Optimal Production Decision in the Closed-Loop Supply Chain Considering Risk-Management and Incentives for Recycling

Production companies in the real world face many decision making situations, such as logistics, scheduling, and data mining in the supply chain. The optimal production decision is also major and important when deciding the optimal rate of product allocation, taking into consideration customers' demands, resource costs, and many conditions surrounding the real market. Many researchers have considered optimal production decision models with the total cost-minimization or the total profit-miaximization arising from the production processes of firms, and recently many mathematical models have been proposed (for example Letmathe & Balakrishnan, 2005; Li & Tirupati, 1997; Morgan & Daniels, 2001; Mula et al., 2006a).


Introduction
Production companies in the real world face many decision making situations, such as logistics, scheduling, and data mining in the supply chain.The optimal production decision is also major and important when deciding the optimal rate of product allocation, taking into consideration customers' demands, resource costs, and many conditions surrounding the real market.Many researchers have considered optimal production decision models with the total cost-minimization or the total profit-miaximization arising from the production processes of firms, and recently many mathematical models have been proposed (for example Letmathe & Balakrishnan, 2005;Li & Tirupati, 1997;Morgan & Daniels, 2001;Mula et al., 2006a).
In production processes, there are many uncertain factors in terms of total volume and customer's demands, such as the occurrence probability of machine breakdown and human error based on historical data, and the ambiguity that derives from the quality of information received and decision makers' intuition.Some recent articles have elaborated on studies of production planning problems including such ambiguous situations (for instance, Mula et al., 2006bMula et al., , 2007;;Vasant, 2000).We (Hasuike & Ishii, 2009a, 2009b) considered optimal production decision models, under both randomness for the return of each product and fuzziness for coefficients of constraints in production processes.
Particularly, in a forward supply chain, the customer is typically the end of the process.However, a closed loop supply chain includes the returns processes and the manufacturer has the intent of capturing additional value and further integrating all supply chain activities.Therefore, closed-loop supply chains include traditional forward supply-chain activities and the additional activities of the reverse supply chain such as product acquisition and reverse logistics.Then, in terms of reverse logistics and closed loop supply chain management, it is important to improve the sustainability of production companies' business considering the reintegration of their returned products into their own production network.The increase of the reintegration rate can be achieved at multiple levels also called recovery paths : reuse, repair, remanufacturing, recycling, etc. (in detail, Lebreton, 2007).Therefore, an incentive, which is the leverage for reintegrating valuable products, is a key factor in the closed roop supply chain management.If the buyback incentive is set higher

Mathematical formulation and notation of parameters
In this section, we introduce the notation used in our proposed models integrating optimal production decision and CSR activities considering the closed-loop supply chain.Furthermore, as a risk measure, we introduce Conditional Value-at-Risk proposed by Rockafellar and Uryasev (2002).CVaR is known as a useful risk measure which is coherent, consistent with the second (or higher) order stochastic dominance, and the consistency with the stochastic dominance implies that minimizing the CVaR never conflicts with maximizing the expectation of any risk-averse utility function.
Through the whole paper, to simplify, we consider the following cases: 1.All products are produced in one manufacture and these products are delivered to p retailers.2. We assume that the leadtime of production is larger than that of consumption, and so the production company must predict the demand in the next period and the amount of production items previously.3. The shortage of each product at the retailers is forbidden as much as possible.

 
Demand and cost parameters , consumers are generally well-affected in the case that production companies actively perform CSR activities.Then, their purchasing powers are certainly increasing but ambiguity due to including consumer's subjectivity.Therefore, we assume that this relation as the following linear relation: where  t ij D is the random demands independent of CSR activities and   is assumed to be a fuzzy number characterized by the L-shape membership function where  is the center value and  is the spread.Therefore, demand  t ij d is presented as a hybrid variable with both randomness and fuzziness.

Identification of cost function
The objects of our proposed model are to maximize the total cost and recycling rate derived from customers' demands, the recycling volume, and the expenses of CSR activities.The total profit function is associated with total profit, transportation costs, production costs, and holding costs.Each profit or cost function is formulated as follows in the multi-item integrated model using the above-listed parameters: a. Profit: Substituting demand relation (1) and performing the equivalent transformation, the feasible solutions set  is equivalently transformed into the following set: Then, the main problem (2) is also equivalently transformed into the following problem:

Ed
of random variable.However, in reality, the wild swing of demands associated with drastic changes of social and economic conditions often happens.In this case, it is obviously impossible that each demand is generally not fixed, and it is rough to degenerate the random distribution into the expected value by neglecting important factors considering fluctuation ranges such as the variance.Therefore, we must c o n s i d e r m o r e a d v a n c e d r i s k m a n a g e m e n t a p p r o a c h e s t o a v o i d f u t i l e s u p p l y c h a i n disruptions, particularly, focusing on the downside risk to decrease uncertainty up to the high-cost as much as possible.

Analytical and efficient solution algorithm based on CVaR
First, we introduce the standard CVaR in the stochastic programming problem.Standard CVaR for randomness was proposed by Rockafellar and Uryasev (2002) as follows: where   , L   is the loss function with fixed value  and random variable  , and  is the confidence level.Then,   fy is the density function of random variable  .

 
VaR   is Value at Risk (VaR) and the standard risk measure in economic and financial fields and used in many practical risk management.Furthermore, we introduce the following function: Rockafellar and Uryasev (2002) proved that the CVaR (10) can be minimized by minimizing the auxiliary function (7).Since the loss function   , L   is convex due to the linearity for fixed demand  , we can obtain the optimal order quantity of our proposed model by solving the following CVaR minimization problem: where probability  is assumed to be a finite discrete random variable like assumptions in this paper, i.e., Furthermore, introducing parameters , ss   , problem (10) is equivalently transformed into the following problem:  This problem includes only linear constraints, but it is hard to find an unique optimal decision for problem (12) analytically due to fuzzy number   and multi-criteria.Therefore, in order to solve this problem analytically, we introduce fuzzy goals as the fuzzy programming approach.

Numerical example
In order to compare our proposed closed-loop supply chain model considering CSR with the previous standard models, we introduce the following simple numerical example.We assume that two periods (t=2), three products (n=3), and three retailers (p=3).To simplify, these products consists of one resource (k=1).Tables 1 and 2 show data of parameters with respect to returns, production cost, expenses for CSR, and demands in all retailers in each period which are assumed to be uniform distributions., respectively.
Then, we solve the proposed model ( 14) considering CSR for each function, and obtain the following optimal production volume and the optimal profit as Table 4.  4., in the case of concave function, whole production volume tends to be larger than the linear and convex-based proposed models.On the contrary, the total profit in the case of concave function is much smaller than linear and convex functions.This means that the cost derived from the CSR activity is not used as promotion activity to increase the customers' demands, but used as the recycling activity to collect old-products.
Fig. 1.L-fuzzy number   2.2 Identification of cost function

cost for recycling item j in period t  t j w : inventory level at end of period t for item j at the manufacture. With respect to the relation between demand  t ij d and expenses of CSR activities  t CSR C
d : demand volume for item j at destination i in period t a : necessary resource volume for item j at kth resource constraint in period t j h : inventory holding cost per unit of item j at the manufacture, which is constant value to each period t u : transportation cost for item j from the manufacture to destination i in period t j v : recycling cost for item j, which is constant value to each period t  C Rt : expenses for CSR activities in period t , solve this problem with fixed value h analytically and efficiently using linear programming approaches such as the Simplex method and Interior point method.Furthermore, using the bisection algorithm on h , we obtain the strict optimal production volume of the main problem (10).Consequently, we have proposed the versatile model for www.intechopen.comOptimal Production Decision in the Closed-Loop Supply Chain Considering Risk-Management and Incentives for Recycling 221 the closed-loop supply chain management considering the CSR activities, and developed the analytical solution algorithm based on the standard linear programming approaches.

Table 3 .
Transportation, holding, andrecycling costs are assumed to be same in two periods.Furthermore, we assume that   is a Optimal solution of production volume From this result, we find that the whole production volume tends to be larger than the standard model due to consideration of CSR.Furthermore, volumes of all products in period 2 of proposed model are much larger than standard model.Results of CSR activities in our proposed model  1 16.4,respectively.This means that the CSR activity for promotion and recycling is mainly performed in the second period.Furthermore, we consider some cases of    

Table 4 .
Optimal solution of production volume to each function    