We present an operational assembly-oriented SCS with in process quality control policy using an open queuing network approach. A nonlinear profit-maximization formulation model is presented, and an enumerative algorithm to solve this problem is provided. For small scale, the approach can obtain efficient approximation solutions. But, for large problems, will need more computing time because of the increased number of alternative policies needed to be evaluated.

Key words:

Supply Chain Management, Quality Control process, Formulation.

Introduction:

We propose an analytically-based queuing network model for an unreliable supply chain with inspection, reworking and scrapping【1】. The present Supply Chain Model consists of several supplying cells【2】. The main components in each cell are assembly machines, robots, and an inspection station【3】. In the assembly-oriented supplying cell, the defective parts which are detected by the inspection station can be dis-assembled (i.e., reworked) if the defects are made in the supplying process of the current cell【4】, then they can be processed just like the original input material from the immediate preceding cell【5】, On the other hand, if defects of parts are caused by the processing from one of the preceding cells, the parts will be scrapped【6】.

Assumptions:

The main assumptions are as follows:

1. After going through a particular cell for processing, the part will not return to that cell again.

2. The local buffer is of infinite size.

3. The processing capacity of each cell is large enough to handle the input, i.e., the aggregate processing time (including the working time, reworking time and inspection time for all part types) multiplied by the aggregate arrival rate of all part type should be less than one.

4. The working time, reworking time and inspection time of parts are mutually independent.

5. The working time and reworking time belong to a general distribution.

6. For each part type, the visitation sequence in the Supply Chain is fixed and given.

7. A part may go through the working area, reworking area and inspection many times, but the processing time in these three processes does not change in each visit.

Assumptions 1 and 6 are practical situations in the supply chain-oriented system because the works of supply chain rarely go back to the stations which were visited before unless they need to be reworked, and also because the routing of an supply chain system for a specified task is difficult to change. The assumption of infinite local size is adopted by many researchers to simplify the solution procedure, to ensure that we reach the steady state of the queuing system, assumption 3 is needed. Assumptions 4 and 7 are required to aggregate the working time, reworking time and inspection time.

An Unreliable Supply Chain Model with Inspection, Reworking and Scrapping:

Let us define the functions of an unreliable Supply Chain Model (SCM) cell and the movements of parts in the cell, and derive preliminary results to the entire SCM. The supplying cell that we consider consists of a working area, a working area, and an inspection station. A part will leave the cell if it passes or skips inspection. If defects are detected, two possibilities arise: if the defect was due to the processing in the current cell, the unit is routed to the reworking area and then circulated back to the working area for processing. Otherwise, it will be scrapped. Inspection is assumed to be 100% reliable and all defective items can be examined to determine if they are from the current cell or from one of the preceding cells.

The Supply Chain Systems:

The Supply Chain systems (SCS) we consider consist of several supplying cells with processing, inspection and reworking functions in each cell. Parts of different types will flow through the SCS following specified routes that visit various cells in the system. As described earlier, a part leaves a cell if it passes or skips inspection. On the other hand, when a defective part is detected, the part will be reworked and circuited back to the working area of the cell if the defects are made in the current cell. However, if the defects are found to be made in one of the preceding cells, the parts will be scrapped from the cell without reworking. This section is further divided into two subsections. In the first section we obtain, for each part type, the expected number of parts that will be reworked and scrapped in each of their visitation cells, the expected number of undetected defective parts, the total flow to the next cell in the visitation sequence, and finally, the average outgoing quality of each part type and of the SCS as a whole. In the second action we examine the waiting time spent by each part in the system due to working, reworking and inspection.

An Unreliable Supply Chain Model:

Although incorporating a quality control process into an unreliable SCM can ensure the quality of the outputs, the trade-off is the inevitable decrease in the throughput of the SCM due to the presence of the quality control process. One of the reasons being that the quality control process can in fact, increase the total processing time of parts and the units of scrapping. Since the in-process quality control system will dynamically affect the performance measures we discuss in the previous section, we have to use a systematic approach to analyzing these effects. Therefore, in this section, we construct a profit-maximization model that seeks a near-optimal (i.e., only a heuristic solution algorithm is proposed here) quality control policy for an unreliable SCM. Relevant revenue and cost parameters are defined as follows.

REVK: the revenue for a type-k part

CIKl: the cost of inspecting a type-k part in the it's lth visitation p cell,

CWK: the waiting cost of a type k part

CSKl: the cost of scrapping a type-k part in its lth visitation cell, with CSkl ＜ CSK,l+1 ,i=1, 2,…,zk-1

CRkl: the cost of reworking a type-k part in the its lth visitation, and

CPkl: the cost of processing a type-k part in the its lth visitation cell.

Let r kj (the inspection rate for type-k parts in cell J) be decision variables, we now show a profit-maximizing model that incorporate the quality control into an unreliable FMS.

MAXIMIZE (1)

where

and (2)

for all k and j .

(3)

In this model, the profit is accounted on per unit time basis and J k represents the total return of type-k parts produced by the SCS. Equation (2) computes the profit generated by type-k, which is the total revenue (the 1st term) minus inspection costs (the 2nd and 3rd terms), waiting costs (the 4th and 5the terms), post-sales failure costs (the 6th term), scrapping costs (the 7th term), and reworking costs (the last term). The above model is a complex nonlinear optimization problem which becomes more cumbersome when we attempt to incorporate the results of the queuing network model into it. In the next section, we use a numerical example to demonstrate the effect of the quality control policy on a two-step and two-digit search procedure is proposed as follows:

STEP1: Search for the optimal inspection rates in the range of 0 and 1 with increment of 0.1. denoting the solutions by r* kj .

STEP2: Search for the optimal inspection rates in the range of r *kj -0.05 and r* kj +0.05 with increment of 0.01.

A Numerical Example:

Consider a SCS with three cells where each cell contains two operating areas, one for working (which also contain an inspection station) and the other for rework two types of parts are processed in the SCS. Three separate cases are examined to illustrate the effect of the quality control function on system characteristics: (1) The SCS is completely reliable (2) the SCS is unreliable and there is no quality control, and (3) the SCS is unreliable and has quality control; 20 percent of the processed parts are inspected.

Conclusion:

In the paper we have obtained the operational characteristics of an assembly-oriented SCS with in process quality control policy use an open queuing network approach. A nonlinear profit-maximization formulation was presented, and an enumerative algorithm to solve this problem was provided. For small scale problem, the approach can be used to obtain efficient approximation solutions. However, for large-scale problems, the proposed search procedure will require more computing time because of the increased number of alternative policies must be evaluated.

References:

【1】Anonymous. “Supply Disruptions May Linger as Quake

aftershock.” http://www.eetimes.com, September 22,1999.

【2】Bowers, M.R., and A. Agarwal. “Lower In-Porcess Inventories and Better On-Time Performance at Tanner Companies, Inc.” Interfaces 25(1995), pp. 30-43.

【3】Federgruen, A., and A. Heching. “Combined Pricing and Inventory Control under Uncertainty.” Operations Research 47(1999), pp. 454-75.

【4】Fisher, M. L. “What Is the Right Supply Chain for Your Product?” Harvard Business Review, March-April 1997, pp. 105-17.

【5】-- --. “Rethinking Distribution: Adaptive Channels.” Harvard Business Review, July-August 1986, pp.112-20.

【6】Stephen Harley, “Transportation: The Cornerstone of Global Supply Chain Management”, Proceeding of CLM Annual Meeting Council of Logistics Management, 1996, pp. 635-641.