Note on ‘ Combining an Improved Multi-delivery Policy into a Single-producer Multi-retailer Integrated Inventory System with Scrap in Production ’

In a recent study, Chiu et al. (2014) employed a mathematical modeling and conventional optimization technique to determine the optimal production-shipment policy for a single-producer multi-retailer integrated inventory system with scrap and an improved product distribution policy. This study replaces their optimization process of using differential calculus with an algebraic derivation. Such a simplified approach enables practitioners, who may have insufficient knowledge of calculus, to manage with ease the real world supply-chain systems.


INTRODUCTION
The most common method for solving the optimal replenishment lot-size problem is the mathematical modeling approach along with differential calculus as an optimization procedure (Tersine, 1994;Hillier and Lieberman, 2001;Nahmias, 2009).Recently, Grubbström and Erdem (1999) introduced an algebraic derivation rather than calculus to determine the Economic Order Quantity (EOQ) model with backlogging over a decade ago.Their algebraic approach successfully found the optimal order quantity for the EOQ model without reference to the first-or second-order derivatives.
Similar methodologies have since been adopted to resolve different aspects of Economic Production Quantity (EPQ) models and various kinds of supply chain systems (Wu and Ouyang, 2003;Wee and Chung, 2006;Francis Leung, 2008;Lin et al., 2008;Chen et al., 2012).This study extends such an algebraic approach to a specific intra-supply chain system studied by Chiu et al. (2014) and demonstrates that the optimal replenishment lot-size and shipment policy along with a simplified formula for the total system cost can be derived without derivatives.

Problem description and formulation:
Reconsider the production-shipment problem for a single-producer multi-retailer integrated inventory system with scrap and an improved product distribution policy as studied by Chiu et al. (2014).In their production-shipment model, annual production rate for a product manufactured by a single production unit is P and manufacturing cost is C per item.An x portion of scrap items may be randomly produced at a rate d during the production process through a 100% quality screening.By not allowing shortages, it is assumed (P -d -λ) >0, where λ is the sum of the demands of all m retailers (i.e., the sum of λ i where i = 1, 2, …, m) and d can be expressed as d = Px.Cost related parameters used in the cost analysis include the following: unit disposal cost C S ; set-up cost per production cycle K; fixed delivery cost K 1i per shipment delivered to retailer location i; unit holding cost h for item retained by the production unit; unit holding cost h 2i for items retained by retailer i; and unit shipping cost C Ti for items shipped to retailer location i.To ease the readability, the following same notations are also adopted:  Under an improved n+1 product distribution policy, one initial delivery of finished products is distributed multiple retailers to meet demand during the production unit's uptime (Fig. 1).Upon the remaining production lot is produced and screened, n fixed quantity installments of the finished products are distributed retailers, at a fixed time interval.Figure 1 illustrates the expected reduction in the production unit's holding costs (in green shaded area) of the proposed model (in blue) in comparison with that of Chiu et al. (2013) model (in black).
Figure 2 depicts the expected reduction in retailers' inventory holding costs (in green shaded area) of the proposed model (in blue) in comparison with that of Chiu et al. (2013) model (in black).
Total production-inventory-delivery costs per replenishment cycle for the proposed model, TC (Q, n+1) consists of the setup cost, variable manufacturing cost, disposal cost, the fixed and variable shipping costs, inventory holding cost incurred in the production unit and stock holding cost incurred in the retailers' locations as follows (Chiu et al., 2014): Taking randomness of scrap rate into account by applying the expected values of x and with further derivations, E [TCU (Q, n+1)] becomes (Chiu et al., 2014): where E i denotes the following: Unlike conventional solution procedure which uses the differential calculus on the cost function E [TCU (Q, n+1)] to derive the optimal production-shipment operating policy (Chiu et al., 2014), this study proposes an alternative two-phase algebraic approach as follows.

METHODOLOGY
Two-phase algebraic approach: In the phase 1, we first derive the optimal number of distribution.It can be seen that Eq. ( 2) has two decision variables, namely Q and n.In Eq. ( 2) there are different forms of decision variables in its Right-Hand Side (RHS), namely Q, Q -1 , Qn -1 and nQ -1 .We first let ν 0 , ν 1 , ν 2 , ν 3 and ν 4 denote the following: Then, Eq. ( 2) can be rearranged as: ( ) ( ) ( ) Further rearrange Eq. ( 9) as: One notes that if the second and fourth terms in RHS of Eq. ( 11) equal zero, then E [TCU (Q, n+1)] can be minimized.That is: ( ) or, By substituting Eq. ( 5) to (8) in Eq. ( 13), one has the optimal number of distribution as: It is noted that Eq. ( 14) is identical to that obtained in Chiu et al. (2014).
In real-life production-shipment applications, the number of deliveries n can only be an integer.Let n + represent the smallest integer greater than or equal to n (i.e., the computational result of Eq. ( 15)) and n -be the largest integer less than or equal to n.Then, it is obviously that the optimal n* is either n + or n -.
In the phase 2, we then derive the optimal replenishment lot-size Q*.By rearranging Eq. ( 2) as a single variable Q cost function as follows: ( ) ( ) ( ) or, ( ) where, Equation ( 16) can be rearranged as: ( ) It can be seen that if the second term in RHS of Eq. ( 18) equals zero, then E [TCU (Q, n+1)] can be minimized.That is: Substituting Eq. ( 5) to (8) in Eq. ( 15) and then in Eq. ( 17), one obtains optimal replenishment lot-size Q* as: It is noted that both Eq. ( 14) and ( 20) are identical to that obtained in Chiu et al. (2014) where the conventional differential calculus was used.Moreover, from Eq. ( 18) one obtains a simplified form for the optimal long-run average cost function E [TCU (Q*, n*+1)] as:
Finally, applying Eq. ( 21) the long-run average cost for the system is obtained as $454,840.These results are identical to that obtained in Chiu et al. (2014).

CONCLUSION
This study presents an algebraic derivation to replace (Chiu et al., 2014) differential calculus method in their optimization procedure for solving a production-shipment problem for a single-producer multi-retailer integrated system with scrap and an improved product distribution policy.Such a straightforward simplified algebraic approach can help practitioners, who may have insufficient knowledge of calculus, to manage with ease the real world supplychain systems.

Fig. 1 :
Fig. 1: Expected reduction in the production unit's holding costs (in green shaded area) of the proposed model (in blue) in comparison with that of Chiu et al. (2013) model (in black)(Chiu et al., 2014)