Stochastic Programming Approach: Uncertain Returns Quality

Product Disposition Decisions on Closed-Loop Supply Chains

10.3 Stochastic Programming Approach: Uncertain Returns Quality

Now, consider the case where the total amount of cores is known in advance, however, their quality levels are uncertain. Thus, the number of cores of qualityiin each period is uncertain, after grading. Relating to our previous model, we now, essen- tially, assume that the grading outcomeri is a random variable. Outcomes should be thought of in terms of higher-level events (e.g., optimistic, average, pessimistic), to allow computational feasibility. Cores can be kept in stock to be graded in the future and graded cores can be kept in stock to be remanufactured in the future. The problem is to determine how many of the available cores to grade, how many of the

Fig. 10.2 Construction of scenarios (j, t)

1,0

1,1

2,1

1,2

2,2

3,2

4,2 p1

1 - p1

t = 0 t = 1 t = 2

75% good

45% good

75% good

45% good 75%

good

45%

good

graded cores to be remanufactured, and how many of them to be salvaged each period (say, dismantling for spare parts). This problem has been addressed by Denizel et al.

(2010), who has formulated it as a multi-stage stochastic program, and we describe their model here. To illustrate the additional complications of uncertain grading outcomes, suppose there are two quality grades (good and bad), and two possible grading outcomes. In outcome 1, which happens with probabilityp₁, the grading process results in 75% of good quality cores, whereas in outcome 2, the grading process results in 45% of good quality cores. We now need to defineJ(t) different scenarios for each time period t, where each scenario corresponds to a possible realization of grading outcomes in time t and previous periods. For example, period 2 has four possible scenarios (J(2)=4) pertaining to the percentage of good quality cores in periods 1 and 2 ((0.75, 0.75), (0.75, 0.45), (0.45, 0.75), (0.45, 0.45)). We denote scenario j in periodtas scenario (j,t).This is shown in Fig.10.2. For a given time period t, define (pred(j),t −1) as the scenario, in period t−1, that is the predecessor of scenario (j,t). For example, in Fig.10.2, pred(1,2)=pred(2,2)=(1,1).

First, the amount of cores to be gradedxis determined; based on that decision, the amounts of cores in each quality grade are revealed. These determine the scenarios in each period. Then for each scenario, the amounts of cores to remanufacture z, salvage and hold in inventory from each quality grade are decided upon. Additional notation is defined below.

(j,t) Scenarioj of time period t,j =1,. . .,J(t);J(0)=1 Parameters

g Unit grading cost

p^(j^,t) Probability of scenario (j,t)

r_i^(j^,t) Fraction of quality-i cores under scenario (j,t) Decision Variables

x_t^(j,t⁻¹⁾ Amount of cores graded at timetunder scenario (j,t−1) z^(j,t)_i Quantity of quality-i cores remanufactured under scenario (j,t)

v_i^(j,t) Quantity of quality-i cores salvaged at end of period under scenario (j,t) Auxiliary Variables

u^(j,t)_i Inventory of quality-i cores at end of period under scenario (j,t)

y^(j,t)+ Inventory of remanufactured products at end of period under scenario (j,t) y^(j,t)− Backlog of remanufactured products at end of period under scenario (j,t) b_t^(j,t⁻¹⁾ Amount of un-graded cores at the end of periodtunder scenario (j,t−1) Our problem of finding the appropriate disposition decision for all periods in the planning horizon, under each probabilistic scenario (assuming zero initial inventories) can thus be formulated as:

max=

T t=1

J(t) j=1

p^(j,t) _I

i=1

((P_r−ci)z_i^(j,t)+siv^(j,t)_i −hiu^(j_i ^,t))−hry^(j,t)+−π y^(j^,t)−

)

− T

t=1 J(t−1)

j=1

p^(j,t)(gx_t^(j^,t⁻¹⁾+hb^(j,t_t ⁻¹⁾) (10.6)

Subject to

y^(pred(j^),t⁻¹⁾⁺−y^(pred(j),t⁻¹⁾⁻−y^(j^,t)++y^(j^,t)−+ I

i=1

z_i^(j^,t) =D^t, ∀(j,t) (10.7) b_t^(j^,t⁻¹⁾+x_t^(j^,t⁻¹⁾−b^(pred(j_t₋₁ ^),t⁻²⁾=B^t,∀t,j =1,...,J(t−1) (10.8) z^(j_i ^,t)+u^(j,t)_i −u^(pred(j_i ^),t⁻¹⁾+v^(j_i ^,t)−r_i^(j^,t)x_t^(pred(j),t⁻¹⁾=0, ∀(j,t) (10.9)

I i=1

k_iz^(j_i ^,t)≤C^t, ∀(j,t) (10.10) y^(j,T⁾=0, j =1,. . .,J(T) (10.11) y^(j^,t)+,y^(j,t)−, z_i^(j^,t), v^(j_i ^,t), u^(j,t)_i ,x_t^(j^,t⁻¹⁾,b_t^(j^,t⁻¹⁾≥0, ∀i, (j,t). (10.12) The objective function (10.6) maximizes total profit. Profit is comprised of revenue from remanufactured products, and salvage revenue minus holding cost for graded cores, holding cost for remanufactured products, backlogging cost, grading cost, and holding cost for ungraded cores. The set of constraints (10.7) are inventory balancing constraints for remanufactured products: for a given scenario (j,t), one can meet demand from the current period’s production and starting inventory of remanufactured products, or demand can be backlogged. Note that all scenarios (j,t) with a common predecessor share the same starting inventoryy^y(pred(j^),t⁻¹⁾⁺. Constraints (10.8) and (10.9) display inventory balance constraints for un-graded and graded cores, respectively. Note that in (10.9), all scenarios (j,t) that share the same predecessor have the same amount of graded coresx_t^(pred(j^),t⁻¹⁾.Constraint

(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 demand 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 remanufactured products, and demand for the salvaging option (spare parts), another modeling approach is necessary, which we present in the next section.

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