Corporate Research and Development Department of Information and Systems Technology Robert Bosch GmbH Box 10 60 50, 70049 Stuttgart, Germany

Chris T. Hendrickson

Department of Civil and Environmental Engineering Carnegie Mellon University Pittsburgh, Pennsylvania 15213-3890

Product take-back of consumer products is generally expensive, especially reverse logistics. In the take-back program for power tools in Germany, costs exceed revenues for recycling power tools. Systematic analysis of take-back alternatives can make take-back policies more attractive. For example, an alternative take-back system for power tools would combine proﬁtable remanufacturing and unproﬁtable materials recycling. The proﬁt from remanufacturing could cover the loss from recycling as well as the costs of reverse logistics, allowing the manufacturer a proﬁt. Remanufacturing requires a continuous ﬂow of returned postconsumer products. By buying back end-of-life products, ﬁrms could control the ﬂow of returned products. We developed a model that allows us to determine the optimal amount to spend on buy-back and the optimal unit cost of reverse logistics. We can use the latter to select a suitable reverse-logistics system for end-of-life products. We apply our model to the remanufacturing take-back concept for power tools, using empirical data on the current take-back program.

M

anufacturers worldwide are increasingly facing responsibility for their products at end of life and must pro-

vide for collection and product recovery or proper disposal. Product take-back is a form of extended producer responsibility

Copyright 䉷 2000 INFORMS 0092-2102/00/3003/0156/$05.00 1526–551X electronic ISSN This paper was refereed.

ENVIRONMENT INDUSTRIES—MACHINERY

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PRODUCT TAKE-BACK (EPR), a policy concept intended to make manufacturing ﬁrms responsible for their products throughout their life cycles [Lifset 1993]. EPR has emerged because traditional environmental regulations that focus on facilities do not deal with situations in which products embody pollution or cause environmental damage at later stages of their life cycles [Davis 1998]. Responsibility for product disposal forces manufacturing ﬁrms to incorporate disposal costs in product prices. This gives manufacturers incentives to design their products for lower costs at end of life, often through reuse or recycling. Besides internalizing waste-management cost, product-take-back policies are intended to reduce waste generation and to increase the use of recycled materials [OECD 1996]. The most common elements of takeback legislation as drafted or implemented around the world are numerical targets for collection and recycling, and time frames for implementation (that is, dates by which manufacturers must achieve collection and recycling targets) [OECD 1996]. Example of Product Takeback: German Power Tools In a cooperation with a leading powertool manufacturer, we studied a voluntary take-back program for power tools in Germany. In the following, we use both dollar and deutschmarks (DM) ﬁgures, as all ﬁnancial data were provided in DM. We use an exchange rate of 1 DM ⳱ $0.556, which was representative at the time of our study. In 1996, some 11 million power tools were sold in Germany, with a value of around $840 million (DM 1.5 billion). The average revenue per power tool amounted to around $76 (DM 136).

In response to drafted legislation and to demonstrate their environmental commitment, the major domestic and foreign manufacturers selling power tools in Germany agreed to voluntarily take back their old power tools free of charge from both private and business customers, starting in 1993. In 1998, the take-back initiative had 13 member ﬁrms, out of 33 power-tool manufacturers offering their products on the German market. Customers can return tools to dealers. The take-back initiative provides dealers with boxes that hold about 200 power tools. Once a box is ﬁlled, the dealer calls a logistics-services provider contracted by the take-back initiative to arrange for its collection. The boxes go to a specialized recycling facility for power tools (SRFPT). There, the manufacturer and the weight of each individual tool are recorded. Then, the tools are disassembled and the individual materials are recycled. Even though the high-grade plastics are reused in power tool manufacture and reclaimed materials are sold on secondary markets, power-tool recycling results in a net loss averaging $2.30 (DM 4.14) per tool. The logistics-services provider charges a weight-based fee that averages $2.40 (DM 4.32) per tool. The resulting cost of $4.70 (DM 8.46) is charged to the member ﬁrms based on the total weights of the returned tools. The net take-back cost is thus 6.2 percent of the unit revenue. This unit cost has to be considered in conjunction with the number of power tools taken back. In 1996, only 61 tons of end-of-life power tools or 20,000 power tools were collected at dealers. This is equivalent to 0.18 percent of the market volume that year. A de-

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KLAUSNER, HENDRICKSON tailed analysis revealed that more than 90 percent of all end-of-life power tools are stored by customers and not—as often suspected—disposed of in municipal solid waste (MSW) [Klausner 1998]. Only 4.1 percent of the market volume appears in MSW. If the return rate increases in the future, total take-back costs would increase, prohibiting the current free-of-charge system. If, for example, 10 percent of the market volume were returned and costs were reduced by 25 percent due to scale economies, the annual net loss would be $3.9 million (DM 7 million). Yet charging customers for take-back is often prohibited in take-back legislation, and internalizing

take-back cost in the new-product price would make the participating tool manufacturers less competitive as long as regulations do not force all manufacturers to take back their products. This loss motivated an investigation of alternative takeback concepts that would reduce costs. We developed a takeback concept that is based on reusing certain high-value components from some products taken back and remanufacturing a certain fraction of the return ﬂow that is characterized by almost no technological obsolescence and a low use intensity [Klausner 1998]. Remanufacturing is particularly appealing for power tools since they are characterized by long technology cycles and low technological obsolescence. Reuse of certain components, such as motor parts, is feasible

due to the uncertain use intensity of power tools, resulting in overengineering for certain user groups in order to meet the needs of other user groups. The core of the take-back concept we developed is a product-integrated device called the Electronic Data Log (EDL). The EDL measures product degradation and computes remaining component lifetime [Klausner, Grimm, Hendrickson, and Horvath 1998]. We developed this device, implemented it in power tools, and tested it extensively. With the EDL in place, remanufacturable power tools and reusable components can be automatically identiﬁed. Products and components that cannot be remanufactured could be recycled for their materials as at present or landﬁlled, both at a loss. The proﬁts from reuse and remanufacturing and sale could pay for the loss from materials recycling. We developed models for computing the minimum return volume for successful EDL implementation [Klausner, Grimm, Hendrickson, and Horvath 1998]. Since the EDL increases manufacturing cost, a certain number of products must be returned so that the savings in product take-back outweigh the higher manufacturing costs. The return volume required is much higher than the return volume currently observed. This turns us to the problem of controlling product-return ﬂow. A New Reverse-Logistics System for Power Tool Takeback A power-tool-remanufacturing facility would require a continuous ﬂow of endof-life power tools. Currently, few customers return end-of-life tools to dealers (only 20,000 tools were returned in 1996). Actively advertising product take-back

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Remanufacturing is particularly appealing for power tools.

PRODUCT TAKE-BACK would probably increase the return rate; however, it might cause the return of obsolete tools not suitable for remanufacturing. Advertising would not offer control of the composition of the product-return ﬂow. A buy-back program (for example, by awarding discounts on new power tools) or more convenient reverse-logistics systems (such as pickup by parcel service) might work better. Take-back should be limited, if possible, to products not exceeding a certain age. Alternatively, buyback could be limited to product models that could be remarketed as remanufactured because of their technology, design, and functionality. Even then, some returns might be in such poor condition that they could not be remanufactured. We analyzed how much a manufacturer should spend on reverse logistics and buy-back to combine proﬁtable remanufacturing with unproﬁtable recycling. The total proﬁt of product take-back is the sum of the proﬁt from remanufacturing plus the (negative) proﬁt from materials recycling minus the costs of reverse logistics (including the buy-back expenditures). The return volume increases as the expenditures for reverse logistics increase, because more sophisticated return systems can be employed, and more can be spent on buy-back. The yield in remanufacturing will increase as the expenditures for reverse logistics increase (the appendix contains the reasoning behind this assumption). With increasing throughput in recycling, the unit recycling cost will decrease. All these effects produce an overall take-back proﬁt (Figure 1). (The appendix contains the equations underlying the model.) Initially, the proﬁt from take-back

increases with increased spending on reverse logistics. The effect of a higher yield and more remanufacturing proﬁt outweighs recycling costs and buy-back spending. At a certain point, the latter two factors outweigh the former, and the unit proﬁt decreases. This point is reached, for example, when the reverse-logistics spending reaches DM 17.72 ($9.85) if the unit proﬁt of remanufacturing is DM 40 ($22) (Figure 1). A sensitivity analysis (appendix) shows that our model is fairly robust for small parameter deviations. Our model can serve as a guide as to which reverse-logistics systems can be set up and what budget can be allocated to buy-back. We assume that take-back can be limited to a certain product age to

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Figure 1: Three levels of proﬁt might be obtained from remanufactured-power-tool sales. With higher reverse logistics and buy-back costs, the proﬁt level can vary from positive to negative. We determined the proﬁt of takeback per product sold as a function of reverse-logistic cost for distinct unit proﬁts of remanufacturing (DM 50, DM 40, and DM 30). The optimal reverse-logistics cost would be DM 17.72 ($9.85) for a unit proﬁt of remanufacturing of DM 40 ($22.00). If the unit cost of reverse transport was DM 4.29 ($2.39) as in the take-back system at present, an amount of DM 13.43 ($7.47) could be spent on buy-back. The corresponding return rate would be 6.2 percent, and the yield would be 66 percent. The proﬁt of product take-back per product sold would amount to DM 0.48 ($0.27).

KLAUSNER, HENDRICKSON avoid the return of completely outdated products. Companies engaged in product take-back can seek maximum proﬁts with speciﬁc buy-back strategies. From an environmental viewpoint, it is probably advantageous to encourage widespread participation in the remanufacturing process to ensure effective use of resources. Conclusions A higher return rate than currently observed in Germany would be desirable to manage a market for remanufactured power tools. A higher return rate is necessary to justify the cost of equipping all manufactured products with an EDL to measure use characteristics. With proﬁts generated from remanufacturing, the manufacturer could afford to offer buy-back plans featuring discounts on new tools, more convenient collection, or even cash payments. We determined the optimal levels for such schemes (Figure 1) for particular parameter values (Table 1). A sensitivValue Parameter assigned rmin

0.4%

rmax CRLO

11% DM 3.00

CRLmax

DM 30.00

yRMmin yRMmax pREC,r min pREC,r max

50% 80% DM ⳮ4.12 DM ⳮ2.06

ity analysis suggests that the proﬁtability is relatively insensitive to our assumed parameter values. By reusing power-tool components, ﬁrms can diminish their overall resource demand and improve industrial practice. It is by a large number of such actions that we can move towards a future of sustainable development. APPENDIX Model for Reverse-Logistics Cost Combining Remanufacturing and Recycling The proﬁt resulting from remanufacturing returned products in a time period (for example, a year) can be quantiﬁed as PRM ⳱ xryRM (pRM ⳮ cRM) ⳱ xryRMpRM

(1)

where x is the number of products sold, r is the return rate (the percentage of products sold that is taken back), yRM is the yield (the fraction of the return ﬂow that is remanufactured; 0 ⱕ yRM ⱕ 1), pRM is the revenue per remanufactured product, and cRM is the unit cost of remanufacturing.

Description of parameter Minimal return rate associated with the least expensive reverse-logistics system without buy-back Maximal achievable return rate Unit reverse-logistics cost associated with the least expensive reverse logistics system Unit reverse-logistics cost at the maximal return rate; this cost includes the average buy-back price paid Minimum yield in remanufacturing Maximal achievable yield in remanufacturing (Negative) unit proﬁt of materials recycling at the minimum return rate (Negative) unit proﬁt of materials recycling at the maximum return rate

Table 1: In our model we assume that the minimal return rate is 0.4 percent, while saturation occurs at a return rate of 11 percent. The corresponding reverse-logistics expenditures are DM 3 ($1.67) and DM 30 ($16.68), respectively. We assume further that the minimal yield in remanufacturing is 50 percent and that the yield cannot exceed 80 percent. Due to scale economies, we assume that material-recycling costs can be reduced from DM 4.12 at the minimal return rate to DM 2.06 at the maximal return rate.

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PRODUCT TAKE-BACK The return rate r refers to the products taken back as percentage of products sold in the same period considered, thereby neglecting the time lag between sale and takeback. We suggest this deﬁnition if the annual changes in sales are small, as is the case for power tools. Alternatively, the return rate could be related to the sales date to account for the time lag between sale and return. In this case, one would use a dynamic model (for example, a difference equation). The advantages of expressing the return rate as percentage of the current sales volume are twofold: First, it directly shows the signiﬁcance of product takeback for the current business processes. Second, while the number of products taken back in the period considered can be easily obtained, data on the detailed composition of the return ﬂow, such as the date of sale for each individual product returned, are hardly available. Yet, relating the return rate to the date of sale is a better indicator for the percentage of products not returned through the take-back program. Because the deﬁnition of the return rate depends upon the objectives associated with a certain model, we choose to express the return rate as percentage of the current sales volume to express the importance of product take-back to the current business processes. We assume the unit proﬁt of remanufacturing, that is, the proﬁt per remanufactured product, pRM ⳮ cRM ⳱ pRM ⬎ 0,

(2)

to be positive. Equation (1) is based on the assumption of linearity, that is, we assume that the unit proﬁt can be linearly scaled by the amount of products. We assume that the cost of remanufacturing is primarily determined by process technology, while the effect of scale economies is only secondary. The total cost of reverse logistics, including the buy-back budget, C, is

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C ⳱ xrcRL ,

(3)

where cRL is the average cost of reverse logistics per returned product. This cost consists of both the cost of collection and transportation and the buy-back price. The assumption of a ﬁxed unit cost per product returned holds for both single-product pickup and the collection of multiple products at the same time. Contracting a parcel service for single-product pickup is associated with a negotiated ﬁxed-unit cost. As the current take-back program shows, contracted logistics services providers also charge a ﬁxed cost per unit returned in case of pickup of a box of endof-life products. One would expect that one unit available per pickup is much more expensive than 500 units available per pickup. Surprisingly, we observed from studying other take-back programs that the charged unit cost is relatively insensitive to the number of products returned if reverse logistics is outsourced, therefore justifying equation (3). For example, the unit cost for single-product pickup of power tools by parcel service would be on the order of $2.70 for a return volume of 500,000 units per year, while a return volume of only 1,000 units would be associated with a unit cost of $4.10. This is due to the fact that a logistics-services provider optimizes truck loads and pickup routes. In contrast, equation (3) would not hold for a dedicated reverse-logistics system operated by the manufacturer. The fraction of products recycled is (1 ⳮ yRM). Let PREC denote the total (negative) proﬁt resulting from recycling, which can be calculated similar to equation (1) as follows: PREC ⳱ xr(1 ⳮ yRM)(pREC ⳮ cREC) ⳱ xr(1 ⳮ yRM)pREC,

(4)

where pREC represents the average revenues resulting from selling the recycled materials, and cREC is the unit cost of mate-

161

KLAUSNER, HENDRICKSON rials recycling (that is, the cost of recycling per product recycled). Using equations (1), (3), and (4), the overall net proﬁt of product take-back per product sold, p, is the sum of the proﬁts of remanufacturing and materials recycling minus the cost of reverse logistics scaled to the number of products sold, that is, p⳱

PRM ⳮ C Ⳮ PREC ⳱ ryRMpRM x ⳮ rcRL Ⳮ r(1 ⳮ yRM)pREC.

more than cRLmax would not result in a higher return rate. Even though it could be argued that the return rate increases nonlinearily as the buy-back price increases, we assume a linear relationship between r and cRL for reasons of simplicity and lack of empirical data on return-rateresponse functions: r⳱

rmax ⳮ rmin (c ⳮ cRL0) Ⳮ rmin , CRLmax ⳮ CRL0 RL cRL0 ⱕ cRL ⱕ cRLmax.

(5)

(6)

Equation (5) can be used to calculate the optimum of reverse logistics and buy-back expenditures cRL, assuming that both return rate r and yield yRM depend on cRL. Ideally, empirical data would be used to ﬁt functions describing the relationship between r, yRM, and cRL using a regression analysis. Unfortunately, we have data only on the current take-back program. Thus, we will assume functional relationships and ﬁt those functions with the data that are available. Relationship Between Return Rate and Reverse-Logistics Cost As the cost of reverse logistics, cRL, increases, the return rate r is likely to increase since a higher amount can be spent on buy-back and more sophisticated reverse-logistics systems (for example, pickup by parcel service) can be employed, thereby providing stronger incentives for customers to return end-of-life products through the take-back program. We assume that there is a lower cost limit cRL0, which reﬂects the cost of the leastexpensive reverse-logistics system without buy-back. At this cost, a return rate rmin can be observed. We also assume an upper limit of the return rate, rmax, below 100 percent, representing saturation. The maximum return rate that can be expected reﬂects the existence of alternative disposal options. The maximal return rate corresponds to the maximum cost of reverse logistics, cRLmax. We assume that spending

While rmin and cRL0 can be derived from studying implemented take-back systems without buy-back, rmax and cRLmax may be obtained from experts’ guesses or experiments. Generally, rmin and cRL0 are associated with less uncertainty than the latter two parameters. We can analyze the impact of parameter deviations on the results obtained from the proﬁt model (5) using the sensitivity analysis presented later. Relationship Between Yield and ReverseLogistics Cost As the incentives to return used products increase, the yield in the remanufacturing process, yRM, is likely to increase based on the following considerations. In a situation without buy-back, we observe a yield yRMmin. Offering buy-back would result in a larger fraction of remanufacturable products returned if take-back is limited to a product age that implies the chance that the product is remanufacturable due to its technology, design, functionality, and so forth. (Products whose technology and functionality qualify them for remanufacturing, however, may not be remanufacturable because they are degraded, damaged, or worn-out.) As the buy-back price increases, the fraction of newer and less-degraded products returned is likely to increase since a high buy-back price would stimulate earlier replacement decisions. Thus, the yield would increase as the buy-back price increases. Similar arguments apply for offer-

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PRODUCT TAKE-BACK ing collection systems that are very convenient for the customer. Owing to a lack of empirical data on the change in yield resulting from a change in buy-back price, we assume a linear relationship similar to equation (6). Then, the yield could be described as yRM ⳱

yRMmax ⳮ yRMmin (cRL ⳮ cRL0) cRL ⳮ cRL0

Ⳮ yRMmin,

cRL0 ⱕ cRL ⱕ cRLmax.

(7)

Relationship Between Proﬁt per Product Recycled and Return Rate As the return ﬂow increases, the negative proﬁt resulting from materials recycling, pREC, is likely to decrease; in other words, the loss of materials recycling will be reduced as the return rate increases due to scale economies. Scale economies in recycling may result from semiautomated disassembly instead of manual disassembly and a reduced overhead cost due to a larger throughput. Instead of using an exponential function describing the increase of the proﬁt of recycling, we use a linear function for reasons of simplicity and lack of empirical data: pREC ⳱

pREC,r max ⳮ pREC,r min (r ⳮ rmin) rmax ⳮ rmin

Ⳮ pREC,r min, rmin ⱕ r ⱕ rmax.

(8)

While the parameter pREC,r max in equation (8) is the negative proﬁt per product recycled at the maximal return rate rmax, pREC,r min is the negative proﬁt per product recycled at the minimal return rate rmin. It is implied that pREC,r max ⬎ pREC,r min. Application of the Model to the Takeback Program Investigated We apply the model to the alternative power-tool take-back concept. We use cost ﬁgures made available to as in deutschmarks (DM). We use an exchange rate of DM 1 ⳱ $0.556 to convert them to dollar ﬁgures. The fraction of the manufacturer’s end-

May–June 2000

of-life power tools in the total return volume translates to a return rate of 0.4 percent. We consider this return rate to be the minimal return rate without any incentives, rmin. The corresponding reverselogistics costs amount to DM 4.29 ($2.39). For multiple product pickup, an amount of cRL0 ⳱ DM 3 ($1.67) would be the lowest unit cost negotiable. Suppose that some 300,000 remanufactured products could be sold annually. If the yield was 80 percent—which represents the maximal achievable yield yRMmax—some 375,000 end-of-life products would have to be returned through the take-back program. This ﬁgure corresponds to a maximal achievable return rate rmax of 11 percent. We estimate that the corresponding reverse-logistics cost amounts to DM 30 ($16.68). Then, equation (6) can be written as r⳱

53 1 (cRL ⳮ DM3) Ⳮ 0.004, 13500 DM

DM3 ⱕ cRL ⱕ DM30.

(9)

Approximately 50 percent of the current return ﬂow could be remanufactured. This ﬁgure represents the lower limit for the yield, yRMmin. It is estimated that no more than 80 percent of the return ﬂow can be remanufactured, thus yRMmax ⳱ 0.8. Using these ﬁgures, equation (7) can be written as yRM ⳱

1 1 (c ⳮ DM3) Ⳮ 0.05, 90 DM RL

DM3 ⱕ cRL ⱕ DM30.

(10)

The proﬁt per product recycled at the present minimal return rate, pREC,r min, is DM ⳮ4.12 ($ⳮ2.29). We estimate that this loss can be reduced by 50 percent at the maximal return rate. Using these ﬁgures in equation (8) leads to pREC ⳱ DM19.44(r ⳮ 0.004) ⳮ DM4.12, 0.004 ⱕ r ⱕ 0.11.

163

(11)

KLAUSNER, HENDRICKSON Table 1 summarizes the parameters we used in applying the model to the alternative power-tool-take-back concept. As an example, consider a unit proﬁt of remanufacturing of DM 40 ($22), which represents a realistic estimate for expensive power-tool models. Then, the optimal reverse-logistics cost would be DM 17.72 ($9.85) (Figure 1). If the unit cost of reverse logistics was DM 4.29 ($2.39) as in the take-back system at present, an amount of DM 13.43 ($7.47) could be spent on buy-back. The corresponding return rate would be 6.2 percent, and the yield would be 66 percent. While the proﬁt of product take-back per product sold would amount to DM 0.48 ($0.27), the proﬁt of product take-back per product returned, p/r, would be DM 7.81 ($4.34). Sensitivity of the Results To get insights on the sensitivity of the model’s results, we analyzed the impact of parameter deviations on the results above. Table 2 shows the absolute sensitivity and both the maximal and the relative deviation of the proﬁt function (5). While the

Parameter

Si

D␣i best case

rmax rmin yRMmax yRMmin pRM pREC,r min pREC,r max

4.26 3.55 1.45 1.21 0.041 0.015 0.011

0.02 0.001 0.02 0.01 DM 0.50 DM 0.05 DM 0.10 7

best-case scenario is based on small parameter deviations, thus implying a high degree of reliance on the parameter estimates, the worst-case scenario represents a very pessimistic view on the certainty of the parameter estimates. If, for example, the maximal return rate rmax was two percent higher or lower than the assumed value of 11 percent, the proﬁt of product take-back per product sold, p, would change by DM 0.09 or 17.75 percent (ﬁrst row in Table 2), where |Si㛳Drmax| ⳱ 4.26*DM 0.02 ⳱ DM 0.09. The proﬁt obtained from the proﬁt model (5) for the nominal case considered changes no more than DM 0.15 or 30 percent if we assure the best-case scenario for all parameters. Hence, in the best-case scenario, there are no chances that let the proﬁt become negative due to parameter deviations. In contrast, in the worst-case scenario, the proﬁt of take-back per product sold has a maximal deviation of DM 1.09 or 227 percent, thus implying the chance that the proﬁt may become nega-

D␣i worst case

|Si㛳D␣i| best case [DM]

|Si㛳D␣i| worst case [DM]

0.10 0.003 0.10 0.05 DM 10.00 DM 1.00 DM 2.00

8.520E-02 3.554E-03 2.897E-02 6.042E-03 2.050E-02 7.561E-04 1.133E-03 0.15

4.260E-01 1.066E-02 1.448E-01 6.042E-02 4.100E-01 1.512E-02 2.267E-02 1.09

兺 |Si㛳D␣i| ⳱ i⳱1

Dp/p0 ⳱

D␣i p0 best case

D␣i p0 worst case

1.775E-01 7.404E-03 6.035E-02 1.259E-02 4.271E-02 1.575E-03 2.361E-03

8.875E-01 2.221E-02 3.018E-01 1.259E-01 8.542E-01 3.150E-02 4.722E-02

0.30

2.27

Si

Si

Table 2: This table indicates the sensitivity of proﬁt from remanufacturing power tools with respect to the various system parameters. The proﬁt is still positive with even a worst-case scenario of parameters. The system parameters are listed in columns one and two. Absolute proﬁt sensitivity shows the change in proﬁt from the base proﬁt of DM 17.72 per item with a unit parameter change. The following columns show best- and worst-case parameter scenarios and their effects on proﬁt.

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PRODUCT TAKE-BACK tive. We concluded that the proﬁt model (5) is robust for small parameter deviations. Yet the parameters must be estimated with a certain accuracy to avoid a negative proﬁt. A sensitivity analysis provides valuable insights into the impact of parameter deviations on the results obtained from the model and hence should be conducted for the nominal cases of interest. More explicitly, we analyzed the sensitivity of the proﬁt model (5) to determine the impact of the uncertainty associated with the parameters on the proﬁt model (5). The proﬁt (5) depends upon the parameter vector ␣ ⳱ (rmax,rmin,yRMmax, yRMmin,pRM, pREC,

r max,pREC, r min),

thus p ⳱ p(cRL,␣).

(12)

Let ␣0 denote the nominal value of the parameter vector and let p(␣0) ⳱ p0 denote the nominal proﬁt function. The absolute sensitivity function is deﬁned as [Frank 1976] Sl(␣0): ⳱

p ␣i

冷

i ⳱ 1, . . . , 7

,

(13)

a⳱a0

for the seven parameters under consideration. The maximal deviation of (12) is deﬁned as [Frank 1976] 7

|Dp| ⳱

7

冷 兺 S D␣ 冷 ⱕ 兺 |S 㛳D␣ |, i⳱1

i

i

i⳱1

i

i

(14)

proﬁt of take-back per product sold, where ln p S¯ i: ⳱ ln ␣i

冷

7

D␣

i S¯ i , 兺 ␣0 i⳱1 i

(16)

is the relative (logarithmic) sensitivity function [Frank 1976]. Table 1 shows the absolute and relative deviations of the results obtained from the proﬁt model (5) for the nominal case c*RL|pRM⳱DM40 ⳱ DM 17.72. References Davis, G. 1998, “Is there a broad principle of EPR?” Proceedings of the International Seminar on Extended Producer Responsibility as a Policy Instrument, Lund, Sweden, pp. 29–33. Frank, P. M. 1976, Empﬁndlichkeitsanalyse dynamischer Systeme, Oldenburg, Mu¨nchen, Germany. Klausner, M. 1998, “Product takeback systems design,” PhD dissertation, Carnegie Mellon University. Klausner, M.; Grimm, W.; Hendrickson, C.; and Horvath, A. 1998, “Sensor-based data recording of use conditions for product takeback,” Proceedings 1998 IEEE International Symposium on Electronics and the Environment, IEEE, Piscataway, New Jersey, pp. 138–143. Lifset, R. J. 1993, “Take it back: Extended producer responsibility as a form of incentivebased environmental policy,” Journal of Resource Management and Technology, Vol. 21, No. 4, pp. 163–175. OECD 1996, “Extended producer responsibility in the OECD area,” working paper No. 66, Phase 1 Report, Paris, France.

(15)

indicates the relative deviation of the

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␣i0 , p0

i ⳱ 1, . . . , 7,

where the |D␣i| represent the absolute parameter deviations. While the maximal deviation (14) determines the dollar value of the maximal proﬁt changes, the relative deviation of the proﬁt model (12), Dp :⳱ p0

a⳱a0

⳱ Si

165