Overview

Chapter 1 Introduction

1.1 Overview

Supply chain management (SCM) was popular in the past years and seems continuing to be so in the future. The main reason of its popularity may be that it extends the traditional operational management in local shop floor to a global context. Basically there are two approaches to model a supply chain (SC), deterministic and stochastic. The former is adopted when all the operational parameters, such as demand arrival rate, processing rate, etc. are certain while the later is used when most of the parameters are uncertain. Material requirement planning (MRP) and lately enterprise resource planning (ERP) are perhaps the most widely used supply control methods to satisfy market demand under a deterministic operational environment. The performance is often judged by the generated supply plan that can deliver the right product with right quantity to right place in right time. Under stochastic environment, the SC performance can be evaluated through stochastic modeling to obtain the steady state system performance. No matter what modeling type the studied problem is, the performance of an SC is often measured by its integrated operational cost and achieved customer service level. It’s well known that in order to save operational cost, two important factors must be addressed: inventory and moving (transportation) costs.

For analytic modeling of a stochastic SC, there are three well-known problem domains for a supply network: infinite buffer, finite-buffer and infinite buffer with planned inventories. The former two problems are suitable for make-to-order (MTO) supply mode, while the last is suitable for make-to-stock (MTS) supply mode. Open or closed queueing network (QN) model is suitable for solving the first kind of problem. Several models are developed to solve the second problem. Notice the blocking effect exists in the second problem and the solution process seems not so straightforward as the first problem. For the third problem, it’s not paid special attention until in the past decade or so (Lee & Zipkin,

2

1992) (we abbreviate it as L & Z hereinafter). However, the MTS supply mode of the third type is a well-known industry application, therefore it shows imperative need for this research direction. It depends on the business process and other factors such as market, capital, physical limitation etc. to choose the suitable supply mode and the accompanied stochastic model for practical study.

In reality, an SC is often shown as a sophisticated supply network with complex operation logic. Under this concern, simulation seems to be the most often used method to analyze a realistic SC, especially when mathematic model is not available. However it becomes time-consuming to build an industry-scaled simulation model. Let’s take a look at fig. 1.1, which is an SC network, composed of 4 basic SC functions: inbound logistics, manufacturing, outbound logistics and distribution. Inside each echelon, there may be several processing stages with serial or parallel configuration and the probability distribution for each service may be arbitrary (for ease of exposition, we only show serial case in fig. 1.1, see the square enclosed by the dotted lines). To let the complex interaction between SC players (contributors) become tractable, an adequate control scheme governing order/replenish behavior for each echelon is necessary if we regard inventory as the main concern. Unfortunately, we don’t know what the optimal order/replenish policy for SC like fig. 1.1 is. Furthermore, the closed-form solution is usually unavailable (Boyacy and Gallego, 2001) even for the simplest tandem supply system with assumed constant supply lead-time. In this study, we do not try to answer these questions. Instead, we “assume” control policies at each supply stage are known in advance. For example, in

“pull-type” control (which is suitable for the situation that the demand is unknown or stochastic), there are base-stock policy, reorder point, order quantity (r, q) policy, reorder and target level (s, S) policy, and KANBAN-card controlled policy etc. Among them, base-stock policy is widely used in industry (L & Z) owing to its simple control logic. It is suitable when economy of sale is not a concern. For example, there is no fixed set-up cost

3

in each ordering cycle.

To model the SC like fig. 1.1, we begin by learning the system dynamic of a base-stock controlled tandem supply system as appeared in (L & Z) and shown below:

Ij = [Sj − Kj]⁺, Bj = [Kj − Sj]⁺, 1 ≤ j ≤ J, (1.1) K1 = N1, Nj = Kj – Bj-1 for j > 1, (1.2) where [x] ⁺ = max(x, 0). J is the stage number of the tandem system. Sj is base-stock level at each stage. Kj isdemand on order, and Bj is backorder level at stage j, and Ij is inventory level at stage j (here assume we have already break the original echelon boundary into a multi-staged tandem form). Nj is the input queue occupancy before each stage j (including the one being served if there is any). (1.2) is due to the property of the underlying system dynamic: [demand on order] = [input queue occupancy] + [backorder level at the previous stage]. We assume ample supply before the first stage, and therefore no backorder from

external supply: K1 = N1. The difficult part of analyzing the above system is that the queueing network will not be a M/M/1 connected system when the planned inventories are added after each stage. Based on (1.1) and (1.2), several methods for performance evaluation of a base-stock controlled SC have been reported recently for the approximation of such SC like recursive method, squared coefficient variation (SCV) of departure process (which will be abbreviated as SCV method hereinafter), quasi-birth-and-death (QBD) method, and matrix computation method. Recursive method starts at stage 1 with K1 = N1, the distribution of N1 is approximated to be that of M/M/1. Specifically

⎩⎨

⎧

≥

−

=

= −

= (1 ), 1.

, 0 , ) 1

( i

i i N P

j i j

j

j ρ ρ

ρ (1.3)

Given Kj, it computes Bj by shift-truncation operation in (1.1). Then, it applies (1.2) to obtain Kj+1 = Nj+1 + Bj, which is just the convolution of product-form approximation.

Motivated by the widely used approximation of the SCV of the departure process from a

4

standard queue (Buzacott and Shanthikumar, 1993), specifically c_d² =(1−ρ²)c_a² +ρ²c_s², SCV method uses the following approximation for the SCV of the departure process from output buffer of stage j, departure process of a standard M/M/1 queue. It then approximates the distribution of input queue occupancy (Buzacott and Shanthikumar, 1993, p76)

⎩⎨

= + (Note here Allen-Cunneen approximation for a

GI/G/1 queue is used to obtain the above formula). This method uses the property:

2 1 , 2 = _{d j}−

aj c

c to recursively find performance measures as the previous method. Specifically, it lets c be the SCV of the inter-arrival time of external demands. It then computes _a²₁ c _d²₁ by (1.4) and K1 = N1 (since B0 = 0) by (1.2), whose distribution is known by (1.5). Given K1, it computes B1 by (1.1), and then it moves to the next stage and recursively call the above procedure until stage J. Notice these two methods only differ in queue occupancy calculation between (1.3) and (1.5). Specifically (1.5) related to the SCV of departure process (1.4). QBD method tries to directly solve the whole SC system by approximating the input buffer to be finite number. Then it uses matrix geometric method (MGM) to solve the finite QBD. However, the computation becomes intractable when J is large, say J > 4.

Instead, herein we use QBD process to decompose each queueing system in an SC and therefore get tractable results. In this study, our evaluation model belongs to the last

5

method, matrix computation. It is essentially equal to recursive method. However, it focuses on response times instead of queue occupancies. It implements the calculation with simple matrix-algebraic manipulations. Herein we combine it with our proposed QBD method and give it another name, matrix analytical method.

The optimization of the SC to obtaining strategic parameter setting for efficient material flow can be categorized as the following methods: classical derivative-based method, enumerative method, meta-heuristic method among others. These methods will be fully explored herein. In this work, we show that through adequate “transformation”, similar or more complex SC like fig. 1.1 can be tackled in adequate mathematic models.

Specifically, in the first phase, we built the evaluation model for an SC by proposing the QBD modeling procedures for solving non-exponential, parallel processing, and single-server based distribution systems. We also discussed possible extension of the evaluation model with (r, q) controlled system dynamic. In the second phase we detailed the analysis of parallel processing under supply and demand uncertainties with the help of Markov-modulated Poisson process (MMPP) models. Finally, we investigated several optimization methods, classical and modern, which may be used in strategic optimization of an SC.

Fig. 1.1 A supply chain network.

6