Ben-Daya et al. (2010) established a joint economic lot-sizing problem (JELP) for a three-layer supply chain with one supplier, one manufacturer, and multiple retailers, and then proposed a heuristic algorithm to obtain the integral values of four discrete variables in the JELP. In this paper, we first complement some shortcomings in Ben-Daya et al. (2010), and then propose a simpler improved alternative algorithm to obtain the four integral decision variables. The proposed algorithm provides not only less CPU time but also less total cost to operate than the algorithm by Ben-Daya et al. (2010). Furthermore, our proposed algorithm can solve certain problems, which cannot be solved by theirs. Finally, the solution obtained by the proposed algorithm is indeed a global optimal solution in each of all instances tested.
To bear a better resemblance to practice, Ben-Daya et al. (2010) considered a joint economic lot-sizing problem (JELP) in a three-layer supply chain with one supplier, one manufacturer, and multiple retailers as follows: The retailers have a common basic cycle time T . The manufacturer has the cycle time Tm=K2TTm=K2T while the supplier has the cycle time Ts=K1Tm=K1(K2T)Ts=K1Tm=K1(K2T). The supplier receives m1m1 equal shipments of raw materials during its cycle time TsTs, transforms them into semi-finished products, and delivers m2m2 equal-sized batches to the manufacturer during the manufacturer's cycle time TmTm. The manufacturer, in turn, transforms those semi-finished products into finished products and ships finished products to each retailer at its order quantity every T units of time. However, the order quantity received by a retailer might be different from those received by the others. Then they established the chain-wide annual total cost (a.k.a., the total cost) as a function of View the MathML sourceK1,K2,T,m1,andm2 using the sum of the costs incurred by the supplier, the manufacturer, and the retailers. They minimized the chain-wide annual total cost in which four variables (i.e., View the MathML sourceK1,K2,m1,andm2) are discrete positive integers, and the other TT is a real number. Notice that their JELP is a nonlinear integer programming (NLIP) model, and thus is hard to find an optimal solution using an exact method. Furthermore, the JELP is complex and computationally intensive even using mathematical software such as LINGO to solve it. By relaxing all integral variables as continuous variables, Ben-Daya et al. (2010) derived a near optimal solution to the problem using an algebraic method of completing perfect square without classical differential calculus techniques. In general, most studies use the classical differential calculus method to obtain the optimal values of the continuous decision variables. However, an algebraic method of perfect squares has been used in optimization problems in the inventory field recently. Examples are Grubbström (1995), Grubbström and Erdem (1999), Cárdenas-Barrón, 2001, Cárdenas-Barrón, 2007 and Cárdenas-Barrón, 2008, and Sphicas (2006), just to name a few. For an up-to-date review on different optimization approaches in inventory lot-sizing problems, see Cárdenas-Barrón (2011).
Ben-Daya et al. (2010) solved the relaxed JELP (i.e., relaxing all discrete integral variables in JELP as continuous real-number variables) by an algebraic method of completing the square, then proposed an algorithm to find the integral values for those four discrete integral variables. However, their proposed integral procedure seems to be computationally expensive. In fact, their algorithm requires to compute the integral variables (View the MathML sourceK1,K2), the continuous variable (TT), and the total cost function for several times. For simplicity, we set ⌈w⌉⌈w⌉ as the smallest integer which is greater than or equal to w . Then their algorithm requires evaluating the values of View the MathML sourceK1,K2,T, and the total cost TC for 4⌈m1⌉⌈m2⌉4⌈m1⌉⌈m2⌉ times, if both View the MathML source⌈m1⌉ and View the MathML source⌈m2⌉ are greater than one. If any of View the MathML source⌈m1⌉ or View the MathML source⌈m2⌉ is equal to 1, then the number of evaluations is less than or equal to View the MathML source4⌈m1⌉⌈m2⌉. For example, if ⌈m1⌉⌈m1⌉=11 and ⌈m2⌉⌈m2⌉=13, then their algorithm requires to compute each of View the MathML sourceK1,K2,T, and the total cost TC for 572 times.
In this paper, we first complement mathematical errors in Ben-Daya et al. (2010) on the optimal basic cycle time T ⁎=View the MathML sourceW/Y and the minimum value for the annual total cost TC =View the MathML source2WY. If α2α2 in Ben-Daya et al. (2010) is negative then both View the MathML sourceK2=α2ϕ2/(ψ2∑Or) in (29) and Y=(K2ϕ2+α2)/2Y=(K2ϕ2+α2)/2 are not real numbers. Consequently, neither optimal basic cycle time T ⁎=View the MathML sourceW/Y nor the annual total cost TC =View the MathML source2WY is a real number. This contradicts to the facts that both T ⁎ and TC are real numbers. Hence, for correctness and completeness, we need to discuss the case in which α2α2< 0. For simplicity, we discuss and illustrate this case using a numerical example as Instance 14 in Section 3 later. We then rearrange the total cost in (27) in Ben-Daya et al. (2010), and then propose a simple integral procedure similar to that by García-Laguna et al. (2010) to obtain the integral values for those four discrete variables m1m1, m2m2,K1K1, and K2K2. In addition, the proposed integral procedure discriminates the situations in which there is only one solution and when there are two solutions for each discrete variable. Furthermore, we not only obtain the integral values for all discrete variables in simple-to-apply closed-form expressions, but also need to compute the value of the continuous variable (TT) only once, instead of View the MathML source4⌈m1⌉⌈m2⌉ times using the algorithm in Ben-Daya et al. (2010).
We have complemented some shortcomings in Ben-Daya et al. (2010). For example, if α2α2 is negative, then neither the optimal basic cycle time T⁎ nor the annual total cost TC in their algorithm is a real number. We have proposed a simple-to-apply alternative algorithm to obtain 4 discrete integral decision variables by an explicitly closed-form solution. In addition, the proposed algorithm not only needs less CPU time but also less total cost than the algorithm by Ben-Daya et al. (2010). Furthermore, our proposed algorithm has solved some problems, which cannot be solved by the algorithm in Ben-Daya et al. (2010). Finally, it is worth mentioning that the proposed algorithm is remarkably good because it has obtained the global optimal solution for all the 25 instances and its initial solution has been the global optimal solution in 11 out of 25 instances.
