0 873.0 764.0 145.0 63.0 177.0 1584.0 397.0 248.0 80.3 319.3 594.0 1184.7 102.6 222.5 746.8 1870.3 1478.1

11491183.6 10766003.4 15746179.0 30162371.2 38996430.5 16906242.4 16646128.9 23167955.5 62303571.3 25527915.7 26208024.3 36719271.6 71058577.8 99288922.5 44172367.5 44030475.8 61392672.4 119007914.2 261370244.6

11.0 34.0 353.0 170.0 250.0 46.0 114.0 84.0 N/A 89.0 66.0 128.0 N/A N/A 80.0 135.9 N/A N/A N/A

11495498.8 10766003.5 15755033.6 30193235.9 39021657.5 17128209.4 16946954.0 23511251.4 N/A 26174613.9 27066686.7 37810463.9 N/A N/A 45183260.7 45266770 N/A N/A N/A

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Gap 1 (%)

Gap 2 (%)

0.03% 0.04% 0.18% 0.00% 0.21% 0.05% 0.00% 0.14% N/A 0.00% 0.01% 0.24% N/A N/A 0.04% N/A N/A N/A N/A

0.04% 0.00% 0.06% 0.10% 0.06% 1.31% 1.81% 1.48% N/A 2.53% 3.29% 2.97% N/A N/A 2.24% 2.81% N/A N/A N/A

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shipping of assets of such magnitude, it may be the case that specific routes and permits may be required, therefore knowing the optimal shipping schedule is important. Because the problem is complex, a heuristic algorithm was developed to solve large problem size efficiently. Specifically, when the number of projects being considered is large, the algorithm may be necessary to generate a feasible solution in a reasonable time. An important managerial implication from the numerical example is that of the effect of shipping cost when the optimal solution is in place. It was observed that having the optimal shipping and replacement schedule mitigates the effects of shipping cost inflation for the realistic numerical example. Even when the cost increased significantly the optimal solution did not change. Shipping cost can be uncertain, yet having the optimal solution may eliminate the concern of these costs in terms of procurement. This same process can be implemented by companies for their specific cases to determine the effect of shipping costs, or derive other managerial implications. As discussed, there is an expected significant economic impact of implementing the presented methodology. Construction, mining, and trucking industries specifically may be able to greatly benefit from solving the presented problem formulation for their specific applications. Future works derived from this problem formulation can be stochastic formulations, multi-purpose machines, or solution method development that is independent of commercial solvers. One stochastic consideration could be the uncertain nature of gaining new projects (potential demand sites). Although the problem is determined over a finite horizon, new demand is commonly realized as companies win new project bids. Also, as seen in the construction industry some machines can perform multiple operations, such as excavators. Therefore, considering operation depended demand when the capabilities of machines matches various types of demands is valuable extension of PMRP.

grandviewresearch.com/industry-analysis/construction-equipment-market-analysis. des-Bordes, E., Büyüktahtakın, İ.E., 2017. Optimizing capital investments under technological change and deterioration: a case study on MRI machine replacement. Eng. Econ. 62 (2), 105–131. Esra Büyüktahtakin, İ., et al., 2014. Parallel asset replacement problem under economies of scale with multiple challengers. Eng. Econ. 59 (4), 237–258. Hartman, J.C., 2004. Multiple asset replacement analysis under variable utilization and stochastic demand. Eur. J. Oper. Res. 159 (1), 145–165. Hartman, J., Ban, J., 2002. The series–parallel replacement problem. Robot. Comput. Integrated Manuf. 18 (3–4), 215–221. Hartman, J.C., Lohmann, J.R., 1997. Multiple options in parallel replacement analysis: buy, lease or rebuild. Eng. Econ. 42 (3), 223–247. Hartman, J.C., Tan, C.H., 2014. Equipment replacement analysis: a literature review and directions for future research. Eng. Econ. 59 (2), 136–153. Hopp, W.J., Jones, P.C., Zydiak, J.L., 1993. A further note on parallel machine replacement. Nav. Res. Logist. 40 (4), 575–579. Jones, P.C., Zydiak, J.L., Hopp, W.J., 1991. Parallel machine replacement. Nav. Res. Logist. 38 (3), 351–365. Keles, P., Hartman, J.C., 2004. Case study: bus fleet replacement. Eng. Econ. 49 (3), 253–278. Leung, L.C., Tanchoco, J., 1986. Multiple machine replacement within an integrated system framework. Eng. Econ. 32 (2), 89–114. Leung, L.C., Tanchoco, J.M.A., 1990. Multiple machine replacement analysis. Eng. Costs Prod. Econ. 20 (3), 265–275. Lotfi, V., Suresh, N.C., 1994. Flexible automation investments: a problem formulation and solution procedure. Eur. J. Oper. Res. 79 (3), 383–404. MachineryTrader.com. CATERPILLAR 320EL for Sale. Available from: https://www. machinerytrader.com/listings/construction-equipment/for-sale/list/category/1031/ excavators/manufacturer/caterpillar/model/320el. McClurg, T., Chand, S., 2002. A parallel machine replacement model. Nav. Res. Logist. 49 (3), 275–287. Montoya-Torres, J.R., et al., 2015. A literature review on the vehicle routing problem with multiple depots. Comput. Ind. Eng. 79, 115–129. Parthanadee, P., Buddhakulsomsiri, J., Charnsethikul, P., 2012. A study of replacement rules for a parallel fleet replacement problem based on user preference utilization pattern and alternative fuel considerations. Comput. Ind. Eng. 63 (1), 46–57. Pdf Printable Us States Map. Owlab.Co: http://owlab.co/pdf-printable-us-states-map-usmap-with-states-pdf-usa/pdf-printable-us-states-map-us-map-with-states-pdf-usainspirationa-us-state-map-blanck-for-kids-us-state-map-blank-pdf-usa-with-game/. Rajagopalan, S., 1998. Capacity expansion and equipment replacement: a unified approach. Oper. Res. 46 (6), 846–857. Rajagopalan, S., Singh, M.R., Morton, T.E., 1998. Capacity expansion and replacement in growing markets with uncertain technological breakthroughs. Manag. Sci. 44 (1), 12–30. Seif, J., Rabbani, M., 2014. Component based life cycle costing in replacement decisions. J. Qual. Maint. Eng. 20 (4), 436–452. Seif, J., Shields, B., Yu, A., 2018. Parallel machine replacement under horizon uncertainty. Eng. Econ. 64 (1), 1–23. Shane, J.S., et al., 2009. Construction project cost escalation factors. J. Manag. Eng. 25 (4), 221–229. Stinson, J.P., Khumawala, B.M., 1987. The replacement of machines in a serially dependent multi-machine production system. Int. J. Prod. Res. 25 (5), 677–688. Tang, J., Tang, K., 1993. A note on parallel machine replacement. Nav. Res. Logist. 40 (4), 569–573. Transport, N.E.A., 2017. Cost to Ship Heavy Equipment. [cited 2018; Available from: https://nxautotransport.com/cost-to-ship-heavy-equipment. Vander Veen, D., 1985. Parallel Replacement under Nonstationary Deterministic Demand. PhD thesis. University of Michigan. VanderVeen, D., 1985. Parallel Replacement under Nonstationary Demand. Unpublished doctoral thesis. Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor.

References Ansaripoor, A.H., Oliveira, F.S., 2018. Flexible lease contracts in the fleet replacement problem with alternative fuel vehicles: a real-options approach. Eur. J. Oper. Res. 266 (1), 316–327. Büyüktahtakin, I.E., Hartman, J.C., 2009. Parallel replacement under technological change and deterioration. In: IIE Annual Conference. Proceedings. Institute of Industrial and Systems Engineers (IISE). Büyüktahtakın, İ.E., Hartman, J.C., 2016. A mixed-integer programming approach to the parallel replacement problem under technological change. Int. J. Prod. Res. 54 (3), 680–695. Caterpillar. 320E L Hydraulic Excavator. [cited 2017; Available from: http://s7d2. scene7.com/is/content/Caterpillar/C719302. Chand, S., McClurg, T., Ward, J., 2000. A model for parallel machine replacement with capacity expansion. Eur. J. Oper. Res. 121 (3), 519–531. Chartrand, G., Lesniak, L., Zhang, P., 2010. Graphs & Digraphs. Chapman and Hall/CRC. Construction Equipment Market Size, Share & Trends Analysis Report by Product Type (Earth Moving, Material Handling, Concrete & Road Construction), by Region, and Segment Forecasts. pp. 2018–2025. Available from: https://www.

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