transportation delays in reverse logistics
TRANSCRIPT
Int. J. Production Economics 143 (2013) 395–402
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics
0925-52
doi:10.1
n Corr
prehod
E-m
robert@
journal homepage: www.elsevier.com/locate/ijpe
Transportation delays in reverse logistics
Marija Bogataj a,c,n, Robert W. Grubbstrom b,c
a University of Ljubljana, SI 1000 Ljubljana, Sloveniab Linkoping Institute of Technology, SE 58183 Linkoping, Swedenc MEDIFAS, Primorski tehnoloski park, Mednarodni prehod 6, Vrtojba, SI 5290 Sempeter pri Gorici, Slovenia
a r t i c l e i n f o
Available online 13 December 2011
Keywords:
Supply chain
Reverse logistics
Material requirements planning
Laplace transform
Input–output analysis
Lead time
Net present value
Extended producer responsibility
Transportation matrix
73/$ - see front matter & 2011 Elsevier B.V. A
016/j.ijpe.2011.12.007
esponding author at: MEDIFAS, Primorski t
6, Vrtojba, SI-5290 Sempeter pri Gorici, Slove
ail addresses: [email protected] (M
grubbstrom.com (R.W. Grubbstrom).
a b s t r a c t
In this paper we extend and apply MRP theory towards reverse logistics including the considerations of
transportation consequences. Our aim is to demonstrate the versatility obtained from using MRP theory
when combining Input–Output Analysis and Laplace transforms. This enables an analysis of a supply
chain including four sub-systems, namely manufacturing, distribution, consumption and reverse
logistics, where the geographical distance between the activities play an important role. The main
focus in this paper is on reverse logistics (recycling, remanufacturing). Especially we wish to model the
evaluation of disposal and reverse activities far away from agglomerations, which often means an
improved environment for nearby inhabitants. This is also illustrated in a numerical example. We use
the Net Present Value as a measure of the economic performance.
Our ambition is to show that supply chain sub-systems may accurately be described using input and
output matrices collected together in corresponding matrices for the system as a whole. Activity levels
in each sub-system govern the speed of the respective processes, and these activity levels, in general,
will be considered as decision variables.
We now analyse reverse logistics activities in which the flows of materials and goods are typically
divergent (arborescent processes), similar to properties of the distribution sub-system, and recent
results on the extensions of basic MRP theory introducing the concepts of output delays and the
generalised output matrix are also introduced here, when modelling the reverse logistics sub-system.
& 2011 Elsevier B.V. All rights reserved.
1. Introduction to the reverse logistics, studied in integratedsupply chain
Optimal decisions on (i) where to produce, (ii) how todistribute the product and (iii) at what time to order or deliveritems in an integrated supply chain can be successfully discussedand evaluated in a transformed environment, where lead timesand other time delays are easily considered. This approach, forinstance, has earlier been applied to the site and capacity selec-tion problem (Bogataj et al., 2011), treating where it is best tolocate a facility and what capacity is needed. Lead times in theentire supply chain can be analysed in the Laplace transformedspace in a compact form using MRP theory.
As in (Grubbstrom et al., 2007) in this paper we classify activitiesof the integrated supply chain into four distinct sub-systems:
–
manufacturing, – physical distribution,ll rights reserved.
ehnoloski park, Mednarodni
nia.
. Bogataj),
–
consumption and – recycling or remanufacturing.Various operating stages in the logistic chain (nodes of thechain) can be represented by simple models of material transfor-mations. In each process cell values are added and/or costsincurred. At each processing stage there is a supply and a demand,and often both are stochastic by nature. However, in this paperwe only consider deterministic processes.
Inventories provide insurance against the risk of shortage ofgoods in each cell of the logistic chain. They are limited by thecapacity of each processing node of the chain and by thetransportation capability of input and output flows, and therebyinfluence all kind of inventory costs.
The flows of items in supply chains influence transportationcosts and costs of activities in logistic nodes of the globaleconomy, and consequently the Net Present Value (NPV) of thecash flows generated by all activities performed in logistic net-works. The cost of item flows between the two processes dependson their location and the transportation mode adopted. Importantin this context is the location of reverse logistics activities anddisposal units, with spillover of negative pollution externalities.Sometimes these effects are not included in the price of items, but
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402396
they create high external costs to nearby agglomerations, andnormally decrease by distance. Arguments for applying the NetPresent Value as a superior criterion for capturing economicconsequences of working capital decisions are found in, e.g.,(Grubbstrom, 1980).
An integrated supply chain includes the purchasing of rawmaterials, manufacturing with assembly and/or extraction andthe distribution of the produced goods to other production units,or to the final clients, where consumption defines the activitylevels. A fourth sub-system, the main subject of our analysis here,is reverse logistics, which we model using the same formalproperties as for the other sub-systems.
In a supply chain, key variables that have to be considered foreach activity are its activity level and timing, the inventory leveland lead times and other delays.
Managers of a supply chain have two main goals: (a) to keepthe level of inventory in the supply chain as low as possiblereducing inventory costs while balancing the these costs againstordering costs; (b) to move the products in their continuallychanging form or location from raw materials to finished goods,and to manage the physical distribution to the final consumer atdifferent locations, or back into remanufacturing or recycling asfast as possible.
We consider the supreme objective to be to achieve themaximal Net Present Value (NPV) of all activities in the supplychain. By releasing wastes into the environment, supply chainsmay avoid costs of mitigation of pollution. If they do not have tobear them, market prices and the resulting NPV do not reflectthese costs accurately, which would encourage a higher con-sumption level. By reallocating reverse activities and disposals tomore distant locations, the cost of transportation increases andexternalities decrease.
Raw materials or components are delivered and stored in rawmaterial warehouses at the production centres. From there, rawmaterials and components are withdrawn as much as they areneeded from the production centres. There raw materials andcomponents are transformed into subassemblies, subassembliesare transformed in semi-finished goods, and so on till the finalproduction centre, where finished products are manufactured.From there finished goods are put into final product storage. Fromstorage centres finished goods are delivered to distributioncentres, or to their own warehouses, or directly to retail outlets,
Fig. 1. Concept of all four parts of an
often at several different locations. There goods are available tothe final consumer.
The fourth sub-system, which can be considered in a similarway as the first (manufacturing) and second (business logistics),is the reverse logistics sub-system. Here the main question iswhen to include activities of the reverse logistics in a supply chainand how the increased percentage of recycling items in produc-tion affects the NPV of the total integrated chain. This isparticularly difficult to evaluate when lead times are varyingand when backlogs appear.
The compact analysis in the frequency domain, in which lead timeperturbations may be studied, might also provide advice aiding indirect evaluations of the European Union (EU) directives on environ-mental protection by recycling and remanufacturing activities.
This research has a close tie with supply chain control inmaterial requirements planning (MRP) studies. Basically, it con-sists of a set of logically related procedures, decision rules andrecords designed to translate a Master Production Schedule intotime-phased net requirements (Orlicky, 1975). To fulfil theplanned coverage of these requirements, a schedule needs to beimplemented for each activity and its inventory.
A modern approach combining Input–Output Analysis andLaplace transforms was introduced by Grubbstrom and hisLinkoping School, for overviews see (Grubbstrom and Bogataj,1998) and (Grubbstrom and Tang, 2000).
This approach gives us good theoretical and practical results alsofor the extension of the analysis to distribution (Bogataj M. andBogataj, 2003) and especially to the reverse logistics part of a supplynetwork first studied in (Grubbstrom et al., 2007), treating trans-portation costs and transportation delays in a less detailed way. Thealgebraic structures for various timing relationships were developedmore thoroughly in (Bogataj and Grubbstrom, in press).
The concept of all four sub-systems of an integrated logisticchain can be illustrated as in Fig. 1.
From one stage to another, physical characteristics of the itemsand their location change step by step. In the process of globalisation,the physical distances between the pairs of production cells orbetween production and distribution cells increase. The restrictionsin local legislation drive out disposal and reverse logistics activitiesfrom some countries to those, where environmental costs are lower,increasing transportation costs and lead times between the activities.Where to locate an activity influences transportation costs and lead
integrated logistics chain system.
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402 397
times for the delivery of items, as well as some environmental costs(including environmental taxes), especially when reverse logisticsactivities have to find their best location.
A rationale for this paper is the extended producer responsibility.This new concept is spreading around the world. The responsibilityof producers has an influence on the extension of models describingsupply chains. This responsibility is internalised. The responsibilityof producers for their products extends to the post-consumer stage.A company or partner in a supply chains must be concerned notonly with making the product and how it functions, but also withwhat will become of the product at the end of its useful life. In thecase of consumer goods, this principle shifts responsibility forrecycling and waste disposal from local government to mainlyprivate actors in the supply chain. It means internalising the costof waste management into product prices. Under such a scheme,citizens pay for waste management as consumers when purchasingproducts, rather than as taxpayers. Through local taxes this changesthe prices. For our purposes, a negative price parameter is intro-duced into our model, mainly to reflect costs of disposal asprescribed by local or governmental taxes (for instance collectedin a way that costs of waste are internalised while letting consumersreturn the waste to the original producer).
The disposal flow depends on the percentage of items returned.It depends on labour and the quality of the product at the end of itsuseful life. The higher the percentage of recycling or remanufactur-ing is or the lesser quality the used item has, the more input oflabour is needed. Both will be added to the final price of products,directly or through local taxes. Reverse logistics activities reduce theamount of materials going to landfills. However, the optimalpercentage of disposal needs to be decided, when optimising theNet Present Value of all activities in the supply chain Unit transpor-tation costs to disposal or to remanufacturing/recycling also influ-ence the optimal decision. The question of optimising the processes,however, is left to future research.
In any case producers must pay for the waste. Therefore they willhave an incentive to make products that are less wasteful and todesign a layout of the reverse logistics sub-system, which will taketransportation costs into consideration. The approach that wesuggest, provides a missing link between the product design andrecycling, a link that is a key for making recycling efficient andeconomic. The movement towards designing for disassembly,
Fig. 2. Integrated system. All flows are vector-valued
developing reverse logistics systems and remanufacturing, are stra-tegies that industries in supply chains have to use in response to thefresh incentives posed by the new European and American ruling.
In this paper, we attempt to extend MRP theory in the totalsupply chain by including reverse logistics and disposal, whentransportation costs and transportation delays from differentdirections of sources play important roles. Transportation costs,earlier treated in a less detailed way in the distribution sub-system by (Grubbstrom et al., 2007) and variations of delays fromdifferent directions of sources (earlier discussed by Bogataj et al.,2011), are now given detailed attention.
2. The closed-loop system made up of four sub-systems
Let us now formalise our system from Fig. 1 into a blockdiagram with four sub-systems as reported in (Grubbstrom et al.,2007) and in (Bogataj and Grubbstrom, 2011), and shown inFig. 2. Below we provide a summary, details are found in thesetwo references. An input–output analysis approach is applied. Forthe original macro-economic method and its later application toprocesses on the company level, see (Leontief, 1928, 1951;Koopmans, 1951; Vazsonyi, 1955, 1958).
There are four sub-systems and the processes are grouped intofour sets of columns, representing manufacturing, distribution,consumption, recycling/remanufacturing, respectively. Items arealso grouped into four categories, depending on which items areinputs into the four sub-systems. The activity vectors are denotedP1, P2, P3 and P4 concerning the respective processes of the sub-systems. Finally, xk denotes input requirements to and yk outputsfrom sub-system k, k¼1, 2, 3, 4.
The input requirements of the system as a whole x may thenbe written as:
x¼
x1
x2
x3
x4
266664
377775¼
H11H12H13H14
H21H22H23H24
H31H32H33H34
H41H42H43H44
266664
377775
P1
P2
P3
P4
266664
377775¼HP, ð1Þ
here, Hkl is the input matrix concerning the need of items required bysub-system k from running processes in the sub-system l. Some italic
with same dimension (Grubbstrom et al., 2007).
Table 1Interpretations of net production sub-vectors.
zk P4l ¼ 1
ðGkl�HklÞPl
Interpretation Comment
z1 G11P1þG14P4�H11P1�H12P2 Net production of items used in and
delivered from manufacturing
‘‘Standard’’ net production, to which recycled usable items G14P4 are added
z2 G22P2�H22P2�H23P3 Net ‘‘production’’ of items distributed Items from output of distribution (less those redistributed), reallocated
geographically (and temporally), compared to inputs
z3 G33P3�H34P4 Net ‘‘production’’ of items consumed Used items and ‘‘new’’ in the sense that they did not exist before as used
z4 G44P4 Net production of recycled items Outputs from recycling activities possible to use as components in production
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402398
elements of H in (1) would be zero matrices in a typicalstandard case.
The outputs of the system as a whole, written y, as the resultsof the activities of the four sub-systems, are written in a similarway (some italic elements typically zero matrices):
y¼
y1
y2
y3
y4
266664
377775¼
G11G12G13G14
G21G22G23G24
G31G32G33G34
G41G42G43G44
266664
377775
P1
P2
P3
P4
266664
377775¼GP, ð2Þ
where G is the overall output matrix, with elements Gkl correspond-ing to the same lth sub-system as Hkl and the same group of items(kth), and with some italic elements typically being zero matrices.
The ‘‘net production’’, written z, is defined as the differencebetween the total outputs and total inputs. If an element isnegative, it needs to be imported, and if positive it can beexported from the system according to the general formula:
z¼
z1
z2
z3
z4
266664
377775¼ y�x¼
y1
y2
y3
y4
266664
377775�
x1
x2
x3
x4
266664
377775¼ ðG�HÞP: ð3Þ
For our closed-loop system, taking the assumptions of zero-valued matrices into account (special structures of G and H), wehave the interpretations as listed in Table 1.
3. Demonstration of the concepts in a basic example
In order to demonstrate our concepts, definitions and relationsfor the system in Section 2, we illustrate our system by thefollowing basic example, taken from (Grubbstrom et al., 2007).
One ‘‘standard’’ product (item no. 1) is manufactured. Onecomponent (item no. 2) and one raw material (item no. 3) areneeded. Raw materials are purchased and components manufac-tured or recycled (30% of the used components are consideredusable after reconditioning). One standard item uses 2 compo-nents and 5 units of raw materials.
We assume no extraction processes concerning production,although the theory allows this as is illustrated by Fig. 2.
The standard product is distributed to two warehouses, and onarrival it becomes item no. 4 or no. 5. Consumers buy items no.4 or no. 5, depending on where they live. All items after use aresent to recycling. A used item is numbered no. 6. A used item maybe recycled, some become usable (the component recovered isthen identical to item no. 2), others are disposed of (item no. 7).
The activities of the system are:
–
Activity 1: Manufacture standard item in one unit. – Activity 2: Distribute standard item to Location 1 in packagesof 100 units.
– Activity 3: Distribute standard item to Location 2 in packagesof 200 units.
–
Activity 4: Consume standard item in Location 1.–
Activity 5: Consume standard item in Location 2. – Activity 6: Recycle used item (to recover component).The number of processes is thus 6 and the number of items 7,making the matrices H and G (7�6)-dimensional. The input andoutput matrices will then be:
100 200
2
5
1
1
1
=H ,
1
0.3
1
1
100 200
0.7
=G . ð4Þ
By way of example, 800 standard units are manufactured, 200are distributed to Location 1 (2 packages), and 600 to Location 2(3 packages). All standard items are consumed, after which theyare recycled. Packages are consumed, but their contents like theitems are recycled.
The activity vector P then becomes
1
2
3
4
800
2
3
2
3
800
= =
P
P
P
P
P
, , ð5Þ
which gives rise to input requirements in the amounts of
0 100 200 800800
006122
000453
2
3
1 2
1 3
1 800800
0
===x HP , ð6Þ
and outputs amounting to
0081800
0.3 2402
03
212
313
100 200 800800
0.7 560
===y GP . ð7Þ
Fig. 3. Illustration of basic numerical example (Grubbstrom et al., 2007).
Fig. 4. Recycling process to which a labour input is added (Grubbstrom et al.,
2007).
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402 399
Thus, net ‘‘production’’ becomes
z = y – x =
800 800 0
240 1600 1360
0 4000 4000
2 2 0
3 3 0
800 800 0
560 0 560
−−
− = . ð8Þ
We make the following interpretation. All standard items areused, 1360 items of components need to be purchased (imported)and 240 come from recycling. Concerning raw materials, 4000units need to be purchased. A surplus of 560 waste items notsuitable for reuse (item 7) need to be disposed of.
The numerical values of all flow vectors are shown in Fig. 3.
3.1. The recycling sub-system
As in (Bogataj and Grubbstrom, 2011), we assume that therecycling/remanufacturing processes have an imported primaryproduction factor in the form of labour. There are thus two inputsinto this sub-system in this basic example, on the one hand, thereturned output from consumption, on the other, labour, and twooutputs (reconditioned components and irreparable componentsto be disposed of), cf. Fig. 4.
The production function for this transformation is assumed tobe of the Cobb–Douglas type (Sandelin, 1976) with non-increas-ing returns to scale, the output of recoverable items type 2 beinga times the input of used items (type 6), where a is the recovery
rate, cf. also (Grubbstrom and Tang, 2006). The recycling processis assumed to be of the cyclical type with a period of T6 and abatch size of P6. When producing a batch P6, an amount of labourL is needed as input. With the Cobb–Douglas function, the amount
of recovered items is thus assumed to follow
aP6 ¼ APd6L
g, ð9Þ
where standard assumptions are assumed concerning parametersA40, g, d40 and 0ogþdr1 giving decreasing returns to scale,and g, do1. The volume of irreparable items from each batch willthen be ð1�aÞP6, which is the batch-size output of item 7.
The input and output matrices showing only elements relevantto the reverse logistics sub-system only are then:
recycling
1
=H , recycling
1
α
α
=
−
G . ð10Þ
4. Modelling of flows in the frequency domain
According to MRP Theory, we may model the flow of inputsand outputs of the recycling sub-system in the following way. For
Fig. 5. Development of recoverable inventory (item no. 6) (Grubbstrom et al.,
2007).
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402400
an introduction to the methodology of applying the Laplacetransform to this type of model, please see (Grubbstrom, 2007).
The outflow of used items from consumed items no. 4, 100-packages, and no. 5, 200-packages as a steady flow generated byconstant activity levels P04 and P05. With s denoting the Laplacefrequency, the transform of the activity levels will be ~P4ðsÞ ¼ P04=s
and ~P5ðsÞ ¼ P05=s, respectively, where a tilde denotes the trans-formed variable
~zðsÞ ¼ £fzðtÞg ¼
Z 1t ¼ 0
zðtÞe�stdt: ð11Þ
for a time function z(t) (Aseltine, 1958). If the activities start attimes t4and t5 if the output delays of consumption are D4 and D5,and the transportation times to recycling are D64 and D65, theoutput flow of item no. 6 from consumption to recycling will havethe transform
~y6ðsÞ ¼ e�sD4 g64~P4ðsÞþe�sD5 g65
~P5ðsÞ ¼ e�sðD4þD64þ t4Þ100P04=s
þe�sðD5þD65þ t5Þ200P05=s, ð12Þ
where the exponential coefficients account for resulting delays.This is also the inflow of used items into recoverable inventory ofthe recycling sub-system. The long run average flow of theseitems, applying the limit value theorem of the Laplace transform(Aseltine, 1958), is found to be
y6ð1Þ ¼ lims-o
s ~y6ðsÞ ¼ lims-o
sðe�sðD4þD64þ t4Þ100P04=s
þe�sðD5þD65þ t5Þ200P05=sÞ ¼ 100P04þ200P05, ð13Þ
as expected.The recycling activity (Process no. 6) is assumed to take place
cyclically in batches of P6 with the period T6, its transform being~P6ðsÞ ¼ P6e�st6=ð1�e�sT6 Þ, if the first batch is completed at t6. Witha lead time for inputs (production time) of t6, it requires useditems type 6 according to h66est6 ~P6ðsÞ ¼ P6esðt6�t6Þ=ð1�e�sT6 Þ,which represents the outflow from recoverable inventory of itemno. 6 (if all this outflow is taken care of).
Hence recoverable inventory develops according to the inte-gral of the difference between in- and out-flows
~R6ðsÞ ¼e�sðD4þD64þ t4Þ100P04=sþe�sðD5þD65þ t5Þ200P05=s�est6 P6e�st6=ð1�e�sT6 Þ
s
!,
ð14Þ
where the division by s corresponds to a time integration. Onceagain, using the limit value theorem, we have the averageinventory level:
R6ð1Þ ¼ lims-0
se�sðD4þD64þ t4Þ100P04=sþe�sðD5þD65þ t5Þ200P05=s�est6 P6e�st6=ð1�e�sT6 Þ
s
!:
ð15Þ
Since a Laurent expansion of 1/(1�e�sT) becomes 1/(sT6)þ1/2�(sT6)/12þO((sT6)2), the long run average inventory will onlybe kept limited, if
100P04þ200P05 ¼ P6=T6, ð16Þ
i.e. that the average inflow equals the average production rate (asexpected).
Assuming this long run balance to be valid, on carrying out thelimit in (15), we easily find the long run average inventory level:
R6ð1Þ ¼ ð100P04þ200P05Þðt6�t6�T6=2Þ�100P04ðD4þD64þt4Þ
�200P05ðD5þD65þt5Þ ð17Þ
In the very simple case that Process 6 starts as soon aspossible, t6¼T6, and all time parameters except t6 and T6 havezero values, we obtain the expected long run level to be half of thebatch size, R6ð1Þ ¼ P6=2, quite as intuitively anticipated.
The development of recoverable inventory is presented in Fig. 5.
Taking the transportation time D26 from recycling to manu-facturing into consideration, the output of recovered items (type2) from Process 6 becomes an inflow into the available inventoryat Process 2, according to ae�sD26 ~P6ðsÞ ¼ ae�sðD26þ t6ÞP6=ð1�e�rT6 Þ
and the disposal outflow of items type 7 will be ð1�aÞe�sD76 ~P6ðsÞ
¼ e�sðD76þ t6Þð1�aÞP6=ð1�e�rT6 Þ, where D76 is the transportationtime from remanufacturing to disposal.
The input of labour to Process 6 is ~LðsÞ ¼ e�st6 L=ð1�e�sT6 Þ,assuming this input to follow the batches.
5. Transportation and ordering costs
According to the Net Present Value Theorem of the Laplacetransform (Grubbstrom, 1967), by exchanging the Laplace fre-quency s in the transform of a cash flow for the continuousinterest rate r, the Net Present Value (NPV) of the cash flow isobtained.
Below, we include the NPV of transportation costs NPVtransportation
expressed in the frequency domain, similarly as for production anddistribution examined in (Bogataj et al., 2011), as:
NPVtransportation ¼�ETð ~PGðrÞþ ~PHðrÞÞ ~PðrÞ, ð18Þ
where ~PGðrÞ and ~PHðrÞ are the two matrices
~PGðrÞ ¼e�rD1 � � � 0
^ & ^
0 � � � e�rDn
264
375
0 0 0 � � � 0
^ & gkjbkjDkj & ^
gn1bn1tn1 � � � gnjbnjDnj � � � 0
264
375,
and
~PHðrÞ ¼0 0 0 � � � 0
^ & hijbijtij & ^
hm1bm1tm1 � � � hmjbmjtmj � � � 0
264
375
ert1 � � � 0
^ & ^
0 � � � ertm
264
375,
and where ET is a row vector of unit components, and bkj aretransportation out-payments per quantity and the time spent intransport, here assuming that these payments are proportional torequired inputs hij or produced outputs gkj and the time deliveringitem i to process j, or delivering from process k the item j to itsdestination.
Ordering costs, viewed as a fixed out-payment at the referencetime of the activity may be included easily in our expressions. Forthe cyclically assumed activities of Process 6, a setup vectorsgnðP6e�rt6=ð1�e�rT6 ÞÞ is defined as the sequence of unitimpulses, when production takes place. With a setup expense ofK6 for each batch from the recycling process, the NPV of thesepayments becomes NPVordering ¼�K6sgnðP6e�rt6= ð1�e�rT6 ÞÞ ¼
�K6e�rt6=ð1�e�rT6 Þ.
6. Economic evaluation of reverse and disposal activitiesincluding transportation and ordering costs
We now introduce prices capturing the values at differentpoint in the flows. For the outflow from consumption, this unit
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402 401
value for item 6 is denoted p6, for the inflow into manufacturingwe assume a value of p2 per unit of item 2, and for disposing item7 we assume a negative value p7 per unit. Furthermore, we denotethe unit labour cost by cL. We are now in a position to formulateoverall economic consequences for the recycling sub-system.
The consequences related to recycling as such (includingordering and disposal) applying the Net Present Value Theorem,will be:
NPVrecyclingþNPVordering ¼ ðp2e�rðD26þD6Þaþp7e�rðD76þD6Þð1�aÞÞ ~P6ðrÞ
�p6ðe�sðD4þD64Þ100 ~P4ðrÞþe�sðD5þD65Þ200 ~P5ðrÞÞ�cL
~LðrÞ�K6sgn ~P6ðrÞ:ð19Þ
Subtracting the additional NPV of transportation costsNPVtransportation and introducing our expressions from (18), weend up with the total consequences NPVtot of reverse and disposalactivities including transportation for the basic case studiedabove as:
NPVtot ¼NPVrecyclingþNPVorderingþNPVtransportation
¼ðððp2�b26D26Þe
�rD26aþðp7�b76D76Þe�rD76 ð1�aÞÞe�rD6 P6�cLL�K6Þe
�st6
1�e�rT6
�e�rðD4þD64þ t4Þ100P04ðp6þb64D64Þþe�rðD5þD65þ t5Þ200P04ðp6þb65D65Þ
r:
ð20Þ
Generalisations to, for instance, more than two sources ofconsumption are obvious. In case transportation is paid at datesdifferent from our assumptions above, this is adjusted for bydiscounting the payments appropriately, for instance by adding afactor e�re76 multiplying b76, where e76 represents the timeadjustment (backwards or forwards in time).
Table 2Increasing time distance to the disposal unit influences timing and transportation
costs, which consequently influences NPVtot.
D76 e�rðD76 þD6Þ NPVtot (�106)
0.03 0.899 229.539
0.06 0.894 136.417
0.12 0.883 �46.484
0.24 0.862 �399.187
7. Numerical example
In this numerical example, we introduce the following valuesfor our parameters and variables. A recycling batch is startedapproximately twice per week T6¼0.01. Let us assume that t6¼0and t6¼T6¼0.01, meaning that the input lead time of recycling isnegligible and that remanufacturing the first recycling batchstarts as soon as there is enough consumed items in recoverableinventory for a batch.
The output delay from consumption (item 6) is assumed to beD6¼0.5. The time spending distances from recycling D26 and D76
are 0.01 and 0.03 time units, respectively, and from consumption(nos. 4 and 5) to recycling the time spending distances D64 andD65 are 0.05 and 0.02 time units, respectively. The output delaysD4, D5 for flows leaving the consumption activities are disre-garded, since they are not considered belonging to the recyclingsub-system.
The amount of items coming out from remanufacturing eitherreturns into production to become part of the inflow to Process 2,or to be disposed of as a surplus of waste items not suitable forreuse (no. 7): g26¼a¼0.3 or g76¼1�a¼0.7, respectively. Asbefore, we have the output matrix elements g64¼100 andg65¼200 for consumption.
The average activity levels of consumption are P04 ¼ 300 andP05 ¼ 100. The batch size adopted at recycling is P6 ¼ 500, whichgives an average outflow of P6=T6 ¼ 500=0:01¼ 50,000, whichequals the average inflow P04g64þP05g65 ¼ 300 �100þ100 �200¼50,000, enabling an equilibrium according to Eq. (16) above.
The cost for transportation of one unit of item per time unit bkj
is equal to 20,000 for all items, (k, j)¼(2,6), (7,6), (6,4), (6,5). Weassume the value of one unit of item 2 to be p2¼8000, the cost fordisposing one unit to be p7¼�100, and the price for obtainingone item from either of the two types of consumption to be p6¼1
(nos. 4 and 5). In general, prices are paid at the delivery time ofthe items.
The fixed costs per recycling cycle equal K6¼200 and the costof one unit of labour is cL¼19. With a technology factor A¼5.621and Cobb–Douglas exponents of d¼g¼1/3 in (9), the amount ofrequired labour for a batch of P6 ¼ 500 becomes L¼ 38. Thecontinuous interest rate is r¼0.2.
The Net Present Value is made up from income by remanu-facturing item no. 2, paying disposal costs, paying for transportingunits either to production or disposal, payments for obtainingused items from either source (nos. 4 and 5), for the transporta-tion of these from consumption to recycling, and payments forrecycling setups and for labour input.
Following the material flow, we first have the NPV of acquiringused items including purchasing and transportation (nos. 4 and 5)
NPVused items ¼�X
j ¼ 4,5
g6je�rðDjþD6jÞðp6þb6jD6jÞP
0j=r
¼�ðg64e�rðD4þD64Þðp6þb64D64ÞP04
þg65e�rðD5þD65Þðp6þb65D65ÞP05Þ=0:2
¼�ð100e�0:2ðD4þ0:05Þð1þ20000U0:05ÞU300
þ200e�0:2ðD5þ0:02Þð1þ20000U0:02ÞU100Þ=0:2
¼�188,595,901
Additionally we have the input NPV cost of labour and ofsetups:
NPVlabour ¼�cLLe�rt6=ð1�e�rT6 Þ ¼�19U38e�0:2U0:01=ð1�e�0:2U0:01Þ
¼ �360,639,
NPVsetups ¼�K6=ð1�e�rT6 Þ ¼�200=ð1�e�0:2U0:01Þ ¼�99,900:
The NPV from outputs of remanufacturing (sales to themanufacturing sub-system less costs for transportation) will be:
NPVremanufactured items ¼ ðp2�b26D26Þe�rðD26þD6ÞaP6e�rt6=ð1�e�rT6 Þ
¼ ð8000�20,000U0,01Þe�0:2ð0:01þ0:5Þ
�0:3U500e�0:2U0:01=ð1�e�0:2U0:01Þ
¼ 527,744,192:
and the NPV of disposals
NPVdisposed items ¼ ðp7�b76D76Þe�rðD67þD6Þð1�aÞP6e�rt6=ð1�e�rT6 Þ
¼ ð�100�20,000U0:03Þe�0:2ð0:03þ0:5Þ
U0:7U500e�0:2U0:01=ð1�e�0:2U0:01Þ ¼�110,069,377:
Hence we obtain the Net Present Value of all activities jointly as:
NPVtot ¼NPVused itemsþNPVlabourþNPVsetupsþNPVremanufactured items
þNPVdisposed items
¼�188,595,901�360,639�99,900þ527,744,192
�110,069,377¼ 229,539,451:
Table 2 and Fig. 6 show how the total NPV decreases when thetime spending distance from recycling to disposal is increased. Inthe table we can also see how the discounting changes because oflonger time spending distances (which for disposal activities workin the reverse direction).
-600
-500
-400
-300
-200
-100
0
100
200
300
0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24
0.010.040.08
Fig. 6. NPVtot as a function of time distance D76 for values of D26¼0.01, 0.04, 0.08.
Vertical scale in multiples of 106.
M. Bogataj, R.W. Grubbstrom / Int. J. Production Economics 143 (2013) 395–402402
8. Conclusions
In this paper we have attempted to extend MRP theory to atotal supply chain including reverse logistics and disposal, whentransportation costs play an important role, applying more detailswhen modelling transportation delays and transportation costscompared to earlier studies (Grubbstrom et al., 2007; Bogatajet al., 2011; Bogataj and Grubbstrom, in press). The combinationof Input–Output Analysis and Laplace transforms enable ananalysis of a complete supply chain including the four sub-systems of manufacturing, distribution, consumption and reverselogistics. Here our analysis has been concentrated on the reverselogistics sub-system.
We have attempted to show that these four sub-systems mayaccurately be described using input and output matrices fromMRP theory, when also transportation delays and transportationcosts are taken into consideration with the aid of transforms. TheNet Present Value as a measure of the economic performance hasbeen applied successfully.
During these developments, we have been giving specialattention to sub-systems in which the flows of materials andgoods are divergent (recycling with disposal). This has necessi-tated MRP theory to be extended to reverse logistics with a coupleof new concepts, namely the output delays, and the generalised
output matrix, recently developed and analysed in detail in(Bogataj and Grubbstrom, in press).
We have also included consequences from when disposal ismodelled as generating consequences in the form of an out-payment at the time of disposal, and, by way of an example,determined economic consequences from having the disposalsink far away in time distance from the location of consumptionand possibly from recycling activities. This has also shown how anincreasing time distance to disposal, to recycling or remanufac-turing influences timing and transportation costs, consequentlyinfluencing the resulting Net Present Value.
Future research is suggested to be organised in a number ofdirections:
a)
to consider uncertainty and risks within the reverse activitiessub-system, such as has begun in (Bogataj D. and Bogataj,2010),b)
to introduce a multi-criteria evaluation of the consequencesof the model, in which conflicts between NPV maximisationand environmental consequences could be made visible forall sub-systems of the model, and in which decisions could beevaluated from economic and environmental perspectivesalso along a long-term time horizon,
c)
for each sub-system to develop equations for supply anddemand functions as functions of supposed prices by meansof maximising the individual sub-system NPV (or utility, forthe consumer) and studying optimisation conditions,d)
thereby to derive economic equilibrium conditions for thesystem as a whole, in particular for establishing equilibriumprices based on parameters of the system structure and theoptimisations conditions,e)
to develop and study comparative-static relationships for theeffects of parameter changes on the equilibrium state,f)
and finally, to include the condition of the Extended ProducerResponsibility (EPR) to consider mutual consequences forsubsystems 1 and 4, when these sub-systems jointly shareeconomic consequences.References
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