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Optimal core acquisition and remanufacturing policies under uncertain core quality fractions
European Journal of Operational Research 210 (2011) 241–248
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Optimal core acquisition and remanufacturing policies under uncertain core quality fractions Ruud H. Teunter a,⇑, Simme Douwe P. Flapper b,1 a b
University of Groningen, Department of Operations, P.O. Box 800, 9700 AV, Groningen, The Netherlands Technische Universiteit Eindhoven, School of Industrial Engineering, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
a r t i c l e
i n f o
Article history: Received 5 November 2009 Accepted 12 June 2010 Available online 18 June 2010 Keywords: Remanufacturing Product acquisition management Quality uncertainty
a b s t r a c t Cores acquired by a remanufacturer are typically highly variable in quality. Even if the expected fractions of the various quality levels are known, then the exact fractions when acquiring cores are still uncertain. Our model incorporates this uncertainty in determining optimal acquisition decisions by considering multiple quality classes and a multinomial quality distribution for an acquired lot. We derive optimal acquisition and remanufacturing policies for both deterministic and uncertain demand. For deterministic demand, we derive a simple closed-form expression for the total expected cost. In a numerical experiment, we highlight the effect of uncertainty in quality fractions on the optimal number of acquired cores and show that the cost error of ignoring uncertainty can be significant. For uncertain demand, we derive optimal newsboy-type solutions for the optimal remanufacture-up-to levels and an approximate expression for the total expected cost given the number of acquired cores. In a further numerical experiment, we explore the effects of demand uncertainty on the optimal acquisition and remanufacturing decisions, and on the total expected cost. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Remanufacturing is a multi-billion dollar industry of growing importance (Corbett and Kleindorfer, 2001; Majumder and Groenvelt, 2001) that has received considerable academic interest over the past two decades. A large number of contributions have dealt with logistics and supply chain management for systems with (manufacturing and) remanufacturing. Reviews in this area are provided by Dekker et al. (2004), Rubio et al. (2008), and Pokharel and Mutha (2009). In remanufacturing, the quality of cores (i.e. products supplied for remanufacturing) can vary significantly, affecting the cost of remanufacturing. Guide et al. (2003) report the case of a mobile phone remanufacturer, ReCellular, that distinguishes six quality grades based on functional and cosmetic criteria with the remanufacturing cost increasing by a factor of about ten from the highest to the lowest quality grade. An important decision in remanufacturing is therefore which of the available cores should be remanufactured. Obviously, cores of the highest quality (grade 1) will be remanufactured first, followed by cores of grade 2, and so on. The availability of cores of the different quality grades clearly depends on the number of cores that ⇑ Corresponding author. Tel.: +31 50 3638617; fax: +31 50 3632032. E-mail addresses: [email protected] (R.H. Teunter), s.d.p.fl[email protected] (S.D.P. Flapper). 1 Tel.: + 31 40 2474385; fax: +31 40 2464596. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.06.015
have been acquired. Hence, the acquisition decision affects the remanufacturing decision, and these two decisions should therefore be analyzed jointly. In this paper, we will do so. The situation that we consider is one where unsorted cores are acquired from third-party brokers or collectors. We remark that this is different from cases where sorting is done first and acquisition prices are quality dependent, as assumed by Guide et al. (2003) in their model. In the situation of acquiring unsorted cores, there is uncertainty about the numbers of cores of the various grades that have been acquired until sorting is carried out, even if good estimates are available for the probabilities that a core is of a certain type, i.e. even if the core quality distribution is known. Very few authors have addressed situations where core quality affects the acquisition and remanufacturing decisions. Their models will be discussed in detail in Section 2. None of them have considered the situation with multiple (more than two) quality classes, as we do in this paper. Because the analysis of the situation with both quality and demand uncertainty is more complex and less insightful than the analysis with quality uncertainty only, we start with the latter analysis and associated numerical investigations. Afterwards, we consider the more realistic case with both types of uncertainty, and perform further numerical investigations. The rest of the paper is organized as follows. In Section 2, we shortly review the related literature. In Section 3, we present the model. In Section 4, we analyze the situation of deterministic
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demand, followed by a numerical experiment in Section 5. The same is done for stochastic demand in Sections 6 and 7. We end in Section 8 with a discussion of generalizations, limitations and directions for further research. 2. Related literature This review is structured as follows. First, in Section 2.1, we shortly discuss the broader setting of supply quality and yield in manufacturing. Section 2.2 continues with the related literature on remanufacturing. Finally, in Section 2.3, we provide a detailed discussion of remanufacturing models that consider the effect of quality uncertainty on the acquisition of cores, and point out the contribution of this paper. 2.1. Quality uncertainty in manufacturing There are a number of contributions that assume it is possible to (partially) correct for undesired yield with respect to quality, by additional ordering or production after observing the yield (e.g. Grosfeld-Nir and Gerchak, 2004) or by using downgrading (e.g. Yano and Lee, 1995). Some authors have explored the setting where the quality of the products ordered or produced can be influenced, e.g. by paying more per product in case of external acquisition (e.g. Bakal and Akcali, 2006 and Liang et al., 2009), by supplier selection (e.g. Grosfeld-Nir and Gerchak, 2004) or by process inspection and order splitting for unreliable production processes (e.g. Yano and Lee, 1995). In this paper, we focus on situations where none of these options are available. 2.2. Quality uncertainty in remanufacturing A number of authors address the complications of quality uncertainty of cores for remanufacturing operations (BloemhofRuwaard et al., 1999; Fleischmann et al., 2000; Guide, 2000; Guide and Jayaraman, 2000; Guide and Srivastava, 1997; Humphrey et al., 1998; Inderfurth, 2005; Ketzenberg et al., 2006; Savaskan et al., 2004; Stanfield et al., 2004; Toktay et al., 2000). Sorting is therefore important (Blackburn et al., 2004; Bloemhof-Ruwaard et al., 1999; Guide et al., 2000; Van Nunen and Zuidwijk, 2004) and several sorting techniques have been proposed (Krikke et al., 1999; Rudi et al., 2000). Quality uncertainty causes extra complications if the remanufacturing process is capacitated (Souza et al., 2002; Guide et al., 2008; Ferguson et al., 2009; Denizel et al., 2007) or if there are stock keeping restrictions (Aras et al., 2004). All these authors study the effect of such system restrictions on the decision whether to remanufacture cores of certain quality or dispose them/sell them ‘‘as is”. Some authors have discussed the use of incentive-systems (Guide and van Wassenhove, 2001; Guide et al., 2003; Klausner and Hendrickson, 2000) or collection options (Savaskan et al., 2004) for influencing the quality distribution of cores. A related contribution is that by Mondol and Mukherjee (2006), who study situations where age affects the quality of cores and analyze when to buy them back. 2.3. Acquisition of cores under uncertain quality Galbreth and Blackburn (2006) assume that there is a continuum of quality levels for cores. They further assume that for any lot of acquired cores, the quality distribution is fixed. They derive optimal acquisition and sorting/remanufacturing policies under both deterministic and uncertain demand.
Galbreth and Blackburn (2010) drop the assumption of a fixed quality distribution, and instead allow the condition of each acquired core in a lot to be random. The condition is assumed to follow a uniform distribution. Only the deterministic demand case is analyzed. Apart from the situation with a continuum of quality levels for cores, they also consider the situation with two discrete classes that can both be remanufactured but at different costs. Zikopoulos and Tagaras (2007) consider a system with two collection sites. There are two grading classes, good and bad (not remanufacturable), and the proportion of good cores for the two sites follows a general joint probability distribution. Demand is uncertain. An expression for the profit function is derived, and conditions are provided under which it is optimal to collect from a single site. Zikopoulos and Tagaras (2008) continue to study the situation with two grading classes, good and bad, and a general probability distribution further. They restrict the analysis to a single collection site, but allow misclassification errors where good cores are classified as bad or vice versa. A comparison of policies with and without sorting before disassembly (that is uncoupled from remanufacturing) provides insights into the economic attractiveness of sorting. Our study is the first that considers acquisition and remanufacturing decisions for situations with multiple (more than two) discrete quality classes when there is quality uncertainty. Moreover, contrary to Galbreth and Blackburn (2006, 2010) who studied a continuum of quality levels, we consider quality and demand uncertainty together. 3. Model We start by listing the different symbols in our model and then describe the model and underlying assumptions. For ease of writing, we refer to cores in a certain quality class i simply as type i cores or cores of type i. Notations A number of acquired cores A* optimal number of acquired cores number of acquired cores that turn out to be of type i Ai B(n,p) binomial distribution for n trials with success probability p acquisition cost per core ca penalty cost per unsatisfied demand cp remanufacturing cost per type i core cri D demand f probability density function of demand F probability distribution function of demand K number of quality classes N(l, r) normal distribution with mean l and standard deviation r pi probability that an acquired core is of type i remanufacture-up-to level for type i cores Ri optimal remanufacture-up-to level for type i cores Ri number of cores of type i that are remanufactured Xi u() standard normal probability density function U() standard normal probability distribution function ul, r() N(l, r) probability density function l expected demand r standard deviation demand
We analyze the problem of how many cores to acquire and of how many cores of the different quality types to remanufacture after sorting. We do so in a single-period setting. The sequence of the events in this period is as follows: acquisition, sorting, remanufacturing, demand realization. All acquired cores are sorted. Guide et al. (2006) and Debo et al. (2004) provide examples of companies that do so.
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The type (quality) of each core is uncertain. We assume that there are K different types, and that the type of a core follows a multinomial distribution with probability pi that a core is of type i for i = 1, 2, . . . , K. This distribution is appropriate when the cores in an acquired lot have little relationship, as is often the case in remanufacturing (Galbreth and Blackburn, 2010). See also Karaer and Lee (2007). The relevant costs are the acquisition cost ca per core, the penalty cost cp per unsatisfied demand and the (expected) remanufacturing cost cri per type i core. As we assume that all acquired cores are sorted, the sorting cost per core can be included in ca. The assumption of a fixed acquisition cost/price reflects the situation of complete competition in the supply market. However, our analysis and results can easily be extended to situations where the acquisition price affects the supply of cores, as in e.g. Guide et al. (2003). We will provide the details of how to do so in Section 8. Note that, since sorting does not take place until after acquisition, it is not possible to differentiate the acquisition price based on the quality of the core as Liang et al. (2009) suggest. We also remark that a net cost for discarding cores (which may be negative if there are recycling revenues) can easily be included by adding that cost to ca and subtracting it from cri . Without loss of generality, we rank the core types from highest quality (type 1) to lowest quality (type K). Quality is reflected by the remanufacturing cost and hence cr1 < cr2 < < crK . With this ranking, it is obviously optimal to remanufacture cores in quality class 1 first, and then cores in class 2, and so on. We therefore consider the class of policies that remanufactures cores in this sequence, and use remanufacture-up-to levels Ri for type i cores. It is intuitively obvious that the only policies that need to be considered are those that only remanufacture cores of a certain quality type if there are no cores left of better quality, i.e. policies with R1 P R2 P RK . The analysis in Section 6 will confirm that this holds for the optimal remanufacture-up-to level levels. Because remanufacturing takes place after sorting, the optimization problem can be decomposed into determining the optimal remanufacturing policy first and the optimal acquisition decision next. In the next section, this will be used to derive optimality conditions for the case of certain demand. The more realistic but also more complex case of uncertain demand will be analyzed in Section 6. 4. Deterministic demand We first introduce some notation that will turn out to be useful for the analysis. Let
" Lk ðMÞ :¼ E min
" k X
## Ai ; M
;
k ¼ 1; . . . ; K
ð1Þ
i¼1
and
lk :¼ A
k X
pi ;
rk
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u k k u X X t :¼ A pi 1 pi :
i¼1
i¼1
i¼1
Using standard algebra and applying the Central Limit Theorem (see Appendix A), (1) can be rewritten as
M lk M lk M lk ; ð2Þ Lk ðMÞ ¼ lk rk u 1U
rk
rk
types 1, . . . , k when demand is D. Similarly, we have that Lk1(D) is the total expected number of remanufactured cores of types 1, . . . , k 1 for demand D. Hence, the expected numbers of remanufactured cores of each type are given by
E½X 1 ¼ L1 ðDÞ and E½X k ¼ Lk ðDÞ Lk1 ðDÞ;
k > 1:
ð3Þ
So the total cost, consisting of acquisition and remanufacturing costs, is
CðAÞ ¼ ca A þ cr1 L1 ðDÞ þ
K X
crk ðLk ðDÞ Lk1 ðDÞÞ:
ð4Þ
k¼2
Obviously, the optimal number of acquired cores A* is the value of A that minimizes (4). 4.1. Approximation that assumes constant sample quality fractions In the next section, we will compare A* to the ‘optimal’ acquisi^ under the assumption that there is no uncertainty tion decision A in the quality of cores, i.e. that the fraction of the acquired cores of type k is always exactly pk. Then (1) gives
" # k X ^Lk ðDÞ ¼ min A pi ; D
ð10 Þ
i¼1
^ can be determined by minimizing and A
^ CðAÞ ¼ ca A þ ^L1 ðDÞ þ
K X
crk ð^Lk ðDÞ ^Lk1 ðDÞÞ:
ð40 Þ
k¼2
5. Numerical results for deterministic demand In this section, we analyze some numerical examples to obtain insight into the effect of uncertainty about the quality of cores on the optimal acquisition decision. Recall that incorporating that uncertainty is one of our main contributions to the existing literature, and hence we are particularly interested in the ‘cost error’ of not doing so. This cost error is calculated as
^ CðA Þ CðAÞ 100%: CðA Þ The examples that we consider are inspired by the real life case of a mobile phone remanufacturer ReCellular that is discussed in detail in Guide et al. (2003). This remanufacturer distinguishes six quality grades. Example 1. In our first example, we use the estimated remanufacturing costs for the different grades in the Recellular case study, as given in Table 1. Each grade is assumed to occur with the same probability. Demand is for 100 remanufactured products (D = 100). In Fig. 1 the optimal number of acquired cores under quality uncertainty (A*) is compared with the optimal number of acquired ^ The acquisicores when assuming constant quality fractions ðAÞ. ^ retion cost per core is varied from 0 to 20 (after which A* and A main constant at 100) with step size 0.5. ^ takes on six different Note from Fig. 1 that the approximation A values only. These are 600/1, 600/2, . . . , 600/6, which correspond to
rk
which is straightforward to calculate (in Excel). If the number of available cores of the k best quality classes (i.e. types 1, . . . , k) is less than demand D, then all those cores will be remanufactured. Otherwise, exactly D of those cores will be remanufactured. Therefore, from the definition in (1) it follows that Lk(D) is the total expected number of remanufactured cores of
Table 1 Settings for Example 1 (D = 100). Quality grade Remanufacturing cost Probability
1 5 1/6
2 20 1/6
3 30 1/6
4 35 1/6
5 40 1/6
6 45 1/6
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Number of acquired cores
700 600 500
Acquisition under quality uncertainty
400 300
Acquisition w ithout quality uncertainty
200 100 0 0
2.5
5
7.5
10
12.5
15
17.5
20
Acquisition cost per core ^ for Example 1. Fig. 1. Number of acquired cores under quality uncertainty (A*) versus without quality uncertainty ðAÞ
remanufacturing cores with quality up to 1, 2, . . . , 6, respectively. The optimum number of acquired cores (A*) also takes on other values and is in fact a continuous function of the acquisition cost that can take on any value equal to or larger than 100. It is also interesting to note that A* can be both smaller and lar^ So, the added uncertainty in the quality of cores does ger than A. not always lead to acquiring more cores. Fig. 2 shows the effect of ignoring the quality uncertainty, i.e. ^ instead of A* cores, on the expected total cost. We of acquiring A remark that we also calculated and included those acquisition ^ jumps in value. We further remark that the cost costs where A error is 10.93% if the acquisition cost is zero. (The cost error axis of Fig. 2 is truncated at a smaller value for presentational reasons.). It appears from Fig. 2 that the maximum cost errors occur at the ^ (see also Fig. 1). left hand side of the ‘intervals’ with a constant A Apparently, the cost of not acquiring enough cores can be more severe than the cost of buying too many. This will be confirmed by the results for Example 2. It also appears that for Example 1 the average/maximum ^ cost errors over the intervals are increasing functions of A. However, the next example will illustrate that this is not true in general.
Example 2. Compared to Example 1, we change the remanufacturing costs to those in Table 2. So, instead of a fairly gradual increase in the remanufacturing cost over the six classes, there are now two groups of classes: three relatively low cost (1–3) and three relatively high cost classes (4– 6), where there is variation within each of these two groups. Fig. 3 shows the optimal number of acquired cores with and without quality uncertainty. The acquisition cost per core is varied ^ remain constant at 100) with from 0 to 30 (after which A* and A step size 0.5. Comparing Figs. 1 and 3 shows that they are quite similar, but ^ = 200 is much longer in Fig. 3. This is exthe interval where A pected, because this is the interval for which (without quality uncertainty) all cores with a low remanufacturing cost but none of the high cost cores are remanufactured. Fig. 4 shows that this is also the interval where the highest cost errors of ignoring quality uncertainty are incurred. The parameter settings for remanufacturing costs in this example imply that the largest cost errors are incurred when exactly 200 cores are acquired (by ignoring quality uncertainty), whereas it is better to acquire more cores in order to avoid the probability of having to remanufacture some of the cores in classes 4–6 with much larger remanufacturing costs.
6.00%
Cost error
5.00%
No quality uncertainty
4.00%
Acquire 600 Acquire 300
3.00%
Acquire 200 Acquire 150
2.00%
Acquire 120 Acquire 100
1.00% 0.00% 0
5
10
15
20
Acquisition cost per core ^ instead of A* cores for Example 1. Fig. 2. Cost error of acquiring A
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The third relevant cost component, for the case with uncertain demand, is the expected number of lost sales. Obviously, this cost P component will depend on X ¼ Kk¼1 X k , the number of remanufactured cores. It is easy to see that the expected number of lost sales, given X, is given by
Table 2 Settings for Example 2 (D = 100). Quality grade Remanufacturing cost Probability
1 5 1/6
2 10 1/6
3 15 1/6
4 45 1/6
5 50 1/6
6 55 1/6
Z 6. Uncertain demand
v ¼X
ðv XÞf ðv Þ dv :
Z
1
v ¼E½X
6.1. Optimal remanufacture-up-to levels
ðv E½XÞf ðv Þ dv :
ð5Þ ca A þ
Note that (5) implies that R1 P R2 P P RK , confirming (see Section 3) that higher quality and therefore lower remanufacturing costs imply a higher optimal remanufacture-up-to level. Note also that the core acquisition cost does not affect the remanufactureup-to levels, since remanufacturing decisions are made after acquisition and sorting.
K X
crk E½X k þ cp
Z
!
1
v ¼E½X
k¼1
ðv E½XÞfðv Þ dv
ca A þ
K X
crk E½X k
k¼1
E½X l
r u
6.2. Optimal acquisition decision
r
E½X l
r
E½X l : 1U
Example 3 has the same settings as Example 1 in Section 4, but demand follows a normal distribution. Expected demand (l) is still
Number of acquired cores
700 600 500
Acquisition under quality uncertainty
400 300
Acquisition w ithout quality uncertainty
200 100 0 5
7.5
10
12.5 15
ð10Þ
7. Numerical results for uncertain demand
ð6Þ
The acquisition cost is linear in the number of acquired cores and, using (6), we can determine the expected remanufacturing cost.
2.5
r
In the next section, we will use this expression to explore the effect of demand uncertainty on the acquisition decision. We remark that for all the parameter settings that are considered in that section, it was verified by simulation (see also Appendix B) that the number of acquired cores is indeed optimal. This confirms that the approximation is indeed very accurate.
What is left is to determine the optimal number of cores to acquire, given the optimal remanufacture-up-to levels determined in Section 6.1. Similar to the derivation of (3) for deterministic demand, it follows from (1) that the expected numbers of remanufactured cores for uncertain demand are
0
ð9Þ
and for normally distributed demand by (see e.g. Axsäter, 2006)
þ cp
E½X 1 ¼ L1 R1 and E½X k ¼ Lk Rk Lk1 Rk ; k > 1:
ð8Þ
Obviously, this approximation becomes more accurate if X varies less. However, as we illustrate in Appendix B for a specific example that will be considered in Section 7, the approximation is very accurate even if the coefficient of variation is large. So, the total expected cost can be approximated for any demand distribution by
The optimal remanufacture-up-to level for type i cores is given by the following well-known newsboy solution (see e.g. Silver et al., 1998) with underage cost u ¼ cp cri for each lost demand and overage cost o ¼ cri for each unsold remanufactured core.
p p c cri c cri u ¼ F 1 : ¼ F 1 p r r p uþo c c i þ ci c
ð7Þ
By taking the expected value over X of (7), the expected number of lost sales is obtained. However, as this is complicated and does not lead to closed-form expressions, we instead approximate the expected number of lost sales by evaluating (7) for the expected value of X, i.e., we approximate (7) by
We now introduce uncertainty in the demand, next to the uncertainty in the quality of acquired cores. Since remanufacturing decisions are taken after acquisition and sorting, we apply a 2-step procedure that determines the optimal remanufacture-up-to levels first (Section 6.1) and the optimal acquisition decision next (Section 6.2).
Ri ¼ F 1
1
17.5
20
22.5
25
27.5
30
Acquisition cost per core ^ for Example 2. Fig. 3. Number of acquired cores under quality uncertainty (A*) versus without quality uncertainty ðAÞ
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6.00%
No quality uncertainty
Cost error
5.00% 4.00%
Acquire 600 Acquire 300
3.00%
Acquire 200 Acquire 150
2.00%
Acquire 120 Acquire 100
1.00% 0.00% 0
5
10
15
20
25
30
Acquisition cost per core ^ instead of A* cores for Example 2. Fig. 4. Cost error of acquiring A
100, but the standard deviation (r) varies from 0 to 40 (step size 5). This allows us to study the effect of demand uncertainty. The lost sales penalty is set at 200. Table 3 shows the effect of demand uncertainty on the optimal policy parameters when the acquisition cost is 10 per core. Using other values for the acquisition cost results in similar results. Table 3 reveals that the optimal number of acquired cores and all optimal remanufacture-up-to levels increase with the standard deviation of demand. So, the ‘safety stock’ of remanufactured cores increases with the uncertainty in demand, as expected. However, as Table 4 shows, the expected number of lost sales and the corresponding penalty cost also increase with the standard deviation of demand. So, the increase in safety stock only partially offsets the increase in demand uncertainty. Next, we explore the effect of demand uncertainty on the cost (error) of ignoring quality uncertainty. Table 5 gives the costs and cost errors for Example 3 with acquisition cost 3 per core. Table 3 Effect of the standard deviation of demand on the optimal number of acquired cores and on the optimal remanufacture-up-to levels for Example 3 with acquisition cost 10 per core.
r
A*
R1
R2
R3
R4
R5
R6
0 5 10 15 20 25 30 35 40
150 156 163 170 176 183 190 196 203
100 110 120 129 139 149 159 169 178
100 106 113 119 126 132 138 145 151
100 105 110 116 121 126 131 136 141
100 105 109 114 119 123 128 133 137
100 104 108 113 117 121 125 129 134
100 104 108 111 115 119 123 126 130
Table 4 Effect of the standard deviation of demand on the expected acquisition, remanufacturing, lost sales and total costs for Example 3 with acquisition cost 10 per core.
r
0 5 10 15 20 25 30 35 40
Expected costs Acquisition
Remanufacturing
Lost sales
Total
1500 1564 1631 1697 1764 1831 1897 1964 2030
2250 2363 2463 2562 2662 2761 2861 2960 3060
0 103 206 308 411 514 617 720 822
3750 4030 4299 4568 4837 5106 5375 5644 5913
Table 5 Effect of ignoring quality uncertainty on the number of acquired cores and the total expected cost for Example 3 with acquisition cost 3 per core.
r
0 5 10 15 20 25 30 35 40
Considering quality uncertainty
Ignoring quality uncertainty
Acquire
Cost
Acquire
Cost
Cost error (%)
300 319 337 356 374 393 412 430 449
2183 2365 2547 2730 2913 3096 3279 3463 3646
325 342 359 376 393 411 428 446 463
2170 2354 2538 2722 2906 3090 3274 3458 3643
0.59 0.47 0.38 0.30 0.24 0.20 0.16 0.13 0.10
It appears from Table 5 that the cost error of ignoring quality uncertainty is decreasing with the uncertainty in demand. This was also observed for other values of the acquisition cost and other examples that we considered. Apparently, ignoring a relatively smaller part of the total uncertainty is less costly. 8. Conclusions and further research In this paper, we studied acquisition and remanufacturing decisions for situations with more than two discrete quality classes when there is quality uncertainty. We first analyzed the deterministic demand case, and showed that ignoring the uncertainty in the quality of cores can lead to acquiring either not enough or too many cores. For the numerical examples that we considered, this can increase cost by more than 5%, although the increase is less than 1% in many cases. The cost error is especially large if the deterministic solution acquires ‘just enough’ high quality cores to avoid the need (on average) to remanufacture lower quality cores at a much higher cost. In comparison, the solution under uncertain quality is to acquire additional cores as safety stock, in case a larger than average percentage turns out to be of lower quality. We then considered the more realistic case with uncertainty in both the quality of the cores and in the demand. We decomposed the problem and first derived newsboy-type optimality conditions for the remanufacture-up-to levels for all core quality classes. Then, we derived a closed-form approximation for the total expected cost, which can easily be implemented in standard spreadsheet packages like Excel to find the (nearly) optimal number of acquired cores. The results showed that increased demand
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Appendix A From (1), we easily get
!þ # " # " j j X X Lj ðDÞ ¼ E Ai E Ai D ; i¼1
It is well-known that, according to the Central Limit Theorem, the binomial distribution is asymptotically normal (if the expected value is at least 5). So, j X
Ai B A;
i¼1
! pi
Nðlj ; rj Þ;
ðA2Þ
where
lj :¼ A
j X
pi ;
rj
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u j j u X X :¼ tA pi 1 pi :
i¼1
i¼1
ðA3Þ
i¼1
Using (A2) and (A3), we can rewrite (A1) as
Lj ðDÞ ¼ lj E½ðNðlj ; rj Þ DÞþ :
ðA4Þ
The expectation term in (A4) is a ‘normal loss function expression’, which can be rewritten (see e.g. Axsäter, 2006) to give
D lj D lj D lj : Lj ðDÞ ¼ lj rj u 1U
rj
rj
rj
Appendix B In this appendix, we illustrate that the approximation of (7) by (8) in Section 6.2 is very accurate. We consider the ‘worst case’ setting in Section 7 with the standard deviation equal to 40, the maximum value considered. Expected demand is 100, the acquisition cost is 10 per core, the lost sales cost is 200 and the other parameter settings are as given in Table 1. We remark that other examples that we considered confirmed the high accuracy and the explanations that follow. First of all, we simulate the situation 1000 times to find that the number of remanufactured cores is always between 133.7 ðR5 Þ and 137.4 ðR4 Þ. Note that this variation is quite small, considering that the variation in the of cores of any pnumber ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi quality types has a standard deviation of 200 ð1=6Þ ð5=6Þ ¼ 5:3. The relatively small variation in the total number of remanufactured cores is explained by the fact that in order to avoid excessive lost sales, more low quality cores are remanufactured if fewer high quality cores are available. This effectively smoothes the total number of remanufactured cores. In the small relevant range (133.7–137.4) for the total number of remanufactured cores, the loss function is almost linear as is illustrated in Fig. 5. Therefore, (8) is a very accurate approximation of (7). The accuracy was further confirmed by an average (over 1,000 runs) simulated penalty cost of 823.8 compared to the approximation of 822.5 that was obtained using (8).
1100 1000 900 800 700 600 130
ðA1Þ
j X i¼1
Expected penalty cost
variation leads to a larger optimal number of acquired cores and larger optimal remanufacture-up-to levels. However, it still leads to more lost sales, showing that the increase in ‘optimal safety stock’ only partially offsets the increase in demand uncertainty. We also observed that the cost error of ignoring quality uncertainty decreases with increasing uncertainty in demand. In other words, the value of using the information on quality uncertainty is smaller if the uncertainty in demand is larger. As remarked in Section 3, our model assumes that an unlimited amount of cores can be acquired at a fixed price, but that this can easily be extended to the situation where a higher price is needed to attract more cores. Indeed, the cost expressions (4), (9) and (10) are easily modified for that case by replacing caA with the relevant core supply as a function of the acquisition price. It is also straightforward to account for situations where the acquisition price influences the quality distribution of supplied cores. Such situations seem realistic, as a higher acquisition price may increase the willingness to return high quality cores. These situations can be dealt with by letting the quality probabilities pi be functions of the acquisition price. In doing so, it should of course be ensured that the probabilities still sum to one whatever the value of the acquisition price. More difficult, but very relevant, extensions would be to analyze situations with multiple suppliers or multiple parties that collect cores. One example of a situation with multiple suppliers is that where airtime providers and charities both supply mobile phones (Guide et al., 2003). The core quality distributions are different for these two supply sources, with airtime providers typically supplying newer phones of higher average quality. A remanufacturer can respond to this and other situations by varying his acquisition price. Future research can address how this is best done. Competition for cores in case of multiple collecting parties is another interesting research direction. Aside from the acquisition pricing, this research could address which collection channels best suit certain competitive environments and how collection can be managed. Interested readers are referred to Mutha and Pokharel (2009) for a recent contribution and review on the design of a collection and distribution network. The value of information on quality uncertainty can also be explored further, by considering situations where the quality levels of cores in an acquired lot are related and hence the distribution for the quality of acquired cores is no longer multinomial. Other directions for further research are to generalize our model by considering situations (a) where a higher acquisition price can provide an incentive for relatively more cores of high quality, (b) with an option to test cores before acquiring them in order to get partial information, and potentially use that information for acquisition price differentiation, (c) with multiple recovery options for cores as in Robotis et al. (2005), and (d) with product life-cycle considerations as in Östlin et al. (2009). Finally, our approach for incorporating quality uncertainty of supplied materials into procurement planning could also be applied to or modified for situations where quality is not related to usage (as is typically the case for cores) but e.g. to using multiple (unreliable) suppliers or to production systems with defects and rework opportunities.
131
132
133
134
135
136
137
138
139
140
Total number of remanufactured cores
i¼1
where x+ = x if x > 0 and x+ = 0 if x 6 0.
Fig. 5. Expected penalty cost of lost sales as a (loss) function of the total number of remanufactured cores.
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Humphrey, A.S., Taylor, G.D., Landers, T.L., 1998. Stock level determination and sensitivity analysis in repair/rework operations. International Journal of Operations & Production Management 18 (6), 612–630. Inderfurth, K., 2005. Impact of uncertainties on recovery behavior in a remanufacturing environment: A numerical analysis. International Journal of Physical Distribution & Logistics Management 35 (5), 318–336. Karaer, O., Lee, H.L., 2007. Managing the reverse channel with RFID-enabled negative demand information. Production and Operations Management 16 (5), 625–645. Ketzenberg, M., van der Laan, E.A., Teunter, R.H., 2006. Value of information in closed loop supply chains. Production and Operations Management 15 (3), 393– 406. Klausner, M., Hendrickson, C.T., 2000. Reverse-logistics strategy for product takeback. Interfaces 30 (3), 156–165. Krikke, H.R., van Harten, A., Schuur, P.C., 1999. Business case Oce: Reverse logistic network re-design for copiers. OR Spektrum 21, 381–409. Liang, Y., Pokharel, S., Lim, G.H., 2009. Pricing used products for remanufacturing. European Journal of Operational Research 193, 390–395. Majumder, P., Groenvelt, H., 2001. Competition in remanufacturing. Production and Operations Management 10 (2), 125–141. Mondol, S., Mukherjee, K., 2006. Buy-back policy decision in managing reverse logistics. International Journal of Logistics Systems and Management 2 (3), 255– 264. Mutha, A., Pokharel, S., 2009. Strategic network design for reverse logistics and remanufacturing using new and old product modules. Computers & Industrial Engineering 56 (1), 334–346. Östlin, J., Sundin, E., Björkman, M., 2009. Product life-cycle implications for remanufacturing strategies. Journal of Cleaner Production 17 (11), 999–1009. Pokharel, S., Mutha, A., 2009. Perspectives in reverse logistics: A review. Resources, Conservation and Recycling 53 (4), 175–182. Robotis, A., Bhattacharya, S., van Wassenhove, L.N., 2005. The effect of remanufacturing on procurement decisions for resellers in secondary markets. European Journal of Operational Research 163 (3), 688–705. Rubio, S., Chamorro, A., Miranda, F.J., 2008. Characteristics of the research on reverse logistics (1995–2005). International Journal of Production Research 46 (4), 1099–1120. Rudi, N., Pyke, D.F., Sporsheim, P.O., 2000. Product recovery at the Norwegian National Insurance Administration. Interfaces 30 (3), 166–179. Savaskan, R.C., Bhattacharya, S., van Wassenhove, L.N., 2004. Closed-loop supply chain models with product remanufacturing. Management Science 50 (2), 239– 252. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, third ed. John Wiley & Sons, Inc., New York. Souza, G.C., Ketzenberg, M., Guide, V.D.R., 2002. Capacitated remanufacturing with service level constraints. Production and Operations Management 11 (2), 231– 248. Stanfield, P.M., Wilson, J.R., King, R.E., 2004. Flexible modeling of correlated operation times with application in product-reuse facilities. International Journal of Production Research 42 (11), 2179–2196. Toktay, L.B., Wein, L.M., Zenios, S.A., 2000. Inventory management of remanufacturable products. Management Science 46 (11), 1412–1426. Van Nunen, J.A.E.E., Zuidwijk, R.A., 2004. E-enabled closed-loop supply chains. California Management Review 46 (2), 40–54. Yano, C.A., Lee, H.L., 1995. Lot sizing with random yields: A review. Operations Research 43 (2), 311–334. Zikopoulos, C., Tagaras, G., 2007. Impact of uncertainty in the quality of returns on the profitability of a single-period refurbishing operation. European Journal of Operational Research 182 (1), 205–225. Zikopoulos, C., Tagaras, G., 2008. On the attractiveness of sorting before disassembly in remanufacturing. IIE Transactions 40 (3), 313–323.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Optimal core acquisition and remanufacturing policies under uncertain core quality fractions Ruud H. Teunter a,⇑, Simme Douwe P. Flapper b,1 a b
University of Groningen, Department of Operations, P.O. Box 800, 9700 AV, Groningen, The Netherlands Technische Universiteit Eindhoven, School of Industrial Engineering, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
a r t i c l e
i n f o
Article history: Received 5 November 2009 Accepted 12 June 2010 Available online 18 June 2010 Keywords: Remanufacturing Product acquisition management Quality uncertainty
a b s t r a c t Cores acquired by a remanufacturer are typically highly variable in quality. Even if the expected fractions of the various quality levels are known, then the exact fractions when acquiring cores are still uncertain. Our model incorporates this uncertainty in determining optimal acquisition decisions by considering multiple quality classes and a multinomial quality distribution for an acquired lot. We derive optimal acquisition and remanufacturing policies for both deterministic and uncertain demand. For deterministic demand, we derive a simple closed-form expression for the total expected cost. In a numerical experiment, we highlight the effect of uncertainty in quality fractions on the optimal number of acquired cores and show that the cost error of ignoring uncertainty can be significant. For uncertain demand, we derive optimal newsboy-type solutions for the optimal remanufacture-up-to levels and an approximate expression for the total expected cost given the number of acquired cores. In a further numerical experiment, we explore the effects of demand uncertainty on the optimal acquisition and remanufacturing decisions, and on the total expected cost. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Remanufacturing is a multi-billion dollar industry of growing importance (Corbett and Kleindorfer, 2001; Majumder and Groenvelt, 2001) that has received considerable academic interest over the past two decades. A large number of contributions have dealt with logistics and supply chain management for systems with (manufacturing and) remanufacturing. Reviews in this area are provided by Dekker et al. (2004), Rubio et al. (2008), and Pokharel and Mutha (2009). In remanufacturing, the quality of cores (i.e. products supplied for remanufacturing) can vary significantly, affecting the cost of remanufacturing. Guide et al. (2003) report the case of a mobile phone remanufacturer, ReCellular, that distinguishes six quality grades based on functional and cosmetic criteria with the remanufacturing cost increasing by a factor of about ten from the highest to the lowest quality grade. An important decision in remanufacturing is therefore which of the available cores should be remanufactured. Obviously, cores of the highest quality (grade 1) will be remanufactured first, followed by cores of grade 2, and so on. The availability of cores of the different quality grades clearly depends on the number of cores that ⇑ Corresponding author. Tel.: +31 50 3638617; fax: +31 50 3632032. E-mail addresses: [email protected] (R.H. Teunter), s.d.p.fl[email protected] (S.D.P. Flapper). 1 Tel.: + 31 40 2474385; fax: +31 40 2464596. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.06.015
have been acquired. Hence, the acquisition decision affects the remanufacturing decision, and these two decisions should therefore be analyzed jointly. In this paper, we will do so. The situation that we consider is one where unsorted cores are acquired from third-party brokers or collectors. We remark that this is different from cases where sorting is done first and acquisition prices are quality dependent, as assumed by Guide et al. (2003) in their model. In the situation of acquiring unsorted cores, there is uncertainty about the numbers of cores of the various grades that have been acquired until sorting is carried out, even if good estimates are available for the probabilities that a core is of a certain type, i.e. even if the core quality distribution is known. Very few authors have addressed situations where core quality affects the acquisition and remanufacturing decisions. Their models will be discussed in detail in Section 2. None of them have considered the situation with multiple (more than two) quality classes, as we do in this paper. Because the analysis of the situation with both quality and demand uncertainty is more complex and less insightful than the analysis with quality uncertainty only, we start with the latter analysis and associated numerical investigations. Afterwards, we consider the more realistic case with both types of uncertainty, and perform further numerical investigations. The rest of the paper is organized as follows. In Section 2, we shortly review the related literature. In Section 3, we present the model. In Section 4, we analyze the situation of deterministic
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demand, followed by a numerical experiment in Section 5. The same is done for stochastic demand in Sections 6 and 7. We end in Section 8 with a discussion of generalizations, limitations and directions for further research. 2. Related literature This review is structured as follows. First, in Section 2.1, we shortly discuss the broader setting of supply quality and yield in manufacturing. Section 2.2 continues with the related literature on remanufacturing. Finally, in Section 2.3, we provide a detailed discussion of remanufacturing models that consider the effect of quality uncertainty on the acquisition of cores, and point out the contribution of this paper. 2.1. Quality uncertainty in manufacturing There are a number of contributions that assume it is possible to (partially) correct for undesired yield with respect to quality, by additional ordering or production after observing the yield (e.g. Grosfeld-Nir and Gerchak, 2004) or by using downgrading (e.g. Yano and Lee, 1995). Some authors have explored the setting where the quality of the products ordered or produced can be influenced, e.g. by paying more per product in case of external acquisition (e.g. Bakal and Akcali, 2006 and Liang et al., 2009), by supplier selection (e.g. Grosfeld-Nir and Gerchak, 2004) or by process inspection and order splitting for unreliable production processes (e.g. Yano and Lee, 1995). In this paper, we focus on situations where none of these options are available. 2.2. Quality uncertainty in remanufacturing A number of authors address the complications of quality uncertainty of cores for remanufacturing operations (BloemhofRuwaard et al., 1999; Fleischmann et al., 2000; Guide, 2000; Guide and Jayaraman, 2000; Guide and Srivastava, 1997; Humphrey et al., 1998; Inderfurth, 2005; Ketzenberg et al., 2006; Savaskan et al., 2004; Stanfield et al., 2004; Toktay et al., 2000). Sorting is therefore important (Blackburn et al., 2004; Bloemhof-Ruwaard et al., 1999; Guide et al., 2000; Van Nunen and Zuidwijk, 2004) and several sorting techniques have been proposed (Krikke et al., 1999; Rudi et al., 2000). Quality uncertainty causes extra complications if the remanufacturing process is capacitated (Souza et al., 2002; Guide et al., 2008; Ferguson et al., 2009; Denizel et al., 2007) or if there are stock keeping restrictions (Aras et al., 2004). All these authors study the effect of such system restrictions on the decision whether to remanufacture cores of certain quality or dispose them/sell them ‘‘as is”. Some authors have discussed the use of incentive-systems (Guide and van Wassenhove, 2001; Guide et al., 2003; Klausner and Hendrickson, 2000) or collection options (Savaskan et al., 2004) for influencing the quality distribution of cores. A related contribution is that by Mondol and Mukherjee (2006), who study situations where age affects the quality of cores and analyze when to buy them back. 2.3. Acquisition of cores under uncertain quality Galbreth and Blackburn (2006) assume that there is a continuum of quality levels for cores. They further assume that for any lot of acquired cores, the quality distribution is fixed. They derive optimal acquisition and sorting/remanufacturing policies under both deterministic and uncertain demand.
Galbreth and Blackburn (2010) drop the assumption of a fixed quality distribution, and instead allow the condition of each acquired core in a lot to be random. The condition is assumed to follow a uniform distribution. Only the deterministic demand case is analyzed. Apart from the situation with a continuum of quality levels for cores, they also consider the situation with two discrete classes that can both be remanufactured but at different costs. Zikopoulos and Tagaras (2007) consider a system with two collection sites. There are two grading classes, good and bad (not remanufacturable), and the proportion of good cores for the two sites follows a general joint probability distribution. Demand is uncertain. An expression for the profit function is derived, and conditions are provided under which it is optimal to collect from a single site. Zikopoulos and Tagaras (2008) continue to study the situation with two grading classes, good and bad, and a general probability distribution further. They restrict the analysis to a single collection site, but allow misclassification errors where good cores are classified as bad or vice versa. A comparison of policies with and without sorting before disassembly (that is uncoupled from remanufacturing) provides insights into the economic attractiveness of sorting. Our study is the first that considers acquisition and remanufacturing decisions for situations with multiple (more than two) discrete quality classes when there is quality uncertainty. Moreover, contrary to Galbreth and Blackburn (2006, 2010) who studied a continuum of quality levels, we consider quality and demand uncertainty together. 3. Model We start by listing the different symbols in our model and then describe the model and underlying assumptions. For ease of writing, we refer to cores in a certain quality class i simply as type i cores or cores of type i. Notations A number of acquired cores A* optimal number of acquired cores number of acquired cores that turn out to be of type i Ai B(n,p) binomial distribution for n trials with success probability p acquisition cost per core ca penalty cost per unsatisfied demand cp remanufacturing cost per type i core cri D demand f probability density function of demand F probability distribution function of demand K number of quality classes N(l, r) normal distribution with mean l and standard deviation r pi probability that an acquired core is of type i remanufacture-up-to level for type i cores Ri optimal remanufacture-up-to level for type i cores Ri number of cores of type i that are remanufactured Xi u() standard normal probability density function U() standard normal probability distribution function ul, r() N(l, r) probability density function l expected demand r standard deviation demand
We analyze the problem of how many cores to acquire and of how many cores of the different quality types to remanufacture after sorting. We do so in a single-period setting. The sequence of the events in this period is as follows: acquisition, sorting, remanufacturing, demand realization. All acquired cores are sorted. Guide et al. (2006) and Debo et al. (2004) provide examples of companies that do so.
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The type (quality) of each core is uncertain. We assume that there are K different types, and that the type of a core follows a multinomial distribution with probability pi that a core is of type i for i = 1, 2, . . . , K. This distribution is appropriate when the cores in an acquired lot have little relationship, as is often the case in remanufacturing (Galbreth and Blackburn, 2010). See also Karaer and Lee (2007). The relevant costs are the acquisition cost ca per core, the penalty cost cp per unsatisfied demand and the (expected) remanufacturing cost cri per type i core. As we assume that all acquired cores are sorted, the sorting cost per core can be included in ca. The assumption of a fixed acquisition cost/price reflects the situation of complete competition in the supply market. However, our analysis and results can easily be extended to situations where the acquisition price affects the supply of cores, as in e.g. Guide et al. (2003). We will provide the details of how to do so in Section 8. Note that, since sorting does not take place until after acquisition, it is not possible to differentiate the acquisition price based on the quality of the core as Liang et al. (2009) suggest. We also remark that a net cost for discarding cores (which may be negative if there are recycling revenues) can easily be included by adding that cost to ca and subtracting it from cri . Without loss of generality, we rank the core types from highest quality (type 1) to lowest quality (type K). Quality is reflected by the remanufacturing cost and hence cr1 < cr2 < < crK . With this ranking, it is obviously optimal to remanufacture cores in quality class 1 first, and then cores in class 2, and so on. We therefore consider the class of policies that remanufactures cores in this sequence, and use remanufacture-up-to levels Ri for type i cores. It is intuitively obvious that the only policies that need to be considered are those that only remanufacture cores of a certain quality type if there are no cores left of better quality, i.e. policies with R1 P R2 P RK . The analysis in Section 6 will confirm that this holds for the optimal remanufacture-up-to level levels. Because remanufacturing takes place after sorting, the optimization problem can be decomposed into determining the optimal remanufacturing policy first and the optimal acquisition decision next. In the next section, this will be used to derive optimality conditions for the case of certain demand. The more realistic but also more complex case of uncertain demand will be analyzed in Section 6. 4. Deterministic demand We first introduce some notation that will turn out to be useful for the analysis. Let
" Lk ðMÞ :¼ E min
" k X
## Ai ; M
;
k ¼ 1; . . . ; K
ð1Þ
i¼1
and
lk :¼ A
k X
pi ;
rk
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u k k u X X t :¼ A pi 1 pi :
i¼1
i¼1
i¼1
Using standard algebra and applying the Central Limit Theorem (see Appendix A), (1) can be rewritten as
M lk M lk M lk ; ð2Þ Lk ðMÞ ¼ lk rk u 1U
rk
rk
types 1, . . . , k when demand is D. Similarly, we have that Lk1(D) is the total expected number of remanufactured cores of types 1, . . . , k 1 for demand D. Hence, the expected numbers of remanufactured cores of each type are given by
E½X 1 ¼ L1 ðDÞ and E½X k ¼ Lk ðDÞ Lk1 ðDÞ;
k > 1:
ð3Þ
So the total cost, consisting of acquisition and remanufacturing costs, is
CðAÞ ¼ ca A þ cr1 L1 ðDÞ þ
K X
crk ðLk ðDÞ Lk1 ðDÞÞ:
ð4Þ
k¼2
Obviously, the optimal number of acquired cores A* is the value of A that minimizes (4). 4.1. Approximation that assumes constant sample quality fractions In the next section, we will compare A* to the ‘optimal’ acquisi^ under the assumption that there is no uncertainty tion decision A in the quality of cores, i.e. that the fraction of the acquired cores of type k is always exactly pk. Then (1) gives
" # k X ^Lk ðDÞ ¼ min A pi ; D
ð10 Þ
i¼1
^ can be determined by minimizing and A
^ CðAÞ ¼ ca A þ ^L1 ðDÞ þ
K X
crk ð^Lk ðDÞ ^Lk1 ðDÞÞ:
ð40 Þ
k¼2
5. Numerical results for deterministic demand In this section, we analyze some numerical examples to obtain insight into the effect of uncertainty about the quality of cores on the optimal acquisition decision. Recall that incorporating that uncertainty is one of our main contributions to the existing literature, and hence we are particularly interested in the ‘cost error’ of not doing so. This cost error is calculated as
^ CðA Þ CðAÞ 100%: CðA Þ The examples that we consider are inspired by the real life case of a mobile phone remanufacturer ReCellular that is discussed in detail in Guide et al. (2003). This remanufacturer distinguishes six quality grades. Example 1. In our first example, we use the estimated remanufacturing costs for the different grades in the Recellular case study, as given in Table 1. Each grade is assumed to occur with the same probability. Demand is for 100 remanufactured products (D = 100). In Fig. 1 the optimal number of acquired cores under quality uncertainty (A*) is compared with the optimal number of acquired ^ The acquisicores when assuming constant quality fractions ðAÞ. ^ retion cost per core is varied from 0 to 20 (after which A* and A main constant at 100) with step size 0.5. ^ takes on six different Note from Fig. 1 that the approximation A values only. These are 600/1, 600/2, . . . , 600/6, which correspond to
rk
which is straightforward to calculate (in Excel). If the number of available cores of the k best quality classes (i.e. types 1, . . . , k) is less than demand D, then all those cores will be remanufactured. Otherwise, exactly D of those cores will be remanufactured. Therefore, from the definition in (1) it follows that Lk(D) is the total expected number of remanufactured cores of
Table 1 Settings for Example 1 (D = 100). Quality grade Remanufacturing cost Probability
1 5 1/6
2 20 1/6
3 30 1/6
4 35 1/6
5 40 1/6
6 45 1/6
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Number of acquired cores
700 600 500
Acquisition under quality uncertainty
400 300
Acquisition w ithout quality uncertainty
200 100 0 0
2.5
5
7.5
10
12.5
15
17.5
20
Acquisition cost per core ^ for Example 1. Fig. 1. Number of acquired cores under quality uncertainty (A*) versus without quality uncertainty ðAÞ
remanufacturing cores with quality up to 1, 2, . . . , 6, respectively. The optimum number of acquired cores (A*) also takes on other values and is in fact a continuous function of the acquisition cost that can take on any value equal to or larger than 100. It is also interesting to note that A* can be both smaller and lar^ So, the added uncertainty in the quality of cores does ger than A. not always lead to acquiring more cores. Fig. 2 shows the effect of ignoring the quality uncertainty, i.e. ^ instead of A* cores, on the expected total cost. We of acquiring A remark that we also calculated and included those acquisition ^ jumps in value. We further remark that the cost costs where A error is 10.93% if the acquisition cost is zero. (The cost error axis of Fig. 2 is truncated at a smaller value for presentational reasons.). It appears from Fig. 2 that the maximum cost errors occur at the ^ (see also Fig. 1). left hand side of the ‘intervals’ with a constant A Apparently, the cost of not acquiring enough cores can be more severe than the cost of buying too many. This will be confirmed by the results for Example 2. It also appears that for Example 1 the average/maximum ^ cost errors over the intervals are increasing functions of A. However, the next example will illustrate that this is not true in general.
Example 2. Compared to Example 1, we change the remanufacturing costs to those in Table 2. So, instead of a fairly gradual increase in the remanufacturing cost over the six classes, there are now two groups of classes: three relatively low cost (1–3) and three relatively high cost classes (4– 6), where there is variation within each of these two groups. Fig. 3 shows the optimal number of acquired cores with and without quality uncertainty. The acquisition cost per core is varied ^ remain constant at 100) with from 0 to 30 (after which A* and A step size 0.5. Comparing Figs. 1 and 3 shows that they are quite similar, but ^ = 200 is much longer in Fig. 3. This is exthe interval where A pected, because this is the interval for which (without quality uncertainty) all cores with a low remanufacturing cost but none of the high cost cores are remanufactured. Fig. 4 shows that this is also the interval where the highest cost errors of ignoring quality uncertainty are incurred. The parameter settings for remanufacturing costs in this example imply that the largest cost errors are incurred when exactly 200 cores are acquired (by ignoring quality uncertainty), whereas it is better to acquire more cores in order to avoid the probability of having to remanufacture some of the cores in classes 4–6 with much larger remanufacturing costs.
6.00%
Cost error
5.00%
No quality uncertainty
4.00%
Acquire 600 Acquire 300
3.00%
Acquire 200 Acquire 150
2.00%
Acquire 120 Acquire 100
1.00% 0.00% 0
5
10
15
20
Acquisition cost per core ^ instead of A* cores for Example 1. Fig. 2. Cost error of acquiring A
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The third relevant cost component, for the case with uncertain demand, is the expected number of lost sales. Obviously, this cost P component will depend on X ¼ Kk¼1 X k , the number of remanufactured cores. It is easy to see that the expected number of lost sales, given X, is given by
Table 2 Settings for Example 2 (D = 100). Quality grade Remanufacturing cost Probability
1 5 1/6
2 10 1/6
3 15 1/6
4 45 1/6
5 50 1/6
6 55 1/6
Z 6. Uncertain demand
v ¼X
ðv XÞf ðv Þ dv :
Z
1
v ¼E½X
6.1. Optimal remanufacture-up-to levels
ðv E½XÞf ðv Þ dv :
ð5Þ ca A þ
Note that (5) implies that R1 P R2 P P RK , confirming (see Section 3) that higher quality and therefore lower remanufacturing costs imply a higher optimal remanufacture-up-to level. Note also that the core acquisition cost does not affect the remanufactureup-to levels, since remanufacturing decisions are made after acquisition and sorting.
K X
crk E½X k þ cp
Z
!
1
v ¼E½X
k¼1
ðv E½XÞfðv Þ dv
ca A þ
K X
crk E½X k
k¼1
E½X l
r u
6.2. Optimal acquisition decision
r
E½X l
r
E½X l : 1U
Example 3 has the same settings as Example 1 in Section 4, but demand follows a normal distribution. Expected demand (l) is still
Number of acquired cores
700 600 500
Acquisition under quality uncertainty
400 300
Acquisition w ithout quality uncertainty
200 100 0 5
7.5
10
12.5 15
ð10Þ
7. Numerical results for uncertain demand
ð6Þ
The acquisition cost is linear in the number of acquired cores and, using (6), we can determine the expected remanufacturing cost.
2.5
r
In the next section, we will use this expression to explore the effect of demand uncertainty on the acquisition decision. We remark that for all the parameter settings that are considered in that section, it was verified by simulation (see also Appendix B) that the number of acquired cores is indeed optimal. This confirms that the approximation is indeed very accurate.
What is left is to determine the optimal number of cores to acquire, given the optimal remanufacture-up-to levels determined in Section 6.1. Similar to the derivation of (3) for deterministic demand, it follows from (1) that the expected numbers of remanufactured cores for uncertain demand are
0
ð9Þ
and for normally distributed demand by (see e.g. Axsäter, 2006)
þ cp
E½X 1 ¼ L1 R1 and E½X k ¼ Lk Rk Lk1 Rk ; k > 1:
ð8Þ
Obviously, this approximation becomes more accurate if X varies less. However, as we illustrate in Appendix B for a specific example that will be considered in Section 7, the approximation is very accurate even if the coefficient of variation is large. So, the total expected cost can be approximated for any demand distribution by
The optimal remanufacture-up-to level for type i cores is given by the following well-known newsboy solution (see e.g. Silver et al., 1998) with underage cost u ¼ cp cri for each lost demand and overage cost o ¼ cri for each unsold remanufactured core.
p p c cri c cri u ¼ F 1 : ¼ F 1 p r r p uþo c c i þ ci c
ð7Þ
By taking the expected value over X of (7), the expected number of lost sales is obtained. However, as this is complicated and does not lead to closed-form expressions, we instead approximate the expected number of lost sales by evaluating (7) for the expected value of X, i.e., we approximate (7) by
We now introduce uncertainty in the demand, next to the uncertainty in the quality of acquired cores. Since remanufacturing decisions are taken after acquisition and sorting, we apply a 2-step procedure that determines the optimal remanufacture-up-to levels first (Section 6.1) and the optimal acquisition decision next (Section 6.2).
Ri ¼ F 1
1
17.5
20
22.5
25
27.5
30
Acquisition cost per core ^ for Example 2. Fig. 3. Number of acquired cores under quality uncertainty (A*) versus without quality uncertainty ðAÞ
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6.00%
No quality uncertainty
Cost error
5.00% 4.00%
Acquire 600 Acquire 300
3.00%
Acquire 200 Acquire 150
2.00%
Acquire 120 Acquire 100
1.00% 0.00% 0
5
10
15
20
25
30
Acquisition cost per core ^ instead of A* cores for Example 2. Fig. 4. Cost error of acquiring A
100, but the standard deviation (r) varies from 0 to 40 (step size 5). This allows us to study the effect of demand uncertainty. The lost sales penalty is set at 200. Table 3 shows the effect of demand uncertainty on the optimal policy parameters when the acquisition cost is 10 per core. Using other values for the acquisition cost results in similar results. Table 3 reveals that the optimal number of acquired cores and all optimal remanufacture-up-to levels increase with the standard deviation of demand. So, the ‘safety stock’ of remanufactured cores increases with the uncertainty in demand, as expected. However, as Table 4 shows, the expected number of lost sales and the corresponding penalty cost also increase with the standard deviation of demand. So, the increase in safety stock only partially offsets the increase in demand uncertainty. Next, we explore the effect of demand uncertainty on the cost (error) of ignoring quality uncertainty. Table 5 gives the costs and cost errors for Example 3 with acquisition cost 3 per core. Table 3 Effect of the standard deviation of demand on the optimal number of acquired cores and on the optimal remanufacture-up-to levels for Example 3 with acquisition cost 10 per core.
r
A*
R1
R2
R3
R4
R5
R6
0 5 10 15 20 25 30 35 40
150 156 163 170 176 183 190 196 203
100 110 120 129 139 149 159 169 178
100 106 113 119 126 132 138 145 151
100 105 110 116 121 126 131 136 141
100 105 109 114 119 123 128 133 137
100 104 108 113 117 121 125 129 134
100 104 108 111 115 119 123 126 130
Table 4 Effect of the standard deviation of demand on the expected acquisition, remanufacturing, lost sales and total costs for Example 3 with acquisition cost 10 per core.
r
0 5 10 15 20 25 30 35 40
Expected costs Acquisition
Remanufacturing
Lost sales
Total
1500 1564 1631 1697 1764 1831 1897 1964 2030
2250 2363 2463 2562 2662 2761 2861 2960 3060
0 103 206 308 411 514 617 720 822
3750 4030 4299 4568 4837 5106 5375 5644 5913
Table 5 Effect of ignoring quality uncertainty on the number of acquired cores and the total expected cost for Example 3 with acquisition cost 3 per core.
r
0 5 10 15 20 25 30 35 40
Considering quality uncertainty
Ignoring quality uncertainty
Acquire
Cost
Acquire
Cost
Cost error (%)
300 319 337 356 374 393 412 430 449
2183 2365 2547 2730 2913 3096 3279 3463 3646
325 342 359 376 393 411 428 446 463
2170 2354 2538 2722 2906 3090 3274 3458 3643
0.59 0.47 0.38 0.30 0.24 0.20 0.16 0.13 0.10
It appears from Table 5 that the cost error of ignoring quality uncertainty is decreasing with the uncertainty in demand. This was also observed for other values of the acquisition cost and other examples that we considered. Apparently, ignoring a relatively smaller part of the total uncertainty is less costly. 8. Conclusions and further research In this paper, we studied acquisition and remanufacturing decisions for situations with more than two discrete quality classes when there is quality uncertainty. We first analyzed the deterministic demand case, and showed that ignoring the uncertainty in the quality of cores can lead to acquiring either not enough or too many cores. For the numerical examples that we considered, this can increase cost by more than 5%, although the increase is less than 1% in many cases. The cost error is especially large if the deterministic solution acquires ‘just enough’ high quality cores to avoid the need (on average) to remanufacture lower quality cores at a much higher cost. In comparison, the solution under uncertain quality is to acquire additional cores as safety stock, in case a larger than average percentage turns out to be of lower quality. We then considered the more realistic case with uncertainty in both the quality of the cores and in the demand. We decomposed the problem and first derived newsboy-type optimality conditions for the remanufacture-up-to levels for all core quality classes. Then, we derived a closed-form approximation for the total expected cost, which can easily be implemented in standard spreadsheet packages like Excel to find the (nearly) optimal number of acquired cores. The results showed that increased demand
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Appendix A From (1), we easily get
!þ # " # " j j X X Lj ðDÞ ¼ E Ai E Ai D ; i¼1
It is well-known that, according to the Central Limit Theorem, the binomial distribution is asymptotically normal (if the expected value is at least 5). So, j X
Ai B A;
i¼1
! pi
Nðlj ; rj Þ;
ðA2Þ
where
lj :¼ A
j X
pi ;
rj
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u j j u X X :¼ tA pi 1 pi :
i¼1
i¼1
ðA3Þ
i¼1
Using (A2) and (A3), we can rewrite (A1) as
Lj ðDÞ ¼ lj E½ðNðlj ; rj Þ DÞþ :
ðA4Þ
The expectation term in (A4) is a ‘normal loss function expression’, which can be rewritten (see e.g. Axsäter, 2006) to give
D lj D lj D lj : Lj ðDÞ ¼ lj rj u 1U
rj
rj
rj
Appendix B In this appendix, we illustrate that the approximation of (7) by (8) in Section 6.2 is very accurate. We consider the ‘worst case’ setting in Section 7 with the standard deviation equal to 40, the maximum value considered. Expected demand is 100, the acquisition cost is 10 per core, the lost sales cost is 200 and the other parameter settings are as given in Table 1. We remark that other examples that we considered confirmed the high accuracy and the explanations that follow. First of all, we simulate the situation 1000 times to find that the number of remanufactured cores is always between 133.7 ðR5 Þ and 137.4 ðR4 Þ. Note that this variation is quite small, considering that the variation in the of cores of any pnumber ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi quality types has a standard deviation of 200 ð1=6Þ ð5=6Þ ¼ 5:3. The relatively small variation in the total number of remanufactured cores is explained by the fact that in order to avoid excessive lost sales, more low quality cores are remanufactured if fewer high quality cores are available. This effectively smoothes the total number of remanufactured cores. In the small relevant range (133.7–137.4) for the total number of remanufactured cores, the loss function is almost linear as is illustrated in Fig. 5. Therefore, (8) is a very accurate approximation of (7). The accuracy was further confirmed by an average (over 1,000 runs) simulated penalty cost of 823.8 compared to the approximation of 822.5 that was obtained using (8).
1100 1000 900 800 700 600 130
ðA1Þ
j X i¼1
Expected penalty cost
variation leads to a larger optimal number of acquired cores and larger optimal remanufacture-up-to levels. However, it still leads to more lost sales, showing that the increase in ‘optimal safety stock’ only partially offsets the increase in demand uncertainty. We also observed that the cost error of ignoring quality uncertainty decreases with increasing uncertainty in demand. In other words, the value of using the information on quality uncertainty is smaller if the uncertainty in demand is larger. As remarked in Section 3, our model assumes that an unlimited amount of cores can be acquired at a fixed price, but that this can easily be extended to the situation where a higher price is needed to attract more cores. Indeed, the cost expressions (4), (9) and (10) are easily modified for that case by replacing caA with the relevant core supply as a function of the acquisition price. It is also straightforward to account for situations where the acquisition price influences the quality distribution of supplied cores. Such situations seem realistic, as a higher acquisition price may increase the willingness to return high quality cores. These situations can be dealt with by letting the quality probabilities pi be functions of the acquisition price. In doing so, it should of course be ensured that the probabilities still sum to one whatever the value of the acquisition price. More difficult, but very relevant, extensions would be to analyze situations with multiple suppliers or multiple parties that collect cores. One example of a situation with multiple suppliers is that where airtime providers and charities both supply mobile phones (Guide et al., 2003). The core quality distributions are different for these two supply sources, with airtime providers typically supplying newer phones of higher average quality. A remanufacturer can respond to this and other situations by varying his acquisition price. Future research can address how this is best done. Competition for cores in case of multiple collecting parties is another interesting research direction. Aside from the acquisition pricing, this research could address which collection channels best suit certain competitive environments and how collection can be managed. Interested readers are referred to Mutha and Pokharel (2009) for a recent contribution and review on the design of a collection and distribution network. The value of information on quality uncertainty can also be explored further, by considering situations where the quality levels of cores in an acquired lot are related and hence the distribution for the quality of acquired cores is no longer multinomial. Other directions for further research are to generalize our model by considering situations (a) where a higher acquisition price can provide an incentive for relatively more cores of high quality, (b) with an option to test cores before acquiring them in order to get partial information, and potentially use that information for acquisition price differentiation, (c) with multiple recovery options for cores as in Robotis et al. (2005), and (d) with product life-cycle considerations as in Östlin et al. (2009). Finally, our approach for incorporating quality uncertainty of supplied materials into procurement planning could also be applied to or modified for situations where quality is not related to usage (as is typically the case for cores) but e.g. to using multiple (unreliable) suppliers or to production systems with defects and rework opportunities.
131
132
133
134
135
136
137
138
139
140
Total number of remanufactured cores
i¼1
where x+ = x if x > 0 and x+ = 0 if x 6 0.
Fig. 5. Expected penalty cost of lost sales as a (loss) function of the total number of remanufactured cores.
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