Production, Manufacturing and Logistics
NP-hard and polynomial cases for the single-item lot sizing problem with batch ordering under capacity reservation contract

https://doi.org/10.1016/j.ejor.2016.07.028Get rights and content

Highlights

  • Single-item lot sizing problem is studied under a capacity reservation contract.

  • Batch deliveries are allowed and the overall replenishment cost is stepwise.

  • Four NP-hard cases are identified and an efficient FPTAS is proposed.

  • Pseudo-polynomial time dynamic programming algorithm is given for the general case.

  • Polynomial time algorithms are proposed under restricted parameters.

Abstract

In this paper, we study the single-item lot sizing problem under a capacity reservation contract. A manufacturer is replenished by an external supplier with batch deliveries and a certain capacity is reserved at the supplier level with an advantageous cost. In addition to the classical ordering and inventory holding costs, for each batch ordered under the reserved capacity a fixed cost per batch is incurred; and for batches exceeding this capacity a higher fixed cost per batch is paid, typically through the purchase from the spot market. We identify various NP-hard cases, propose a pseudo-polynomial time dynamic programming algorithm under arbitrary parameters, show that the problem admits an FPTAS and give polynomial time algorithms for special cases. We finally state a list of open problems for further research.

Introduction

We consider the single-item lot-sizing problem where the quantities are delivered by batch (e.g. pallet, container or truck) from an external supplier. A long-term contract is established between the manufacturer and the external supplier in which the costs and the reserved capacities are specified. Up to a certain level of the reserved capacity, the batches have a certain advantageous fixed cost, and once this capacity is exceeded the supplier proposes a second price, often more expensive but without any capacity limit. This type of contract is known in the literature as the capacity reservation contract. In the classical single-item lot sizing problem (LSP), the main decisions are which quantity and when to order over a finite horizon in order to satisfy the discrete and deterministic demand for one type of item, while minimizing the total cost of ordering and storage. In our problem, in addition to those classical decisions, we also determine the number of batches to replenish below and above the reserved capacity in each period. In addition to the classical LSP costs, we have two types of fixed costs per batch, respectively for the batches up to the reserved capacity and beyond this capacity. These latters make the cost function stepwise (also called staircase, multiple setup, stepwise or truckload discount cost). In the remainder of this paper, we call this problem LSP-BCR for Lot Sizing Problem with Batch ordering and a Capacity Reservation contract.

Although we describe a context of replenishment and a relation between a manufacturer and a supplier, our scope is not limited to the procurement activities. The same problem described above can also be encountered in a production system where the ordered quantities may refer to the production quantities, the batch production is substituted for batch delivery and the reserved capacity at the supplier may refer to the own capacity of the manufacturer. Once this production capacity is exceeded, the batches can be procured from outsourcing or via overproduction at a higher cost. In the remainder of this paper we use the terminology of replenishment. Note that the term ordering can easily be replaced by procurement, replenishment or production according to the context of study.

For the LSP-BCR considered in this paper, the manufacturer accepts to pay an initial fixed cost C0 for the signed contract and reserves a capacity Rt for each period t, where Rt is the number of batches reserved. Each batch has a given fixed size Vt in period t. When the number of ordered batches in period t is less than Rt, the manufacturer accepts to pay at per batch; otherwise, a cost of bt is incurred for each extra batch. These additional batches are typically purchased from the spot market. The cost bt can also be considered as a second price proposed by the supplier if the initial reserved capacity is exceeded. For any positive amount ordered, a setup cost Kt is also paid in addition to a unit replenishment cost pt for each unit ordered. The cost structure is illustrated in Fig. 1 and the replenishment cost function is given by Eq. (1).

The ordering cost qt(xt) in period t for xt purchased units is thus considered as follows: qt(xt)={0ifxt=0Kt+atxtVt+ptxtif0<xtVtRtKt+atRt+bt(xtVtRt)+ptxtifxtVt>Rt

This cost structure is first considered by Van Norden and Van de Velde (2005), where the authors study a multi-item LSP-BCR. Here, we study the single-item version of this problem. In most practical applications, the spot market cost bt is assumed to be higher than the initially negotiated and discounted cost at. However, in this paper we extend our theoretical results to the more general cases assuming arbitrary costs. For instance, if the cost bt is lower than the cost at in a period t, in an optimal solution we will never order the batches at a cost at and the reserved capacity Rt can be assumed to be zero at t. Note that if at > bt for all t, which is an extreme case, then the contract is not signed between the manufacturer and the supplier, thus the initial contract cost C0 is not paid.

Contributions of the paper. The contribution of this paper is to study the complexity of different classes of LSP-BCR under a stepwise cost structure for the single-item case, and to propose efficient algorithms for the polynomially solvable cases. We identify NP-hard cases and propose a pseudo-polynomial time dynamic programming algorithm for the general case, which makes our problem NP-hard in the ordinary sense. We also use a result from the literature to show that LSP-BCR admits an FPTAS. To the best of our knowledge the LSP-BCR has not been theoretically studied yet for both the single-item case and the case of a stepwise ordering cost (due to the batch ordering). The same cost structure can also be adapted to production models with additional overproduction costs per batch or with an outsourcing option having stepwise costs. The reader can find a summary of our theoretical results in Tables 1 and 2. In both tables we use the modified unit replenishment cost pt, which is equal to pt=pt+i=tThi where ht is the unit holding cost incurred for each unit kept in stock at the end of t. This transformation is explained in Section 3. Note that in Tables 1 and 2, the notations used for problem parameters are either stationary (K, a, b, p′, R, V) or time-dependent (Kt,at,bt,pt,Rt,Vt).

Section snippets

Literature review

In this section we position our study within the lot sizing literature. We first give the acronyms that are used throughout this paper.

  • WW: Uncapacitated Lot Sizing Problem with Wagner and Whitin cost structure (non-speculative costs)

  • ULSP: Uncapacitated Lot Sizing Problem

  • CLSP: Capacitated Lot Sizing Problem

  • ULSP-B: Uncapacitated Lot Sizing Problem with Batch ordering

  • CLSP-B: Capacitated Lot Sizing Problem with Batch ordering

  • LSP-CR: LSP under the Capacity Reservation contract

  • LSP-BCR: LSP with

Mathematical description

A mixed integer linear programming (MILP) formulation of the problem is stated below, where xt represents the quantity ordered in period t, yt is the binary setup variable and st the stock level at the end of period t. We assume that the initial and the final stock levels are zero (s0=0 and sT=0). Let At and Bt be the number of batches ordered in period t at a cost at and bt respectively. We denote by Vt the batch size in period t and by Rt the reserved capacity in period t (the number of

NP-hard cases of the LSP-BCR

We have identified four classes of LSP-BCR as NP-hard in the ordinary sense. First of all we begin by presenting some reductions of the LSP-BCR problem to the known lot sizing problems, under various assumptions. For instance, the cases where (a=+ and R taking any arbitrary value) or (R=0 and a taking any arbitrary value) correspond to the ULSP-BCR. In both cases, no batch will be reserved with the cost a in an optimal solution. Observe that when the reserved capacity is null (R=0), the cost a

Pseudo-polynomial time algorithm

As the simple cases studied above are NP-hard, the general case of the problem is also NP-hard. However, the pseudo-polynomial time dynamic programming algorithm proposed by Florian et al. (1980) can be adapted to solve the general case of our problem. Thus, this implies that the general problem is NP-hard in the ordinary sense. In the following, we first give the theoretical details of the dynamic programming algorithm. Since its complexity is pseudo-polynomial due to its dependence on the

Problem LSP-BCR belongs to FPTAS

In this section we establish the existence of a fully polynomial time approximation scheme (FPTAS) for LSP-BCR. Recall that an FPTAS is a collection of approximation algorithms such that, for any given ϵ > 0, the problem can be approximated within a performance guarantee of ϵ, in time complexity polynomial in the instance size and in 1/ϵ. In the literature, the first result on the fully polynomial approximation schemes for the single-item capacitated lot sizing problem is given in van Hoesel

Polynomial cases

We first give some optimality properties and general assumptions, and then propose polynomial time algorithms for the special cases of LSP-BCR. Below, some important hypotheses are made in this section:

  • The batch sizes are assumed to be stationary, otherwise we have shown in Section 4 that the problem becomes NP-hard with only one of the time-dependent cost parameters.

  • We allow incomplete batches, also called fractional batches.

  • Although in many practical situations the spot cost b is higher than

Conclusion and perspectives

We theoretically studied the lot sizing problem under the capacity reservation policy. We particularly considered the single-item case where the procurement cost has a stepwise pattern. Note that this specific piecewise cost structure represents many real-life situations, such as production per batch, transportation by container or trucks, deliveries by pallets, etc. Regarding the capacity reservation contracts, there are many real applications mostly in high-tech industry leading to

Acknowledgments

We are grateful to Christophe Rapine and to the anonymous referees for their careful reading and their suggestions which helped to improve the presentation of this paper.

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