ALFRED P. SLOAN SCHOOL OF MANAGEMENT

SHORTEST ROUTE MODELS FOR THE

ALLOCATION OF INSPECTION EFFORT

ON A PRODUCTION LINE : : -o. iiJST.TECH.

by

Leon S. White

Sloan School of Management, M.I.T.

August 11, 1967

Working Paper 278-67

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SEP 18 1967

D^V/EY LiCRARY

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

Hi) 5.^

^0. cP7r-^7

RECEIVED

OCT 2 1967

M. I. T. LIbKrtKlhS

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ABSTRACT

Two shortest route models for determining where to allocate inspection

effort on a production line are developed for the cases where this effort

is unlimited or limited in its availability. A production line is defined

as an ordered sequence of production stages, each stage consisting of a

manufacturing operation followed by a potential inspection station. Items

flow through the line in batches and may incur defects at any stage. De-

fects are assumed to be repairable or non-repairable. The defect generating

process at any stage is taken to be an independent Bernoulli process with a

known parameter. Two levels of inspection effort may be applied at any

stage: no inspection or 100% inspection. Thus, both models are used to de-

termine the stages at which batches are to be 100% inspected. A general cost

structure is postulated which includes fixed and variable costs of inspection,

a cost of repair, a cost of disposal, a cost of processing, and a cost of an

undetected defect. The first three of these costs may depend on the most

recent as well as the present inspection point. An expected cost per batch

criterion is used to determine an optimal inspection plan. Examples are in-

cluded.

i)'>7:y^H-7

Shortest Route Models for the Allocation of Inspection

Effort on a Production Line

by Leon S. White

Sloan School of Management, M.I.T,

1. Introduction

In this paper we develop a general deterministic model for the alloca-

tion of inspection effort on a production line. The model has the analytical

form of a shortest route model; thus several very fast algorithms are avail-

able for generating computational results, [2], [3], In addition, we show

how the basic model may be extended to the case where the amount of available

inspection effort is limited.

We define a production line as an ordered sequence of L production

stages, each stage consisting of a manufacturing operation followed by a po-

Work reported herein was supported (in part) by the Office of Naval

Research under contract Nonr-3963 (06), NR-276-00'+ administered by the

Operations Research Center, M.I.T. Reproduction in whole or in part is

permitted for any purpose of the United States Government.

u

The author is indebted to E. Brown for his helpful comments and sug-

gestions.

tential inspection station.^ A five stage production line is illustrated

in Figure 1. The produce being manufactured is assumed to enter stage 1

of the production line in batches of size B>1. As the items within a batch

move through the manufacturing operations they may incur defects. A defect

created in any item at stage n will be called a type n defect. The defect

generating process at the n*^ stage is viewed as an independent Bernoulli

process with a known parameter b , the probability of producing a defect of

type n. The models to be developed allow for repairable and non-repairable

defects. We let R(n) denote the subset of defects within the set {l,2,.,.,n}

that are repairable at stage n. We also define

R(n) = {1,2, ...,n} - R(n)

i.e., R(n) is the subset of defects within {l,2,...,n} that are not repair-

able at stage n.

If all defects 1,2,.,.,L are repairable, we assume that the net flow

through each production stage is B items; in other words, we assume the ex-

istence of a repair facility that can supply non-defective items to replace

any defectives found during the inspection of a batch at any stage. On the

other hand, if some defects are taken to be non-repairable, then any items

found to have such defects are scrapped, possibly with some salvage value.

Items found to have only repairable defects are again assumed to be replaced

with non-defective items supplied from a repair facility. Thus, for the sit-

uations where some defects are repairable and others are not, a batch that

^In certain cases it may be convenient to think of the first manufac-

turing stage as the agf^regation of all operations that come before the

start of the production line, and of the first potential inspection station

as an incoming inspection point.

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contains B items at the start of the line will be reduced in size at the

inspection stations where non-repairable defective items are discarded.

Models in which all defects are non-repairable have been considered by

Lindsay and Bishop l^ ] and Pruzan and Jackson L5J. Beightler and Mitten

[1] and White L6j have treated cases where all defects are repairable.

The models presented in this paper include these extremes as special cases.

Consider now a class K of multistage inspection plans of the form

(k, ,k_, . . , ,kj ) , where k =0,1,...,B is the number of items to be inspected

in each batch passing through inspection station n and B is the batch size.

Lindsay and Bishop [4] and White [6] have shown for the cases of no repair-

able defects and all repairable defects, respectively, and for a fairly

simple cost structure, that an optimal plan — one that minimizes the expected

cost per processed batch — within the class K has the form k =0 or B .i.e., at

n n» •

any inspection point arriving batches are either passed without inspection or

they are inspected 100%. In this paper we only consider inspection plans of

the "all or nothing" form. The class of such plans will be denoted by K .

The question as to whether K" contains an optimal plan within K for our models

is not investigated.

In what follows, we shall show that the problem of finding an optimal plan

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within K for a rather general cost structure may be formulated as a shortest

route problem. We shall also show that a shortest route model may be formu-

lated for the case where at most T

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