A computational analysis of lower bounds for the economic lot sizing problem in remanufacturing with separate setups

This paper considers economic lot sizing problem in remanufacturing with separate setup (ELSRs), where remanufactured and new products are produced on dedicated production lines. Since this problem is NP-hard in general, which leads to computationally inefficient and low-quality of solutions, we present (a) a multicommodity formulation and (b) a strengthened formulation based on a priori addition of valid inequalities in the space of original variables, which are then compared with the Wagner-Whitin based formulation available in the literature. Computational experiments on a large number of test data sets are performed to evaluate the different approaches. The numerical results show that our strengthened formulation outperforms all the other tested approaches in terms of linear relaxation bounds. Finally, we conclude with future research directions.


Introduction
In this paper, we investigate uncapacitated economic lot sizing problem in remanufacturing with separate setups (ELSRs) that arise in production planning.The problem is to find an effective production plan that meets demand for remanufactured and new products on time as minimizes total setup, production and inventory holding costs.Richter and Sombrutzki [1] and Richter and Weber [2] studied the reverse version of classical Wagner and Whitin algorithm in the case of a large quantity of low inventory cost of used products and solved them in a polynomial time.However, most studies have proven that ELSRs problem is NP-hard problem in general ( [3] and [4]).This NP-hard problem leads to computational inefficient, that is highly computation times, large duality gaps and low-quality of solutions.Therefore, developing and improving solution procedures for ELSR problem is vital.
To the best of our knowledge, Retel Helmrich et al. [4] were the first to present a good mixed integer programming formulation for separate setups case and proposed several alternative formulations such as a shortest path formulation, a partial shortest path formulation and an adaptation of the (l, S, W W ) inequalities for the classic problem with Wagner-Whitin costs to tighten the original formulation.They found that a (partial) shortest path formulation outperforms the original formulation in terms of LP gaps, MIP computation times and number of optimal solutions.Currently, Cunha and Melo [5] studied the same problems proposed a multicommodity reformulation and some new valid inequalities.The results showed that their In this paper, we aim to further improve the lower bounds by introducing several families of valid inequalities and also a multicommodity reformulation, derived differently from Cunha and Melo [5].The remainder of this paper is structured as follows: Section 2 discusses the original problem formulation.A new multicommodity reformulation is introduced in Section 3 and a Wagner-Whitin based formulation of Cunha and Melo [5] with new valid inequalities namely (l, S) − like inequalities are proposed in Section 4. Section 5 presents computational results that show average improvements of existing and proposed valid inequalities.Lastly, Section 6 concludes this paper with suggestions and future directions.

Problem formulation
We refer the original formulation of ELSRs problem from Cunha and Melo [5].Let N T be number of periods.Both returns and demands are assumed to be deterministic over a finite planning horizon.The demands can be fulfilled by either newly produced products or remanufactured products such that total costs are minimized.Then, let x r t and x m t represent the number of remanufactured and new products produced in period t, respectively; y r t and y m t , indicate setup forcing constraint of remanufacturing and manufacturing processes in period t, respectively (1 if there is a production incur, 0 otherwise).
Several parameters are considered such as p r t and p m t are unit production cost of remanufacturing and manufacturing in period t, respectively; h r t and h s t are unit holding cost for inventory of product returns and serviceables in period t, respectively, K r t and K m t are the unit separate setup cost for remanufacturing and manufacturing in period t, respectively with positive production x r t , x m t ≥ 0. Lastly, the total demand for remanufactured and new products and the total returns for product returns between t and t is D The objective (1) is to minimize the total of setup costs, production costs for remanufacturing and manufacturing processes; and holding costs for product returns and serviceable products.Constraints (2) and (3) represent flow conversation (inventory balance) for product returns and serviceable products, respectively.Constraints (4) and ( 5) are setup forcing constraint for remanufacturing and manufacturing, respectively.Next, (6) provide the integrality of remanufacturing and manufacturing.Then, (7) denotes nonnegativity requirements of new and remanufactured products, and inventory variables of product returns and serviceable products.Lastly, without loss of generality, we assume no initial inventory for product returns and inventory of serviceable products on hand as stated in constraint (8).From this, we obtain the feasible region of the basic formulation for ELSRs as X ss = {(x r , x m , y r , y m , I r , I s )|( 2

A multicommodity reformulation
The original formulation of ELSRs problem can be reformulated as a multicommodity (MC) reformulation.This alternative approach, defined by Rardin and Wolsey [6] had been found to be effective in tightening the formulation of fixed-charge network flow problems.Cunha and Melo [5] was the first ones to introduce multicommodity reformulation technique for ELSRs problem.However, we develop an alternative way of developing MC formulation.We decompose the production flow for both remanufacturing, x r t and manufacturing, x m t as functions of their destination nodes (return and demand periods) at t, t + 1, .., n..The inventory flow for both product returns, I r t and serviceable products, I s t are also decomposed at t+1, t+2, .., n.Unlike a classical lot sizing problem, we consider two types of commodities which are: Commodity, A t represents the return delivered onto the system in period t, where t ≤ t', and Commodity B t corresponds the demand delivered onto the system in period t , where t ≤ t'.Now, we define new decision variables as follows.u sr t,t is the amount of remanufactured products in period t of commodity B t , u sm t,t is the amount of new products in period t of commodity B t , v rp t,t is the inventory of product returns at the end of period t of commodity A t , v sp t,t is the inventory of serviceable products at the end of period t of commodity B t .and u rr t,t is the amount of remanufactured products in period t produced from commodity A t .Note that both inventory variables, v rp t,t = 0 and v sp t,t = 0 for all t = 1, .., n do not exist as the commodity cannot be both returned to the system or delivered and hold in stock in period t, respectively.Also, the inventory stock of both product returns and serviceable products at the end of the planning horizon are assumed to be zero.Then, we add the following constraints into the original formulation.
u sr , u sm , u rr , v rp , v sp ≥ 0 (18) Constraints ( 9) and (10) represent the relationship between old and new variables.Constraints (11) -(13) are setup forcing constraints.Constraints ( 14) and (15) are inventory flow balance for serviceable products and constraint (16) is for inventory flow balance for product returns.Constraint (17) links the variables between u rr t,t and u sr t,t .Lastly, (18) provides the nonnegativity constraints.Then, the feasible region and objective function of this new formulation are X ss M C = {(x r , x m , y r , y m , I r , I s , u rr , u sr , u sm , v rp , v sp )|( 2) − (8), ( 9) − (18)} and Z ss M C = min {(1)|(x r , x m , y r , y m , I r , I s , u rr , u sr , u sm , v rp , v sp ) ∈ X ss M C } , respectively.

Valid inequalities
Adding valid inequalities a priori to the original formulation provides a tightened formulation that improves the lower bounds and computation times.This section discusses existing and new valid inequalities for ELSRs.

Existing Valid Inequalities
The first family of valid inequalities is derived by Retel and Helmrich [4], the second one is the improvised version of Retel and Helmrich [4] and the last family of valid inequality is the new valid inequalities, both derived by Cunha and Melo [5].The first family of valid inequalities is related to the returns and the last two families of valid inequalities are associated to the demands.
where, d t,l = l t =t d t .d represents the minimum demands that must be satisfied by new products given that limited amount of returned products.It can be determined as d t = max(0, d t −rr t ).rr t denotes the residual amount of returned products in period t given that the maximum amount of demand can be fulfilled from returned products.It can be calculated as:

New Valid Inequalities
Now, we propose four families of valid inequalities namely ( , S) − like inequalities for ELSRs problem, initially introduced by Barany et al. [7] for single-item uncapacitated problem.
Proposition 1.For any 1 ≤ k ≤ ≤ n, suppose that L = {k, .., } and S ⊆ L, then the following inequalities are valid for X ss : Proof.Consider a point (x r , x m , y r , y m , I r , I s )∈ X ss .If i∈S y r i = 0, then x r i = 0, ∀i ∈ S and I r k−1 ≥ 0, hence the inequality is satisfied.Let i∈S y r i ≥ 1 and p = max {i ∈ S} .Then The first inequality follows the definition S and the nonnegativity of x r i , second inequality shows the constraint of flow conversation for product returns and lastly using y r p = 1 and the nonnegativity of y r i .As regards the remaining inequalities, the proofs are straightforward as in Pochet and Wolsey [8] and therefore omitted here for the sake of brevity.
Lastly, we define a new feasible region of ELSRs problem associated with these existing and new families of valid inequalities as X ss N = {(x r , x m , y r , y m , I r , I s )|( 2) − ( 8), ( 21) − (25)} and the objective function is Z ss N = min {(1)|(x r , x m , y r , y m , I r , I s ) ∈ X ss N }.Note that only new valid inequalities ( 21) is included into our problem formulation as the remaining two valid inequalities are similar to our valid inequalities.As these formulations contain an exponential number of valid inequalities and therefore we use them in a cutting plane approach by solving simple polynomial separation algorithm as in Barany et al. [7] and Akartunalı [9].

Computational testing of lower bounds
This section presents computational analysis of lower bounds for ELSRs problem, where the strength of different lower bounding techniques, the multicommodity reformulation, the existing Wagner-Whitin based formulation and our strengthened formulation are tested using a great extent of data sets available from the literature.We run 360 test instances obtained from [4] on a PC with Intel (R) Core(TM) i7-4500U CPU 2.40 GHz processor and 8 GB RAM.All problems are solved by FICO (R) Xpress Optimization Suite in the Mosel modelling language version 7.9 without any solver cuts.The default time is set to 600 seconds for each test instance.
The planning horizons are 25, 50 and 75 periods.The demands are drawn randomly from a normal distribution with mean, µ = 100, and standard deviation, σ = 50.We also assume the returns parameter is normally distributed with three different parameter settings: low return(µ = 10, σ = 5), medium return (µ = 50, σ = 25), and high return (µ = 90, σ = 45).This gives us nine possible parameter combination settings, where each is replicated 10 times, resulting in 90 different data sets.We assume that the demands and the returns values are nonnegative and the cost parameters are time-invariant.The setup costs for both remanufacturing and manufacturing take the values of 125, 250, 500 and 1000.Then, the holding costs for both product returns and serviceable products are equal to 1 for all test instances.Lastly, the production costs for both remanufacturing and manufacturing are assumed to be zero.Now, we present the pairwise comparisons of lower bounds in terms of average improvement (in percentage), summarized in Table 1.The first column represents return variability, namely low, medium and high returns.This is followed by the number of periods, n, and the variation of setup costs.The "N vs CM" indicates that the average improvement of lower bounds from Cunha and Melo bound to our New bound.For each test, we calculate the average improvement of lower bounds (%) regardless of the solution optimality as AI (%)= N bound−M C bound N bound × 100.Firstly, we discuss the average improvement (AI) from the CM bound to our new valid inequalities, N bound.Given the large number of time periods (e.g.50 and 75), the AI from the CM bound to the N bound deteriorates when the return variability is increased.Note that the problem mostly cannot find an optimal solution for these larger data sets within allocated times, 600 seconds.Additionally, the larger the returns the lower the AI.This is because, the family of valid inequalities involving returns introduced by Cunha and Melo [5] and also considered in our study become more effective than others in improving the lower bounds since remanufacturing operation dominates the total production.As a result, CM bound improves slightly by our N bound, which is only about 4% maximum on average.regards to a low return scenario, the N bound improves the CM bound significantly, up to 11% on average.One of the reasons we obtain a large AI is because our new valid inequalities find optimal solutions in all data tested.Besides, the second inequality (20) involving demands of Cunha and Melo [5] does not include production during the first period, causing demand in this period to not be satisfied.In addition, with low returns, manufacturing will dominate production over remanufacturing.This causes a valid inequality involving returns to become less effective.Lastly, we examine the average improvement from the N bound to MC bound.Unlike previous literatures, the findings show promising results that MC bound in general improves by the N bound, up to approximately 2% for all periods.Specifically, in the case of low and high returns, the N bound shows less significant average improvement over the MC bound, which is less than 1% improvement as compared to medium returns.Obviously, there is no significant pattern of the results can be reported.However, we can say that the lower bound provided by MC is at least as strong as the our N bound.

Table 1 .
[5]n percentage improvement of lower bounds for ELSRs problem.N bound: our new valid inequalities bound, CM bound: valid inequalities bound by Cunha and Melo[5]6.ConclusionThis paper investigates economic lot sizing problem in remanufacturing with separate setup (ELSRs), which is NP-hard in general.We propose several families of valid inequalities to further improve the lower bounds obtained by previous research.The results indicate that our new valid inequalities outperforms all the other tested approaches in the literature in terms of linear relaxation bounds.For future research, it is interesting to use MIP-based heuristic or any other types of heuristic approaches to obtain good solutions in the least amount of time. *