Uncertain Demands and Return Quantity

Product Disposition Decisions on Closed-Loop Supply Chains

10.4 Uncertain Demands and Return Quantity

(10.10) regards the capacity constraint for periodt.Finally, constraint (10.11) ensures that the firm cannot produce in excess of demand over the planning horizon, which is necessary because the objective function maximizes profit, which is based on production.

The formulation above can be easily modified to accommodate the case where de- mand for remanufactured products is uncertain. To that end, we redefine an outcome in any period to be the joint realization of the random grading process and random demand in that period. For example, if there are two possible grading realizations (good and bad), and three possible demand realizations (high, medium, and low), then each period has 2×3=6 probabilistic outcomes. The only change in the above formulation is thatD^tis replaced withD^(j,t)in (10.7). This simple modeling change, however, results in a much larger problem, because the number of decision variables increases exponentially in the number of scenarios.

Denizel et al. are able to solve problems of realistic size within reasonable compu- tation times. For a six-period planning horizon, three quality grades, and five different grading outcomes, the problem has 222,647 variables and 101,556 constraints, which can be solved in Cplex in about one minute. Through an extensive numerical study based on industry data, Denizel et al. find that a firm’s profit is heavily influenced by the shape of the remanufacturing cost curve as it relates to the quality of the core (convex increasing, linear, or concave increasing in the lower quality of the core), the profit margin of the salvaging disposition option, and the cost of grading. Other model parameters (such as shape of demand curve, backlogging and penalty cost) are not as critical. They conclude that firms should take actions toward reducing grading costs (through technology such as electronic data logs, which register usage, found in many products), and increasing the margin of the salvaging option (through use for spare parts, for example, as opposed to materials recycling).

The models presented so far assume a deterministic product returns stream, so that returns can be forecasted for the entire planning horizon with a reasonable degree of accuracy. Again, this assumption is reasonable if returns originate from leasing operations. Although both stochastic returns and demands can be incorporated into the stochastic programming formulation above, this option is not practical due to computational times, at least for a multi-period planning horizon. In cases where there are multiple sources of significant uncertainty: returns, demand for remanufac- tured products, and demand for the salvaging option (spare parts), another modeling approach is necessary, which we present in the next section.

simplifications—for example, to assume unlimited remanufacturing capacity, and to assume a single quality grade for product returns—so as to formulate the disposition decision problem as a Markov decision process.

We illustrate the model by Inderfurth et al. (2001), which is reasonably general.

There arenpossible disposition decisions for stochastic product returnsB^t, received at periodt.Again, product returns are not categorized into different quality grades.

Each disposition option has associated with it a demand stream. For example, dis- position option 1 could be remanufacturing (with associated demand stream being demand for remanufactured products), disposition option 2 could be dismantling (with associated demand stream being demand for a single spare part originated from dismantling), and so forth. Assume that the firm can also dispose (say, through material recycling) returns it does not want to assign to one of the diposition options, at a unit cost ofd.Formally, disposition optionj faces stochastic demandD_j^t at time t, has processing lead time ofλj periods, unit processing costcj, and unit shortage costπj.Unit holding cost for returns ish, and holding cost for option-jprocessed unit ish_j.The firm then decides on the disposition quantities zj t, and disposal quantity vt

at timet.Denoting byαthe one-period discount factor, the firm solves the following Markov decision process:

T C= min

v_t,z_1t,...,znt

E

⎡

⎣^T

t=1

α^t−1

⎡

⎣dvt+ n j=1

cjzj t+h

⎛

⎝ut−1−vt− n j=1

zj t+B^t

⎞

⎠

+ n j=1

h_j'

y_j,t−1+zj,t−λj −D^t_j(₊ +

n j=1

π_j'

D_j^t −y_j,t−1−zj,t−λj

(₊⎤

⎦

⎤

⎦.

(10.13) In (10.13), the expectation is taken with respect toB^t,D₁^t,. . .,D_n^t; ut andyit are, just like before, the ending inventory of returns and processed units for optionj at timet, respectively; they satisfy balancing equations similar to (10.2) and (10.3):

y_{j t} =y_j,t−1+z_j,t−λj −D_j^t, and ut =ut−1−vt−n

j=1zj t+B^t.Inderfurth et al.

(2001) show that a heuristic that sets a base stock policy for returns and base stock policies for each of the reuse options perform well. (A base stock policy for an option j maintains inventory position of processed units constant at z^∗_j, where inventory position is defined as inventory on-hand minus backorders plus in-processing, or in transit, inventory.) However, it is necessary to specify an allocation rule, in case returns are not sufficient to meet demand for all disposition options in a period. They show that if returns are allocated to different disposition options based on their mean demands, then the performance of their heuristic is reasonably good.

The formulation above is reasonably general, and suitable to several situations.

The modeling approach above is not appropriate, however, to a situation where profitabilities across different disposition differ significantly, because it allocates (potentially scarce) returns based on mean demands as opposed to expected marginal profits across disposition options. It is also not appropriate to a situation where the dismantling option produces more than one spare part (each with its own demand

stream), since a disposition option only has one probability distribution of demand associated with it in the formulation above. Ferguson et al. (2011) propose an ap- proach to address these situations. They provide a formulation and solution structure for a one-period problem with two disposition options (remanufacturing and disman- tling for spare parts), which we describe here. They also provide a solution structure for the multi-period problem in the special case where dismantling only results in one type of part. The problem with multiple parts originating from dismantling in a multi-period setting is significantly more complex, and has not yet been solved.

We here describe their one-period model. At the beginning of the period, the firm receivesBreturns, whereB is a random variable with continuous probability density function (pdf) denoted byfB(·), and continuous cumulative distribution function (cdf) denoted byFB(·).The firm then decides upon the number of units to be remanufactured zr, at a unit costcr, and the number of units to be dismantled zd, at a unit costcd.Demand for remanufactured products, sold at a unit pricePr, is a random variableDr with cdf denoted byFr.Each return dismantled results in a_i parts of typei.Demand for spare parts of typei is a random variableD_i with meanμ_i and cdfF_i.Returns that are not remanufactured or dismantled, given by B−zr−zd, are disposed of at a cost normalized to zero. Remanufactured units that are not sold are salvaged at a unit value ofs_r.Demand for partithat is not met is assessed a unit penalty costπ_i, which can be though of as a higher cost of obtaining the part through an alternative supplier. Salvage value for all excess parts is zero.

They assume the interesting case wheresr ≤ cr < Pr, i.e. the unit salvage value does not exceed the unit remanufacturing cost, which is smaller than the unit sales price for remanufactured products. Likewise, they assumecd <

iπiai, i.e. the unit dismantling cost is smaller than the sum of the penalty costs avoided by dismantling one return.

The firm maximizes its one-period expected profit II:

max zr+zd ≤B

zr, zd≥0

=E

*

(Pr−sr)min{Dr, zr} +(sr−cr)zr

−

i

πi(Di−min{Di,aizd})

+−cdzd. (10.14)

Considering thatE[min{D, z}]=z−z

0 F(u)du, whereF(·) is the cdf ofD, (14) becomes:

max zr+zd ≤B

zr, zd≥0

=(Pr−sr)

$ zr−

zr 0

Fr(u)du

%

+(sr−cr)zr

−

i

πiμi−πi

$ aizd −

a_iz_d 0

Fi(u)du

%

−cdzd. (10.15) The optimal solution of this problem is described in Lemma 1 below:

Π_d(z)

Π_r(B-z)

Expected marginal contribution 0 Allocated

quantity

z~r

z~d

B zd

ˆ

Fig. 10.3 Structure of the optimal disposition policy

Lemma 1 Denote by

_d(z)=

i

π_ia_i(1−F_i(aiz))−c_d, r(z)=(Pr−v)(1−Fr(z))+sr−cr.

Then the optimal solution to the disposition problem (10.15) can be written as

(z^∗_d, z^∗_r)=

⎧⎨

⎩

(min{B,z˜d}, 0) if_d(B)> _r(0), (0, min{B,z˜r}) ifd(0)< r(B), (min{˜zd,ˆzd}, min{˜zr,B− ˆzd}) else,

where ˜zd and z˜r solve

d(z) = 0 and

r(z) = 0, respectively, and zˆd solves

d(z)=

r(B−z).

Lemma 1 postulates that if there are enough returns, the firm can satisfy demand for dismantling and remanufacturing at the levels˜zdand˜zr, respectively, that set their respective marginal profits equal to zero. If there are not enough returns, however, the optimal dismantling quantity is set at a quantity that its marginal profit is equal to the marginal profit of remanufacturing.

Figure10.3depicts the optimal disposition policy, which is determined by the expected marginal contributions (and not by the respective mean demands, as in Inderfurth et al. (2001). The optimal decision is not a critical-level policy (where the quantity of returns allocated to the “cheaper” option equals the amount exceeding a certain threshold); this is because demand for both options are uncertain. In fact, if demand for parts is known and deterministic (and can be satisfied by dismantlingdd

returns), then the optimal disposition decision is a critical-level policy:

Corollary 1 Assume thatD_i =d_da_i w.p. 1, for alli, and letπ_d =

ia_iπ_i.Then the optimal remanufacturing and dismantling quantities satisfy:

(z^∗_d, z^∗_r)=

#(B−min{B,zˇr}, min{B,zˇr}) ifB≤dd+ ˇzr, (dd, min{˜zr,B−d_d}) else,

wherezˇr=F_r⁻¹(1−(cr−sr−πd+cd)/(Pr−sr))and˜zris as defined in Lemma 1.

When demand for spare parts is deterministic, the firm still determines the op- timal amounts of remanufacturing and dismantling by comparing their respective expected marginal profits. The marginal profit of dismantling is constant and equal toπ_d−c_d, for a quantity up tod_d.The quantityzˇr is the remanufacturing quantity for which the expected marginal profit equals the marginal dismantling profit. Note thatzˇr is independent ofB and is obtained as a Newsvendor solution. This is the amount of returns protected for the “high-margin customers”, i.e. for remanufactur- ing, similarly to the two-class revenue management problem. Returns in excess of ˇ

zrare available for dismantling, up to a maximumdd.Any remaining returns should again be remanufactured, as long as the expected marginal profit remains positive.

Ferguson et al. (2011) provide a multi-period extension of this problem in the special case where there is only part type that originates from dismantling. Modeling multiple parts with very different demands introduces additional complexities, due to the interdependence between the dismantling decision and the different inventories levels of the different parts and due to the increasing state space dimensionality in the multi-period Markov decision problem.