two-level cost-optimization production control model under random disturbances

Mathematics and Computers in Simulation 52 (2000) 381–398

Two-level cost-optimization production controlmodel under random disturbances

Dimitri Golenko-Ginzburga,∗, Gonik Aharonb,1, Sitniakovski Shimona

a Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israelb Luxembourg Industries, P.O. Box 205, Arad 89101, Israel

Received 1 January 2000; accepted 1 April 2000

Abstract

A two-level flexible manufacturing system is considered to be composed of several different production unitsUi , 1≤i≤n, at the lower level and a section at the upper one. The section is required to produce a given targetamountV by a given due dateD subject to a chance constraint, i.e. the least permissible probabilityp of meetingthe target on time is pregiven. Each production unitUi has several possible speedsvi1, vi2, . . . , vim, which aresubject to random disturbances. The unit’s output can be measured only at preset inspection (control) points. Thetarget amount is gauged by a single measure and may be rescheduled among the production units. For each unit, theaverage manufacturing costs per time unit for each production speed and the average cost of performing a singleinspection at a control point to observe the actual output at that point, are given.

We recently have developed a cost-optimization on-line control model which for a single production unit de-termines both control points and speeds to be introduced at those points, in order to minimize the unit’s expenseswithin the planning horizon, subject to the chance constraint. We present a two-level on-line control model underrandom disturbances, which centers on minimizing the section’s expenses subject to the chance constraint. Thesuggested two-level heuristic algorithm is based on rescheduling the section’s target among the production unitsboth att=0, when the system starts functioning, and at each emergency point, when it is anticipated that a certainunit is unable to meet its local target on time subject to a chance constraint. At any emergency pointt the remainingsection’s targetVt is rescheduled among the units; thus, new local targetsVit , 1≤i≤n,

∑iVit = Vt , are determined.

New local chance constraint valuespit are determined too. Those values enable the system to meet its overall targetat the due date subject to the pregiven chance constraintp.

A numerical example is given. Extensive experimentation has been undertaken to illustrate the efficiency of thealgorithm. © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.

Keywords:Production speed; Cost-optimization; Target amount reassignment; Chance constraint; On-line control; Inspectionpoint

∗ Corresponding author. Tel.:+972-7-6461434; fax:+972-7-6472950.E-mail addresses:[email protected] (G. Aharon), [email protected] (S. Shimon)

1 Tel.: +972-7-9957369.

0378-4754/00/$20.00 © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0378-4754(00)00163-4

382 D. Golenko-Ginzburg et al. / Mathematics and Computers in Simulation 52 (2000) 381–398

1. Introduction

In recent years the problem associated with developing multilevel on-line production control mod-els under random disturbances for flexible manufacturing systems has been discussed in the literature[1,3–6,8–13,15,16]. Most of those investigations deal with not fully automated plants of ‘man-machine’type where the output cannot be measured continuously on-line, but only at preset control points. Themain idea of the interaction problems between different levels in hierarchical control systems is basedon the conception of emergency introduced by Golenko-Ginzburg and co-workers [1,9–12]. By using theidea that hierarchical levels can interact only in special situations, the so-called emergency points, onecan decompose a general and complex multi-level problem of optimal production control into a sequenceof one-level problems.

Two different optimization cases are usually considered:1. Case with a conflicting two-criteria objective, namely, to maximize the probability of completing the

production on the due date, and to minimize the number of control points; but the first criterion isdominant.

2. The objective is to maximize the expected net profit.Note that most of the papers outlined above do not implement a chance constraint in the on-line

production control model. In our opinion, minimizing the system’s expenses to meet the target on time,i.e. at a given due date, is not to be the only goal in the course of the long-term cooperation with variouscustomers. To honor the company’s good name, an additional requirement has to be inserted in the model:the production system has to meet its due date on time with a pregiven confidence probability. Thus, achance constraint has to be implemented in the control model.

A two-level flexible manufacturing system is considered to be composed of several different productionunits Ui , 1≤i≤n, at the lower level and a section at the upper one. The section is required to producea given target amountV by a given due dateD subject to a chance constraint, i.e. the least permissibleprobability p of meeting the target on time is pregiven. Each production unitUi has several possiblespeedsvi1, vi2, . . . , vim, which are subject to random disturbances. The unit’s output can be measuredonly at preset inspection (control) points. The target amount is gauged by a single measure and may berescheduled among the production units. For each unit, the average manufacturing costs per time unit foreach production speed and the average cost of performing a single inspection at a control point to observethe actual output at that point, are given.

Golenko-Ginzburg et al. [7] have developed a cost-optimization on-line control model which for asingle production unit determines both control points and speeds to be introduced at those points, inorder to minimize the unit’s expenses within the planning horizon, subject to the chance constraint. Wepresent a two-level on-line control model under random disturbances, which centers on minimizing thesection’s expenses subject to the chance constraint. The suggested two-level heuristic algorithm is basedon rescheduling the section’s target among the production units both att=0, when the system startsfunctioning, and at each emergency point, when it is anticipated that a certain unit is unable to meet itslocal target on time subject to a chance constraint. At any emergency pointt the remaining section’s targetVt is rescheduled among the units; thus, new local targetsVit , 1 ≤ i ≤ n,

∑iVit = Vt , are determined.

New local chance constraint valuespit are determined too. Those values enable the system to meet itsoverall target at the due date subject to the pregiven chance constraintp.

After reassigning to each unitUi its new targetVit and the chance constraint valuepit , the units firstwork independently and are controlled separately. At eachkth control pointtik of unit Ui , given the actual

D. Golenko-Ginzburg et al. / Mathematics and Computers in Simulation 52 (2000) 381–398 383

amount already produced, decision-making centers on determining both the next control pointti,k+1andthe indexj of the new speedvij to proceed with up to that point, 1≤j≤m. The on-line control at eachproduction unit proceeds either until the next emergency point, or until the due dateD.

Rescheduling the remaining section’s target amountVt among the production units is carried out byusing heuristic procedures. Determining chance constraint valuespit is carried out by using a cycliccoordinate descent method in combination with a two-level simulation model.

2. The control model

A two-level control model is suggested where each level faces a stochastic optimization problem.

2.1. The problem at the section level (Problem A)

At each emergency pointt = temq , 1 ≤ q ≤ Nem, tem

1 = 0, determine local production plansVit ,1≤i≤n, together with local chance constraintspit , in order to minimize the expected total expenses

min{Vit ,pit }

C (1)

subject to the chance constraint

Pr{V f (D) ≥ V } ≥ p. (2)

Note that random valueC satisfies

C =n∑

i=1

Ni−1∑k=0

[Cisik(ti,k+1 − tik)] +

n∑i=1

(Ni − 1)Cins

+NemCem + [C∗ + C∗∗(F − D)]δ + C∗∗∗(D − F)(1 − δ), (3)

where

δ ={

1 if F > D

0 otherwise,(4)

and values{sik} and{tik} are obtained by solving Problem B at the production unit level.Values{Vit} at each emergency pointt, including t=0, are determined according to a widely used

heuristic procedure [4,12], namely

Vit = Vt

vim∑ni=1vim

, (5)

wherevim is themaximalmanufacturing speed which can be introduced for unitUi .As to values{pit}, they are determined by using one of the classical search procedures for optimizing

a multi-dimensional non-linear function, e.g. a cyclic coordinate descent algorithm [3–5,14]. The searchprocedure is carried out via simulation, by undertaking numerous realizations of a simulation model atthe lower level in order to obtain representative statistics. The simulation model represents the process ofmanufacturing for several production unitsUi with input values{Vit} and{pit}, between two adjacent


emergency pointstemq andtem

q+1. In the case of a routine emergency call the problem at the section levelis resolved, new values{Vit} and{pit} are determined, and the manufacturing process proceeds at thelower level, for each unitUi independently.

2.2. The problem at the unit (manufacturing) level (Problem B)

The cost-optimization control model for a single production unit has been formulated in [2,7]. We havemodified that problem for the case of several production units with additional cost parametersCem, C∗,C∗∗ andC∗∗∗.

For the case of an independent production unitUi , given the input valuesVit , pit , d, 1 and vij ,

1 ≤ j ≤ m, the problem is to determine both control points{tik} and production speeds{visik} to

minimize the manufacturing expenses

J = min{tik ,visik

}

{Ni−1∑k=0

[Cisik(ti,k+1 − tik)] + NiCins

}(6)

subject to

Pr{V f

i (D) ≥ Vit

} ≥ pit , (7)

ti0 = t, (8)

tiNi= min

Ti

[Ti : Pr{V f

i (Ti) ≥ Vit}], (9)

ti,k+1 − tik ≥ d, (10)

D − tik ≥ 1, 0 ≤ k ≤ Ni − 1, (11)

sik = j = min1≤q≤m

q ∀q : Wp[V fi (t), Vit , q] ≤ D. (12)

Restriction (8) means that after reallocating target amounts at the routine emergency pointt, the startingmoment to proceed manufacturing, i.e. the first control point to undertake decision-making and to deter-minesi0 andti1, is t. Note that at all emergency points the remaining target amount, as well as the duedate, are updated, i.e. the ordinatet=0 is shifted to the right. Restriction (9) means that the last inspectionpoint is the moment target amountVit is reached. Restrictions (10) and (11) ensure the closeness betweentwo consecutive control points, as well as the closeness of the routine inspection point to the due date.Restriction (12) means that the speed to be chosen at any routine control pointtik must not exceed theminimal speedwhich guarantees meeting the deadlineD on time, subject to the chance constraint (7).

The general idea of solving problems (6)–(12), which is a very complicated stochastic optimizationproblem, is as follows. At each control pointtik decision-making centers on the assumption [2,9–12] thatthere is not more than one additional control point before the due date. Two speeds have to be chosen atpoint tik [2]:1. Speedvij1, j1=sik, which has to be actually introduced at pointtik up to the next control pointti ,k+1.2. Speedvij2, j2=si ,k+1, which is forecast to be introduced at control pointti ,k+1within the period

[ti ,k+1,D].


Thus,j1 is determined in accordance to Eq. (12) andj2 is determined by honoring chance constraint (7).In [2,7] at each routine control pointtik, all possible couples are singled out. The couple, which deliversthe minimum of forecasted manufacturing and control expenses, has to be chosen. Since couple (j1, j2),together with the inspected valueV f

i (tik) and valuesD andVit , fully determines the next control pointti ,k+1, speedvij1 is introduced within the period [tik, ti ,k+1]. At momentti ,k+1decision-making has to becarried out anew.

3. The general idea of the two-level heuristic algorithm

The general idea of the suggested heuristic algorithm is as follows: at each routine emergency pointtemq ,

q=1, 2,. . . ,Nem, decision-making centers on minimizing thefuture costsfrom pointtemq until F, including

the penalty and the storage costs. The costs representing the past (interval [0,temq ]) are not relevant for

this on-line control problem, and there is no need to remember the past decisions [9]. The only relevantinformation to be stored istem

q andV fi (tem

q ). Thus, decision-making at the section level is carried out onlyat emergency pointstem

q including the momentt=0 the system starts manufacturing.Decision-making at the section level at each routine emergency momentt=tem

q centers on determiningboth new chance constraint values{pit} and new target amountsVit for the remaining planning horizon[t,D]. Values{pit} are obtained via simulation, by a combination of a search algorithm and an on-lineon-level control algorithm for several production units. The latter work independently and are controlledseparately at inspection points. It is generally assumed that at the beginning of the work all the availableresources are previously allocated among the production units. Those resources remain unchanged withinthe planning horizon, i.e. no resource reallocation is performed. Thus, the corresponding production speedsvij for each unitUi remain unchanged too.

If for a certain unitUi at a routine inspection pointtik it is anticipated that the unit cannot meet its targetVit on time subject to the previously determined chance constraintpit , an emergency is then called, anddecision-making is affected at the section level. The remaining production planVt at t=tik, together withthe remaining timeDtik = D − tik, is then updated. New quasi-optimal values{pit}, t=tik, together withnew target amounts{Vit}, are then determined. The newly corrected plan is assigned to all productionunits, and the manufacturing process proceeds further, until either the new emergency point or until themoment the target amount is completed. Thus, decision-making at the section level centers on numerousrecalculations of the section’s production plan subject to the chance constraint. This is carried out by usinga forecasting simulation model with input values{Vit ,pit}, t=tik. The matrixZ={Vit ,pit}which deliversthe minimum of total accumulated costs subject to the chance constraintp, is taken as theoptimal correctedplan. Afterwards, that corrected plan is passed to the production units, and on-line decision-making iscarried out at the manufacturing level.

4. The enlarged step-by-step procedure of the heuristic algorithm

The enlarged step-by-step procedure of the two-level heuristic algorithm is as follows:Step1.At time t=0 the input data is externally given:

• the section’s target amountV,


• the due dateD,• the chance constraintp,• speedsvij , 1≤i≤n, 1≤j≤m,• Cost valuesCij , Cins, Cem, C∗, C∗∗ andC∗∗∗.

Step2.Sett=0.Step3.Determine valuesVit by using Eq. (5).Step4.Arrange a cyclic coordinate descent search [3–6,14] in ann-dimensional space of optimized variables

pit , 1≤i≤n. Choose the initial search pointEX={pit}=(p, . . . ,p), i.e. setpit=p for each production unitUi .For each current coordinatexi=pit arrange a local search with a pregiven search incrementhi ,xi±hi ⇒ xi

to determine the direction of the function’sC decline. If a routine coordinatexi changes its value, valuesx1, . . . ,xi−1, which have been optimized before, remain unchanged, while valuesxi+1, . . . ,xn are preset.Thus, we optimize each routine variable separately while all the previous variables have already beenoptimized and the next variables are waiting for their turn. Numerous simulation runs at each search pointEXare to be undertaken at Step 5 to obtain representative statistics. When alln variablespit are optimizedthat means that we have realized the first iteration, i.e. all quasi-optimal variablesp∗

it are determined.Diminish the search stephi and proceed to minimizeC cyclically with respect to coordinate variablesand honoring chance constraint (2). The simulation model for a single production unit is outlined in [2,7]while an extension for the case of several units is outlined below.

Step 4 terminates when the relative difference between two adjacent iterationsCu andCu+1 becomesless than the pregiven toleranceε>0.

Two different cases have to be considered:1. In the course of realizing Step 4 the quasi-optimal solution of problems (1)–(5) has not been obtained,

i.e. a feasible solution honoring chance constraint (2) cannot be found. Go to Step 7.2. A quasi-optimal solution{Vit ,pit} honoring Eq. (2) has been determined. ValuesVit andpit are passed

to unitsUi , 1≤i≤n, and the manufacturing process begins. Go to Step 5.Step5.Simulates an on-line control between two adjacent emergency pointstem

q andtemq+1, 0≤tem

q <temq+1<D. The

input values of the simulation model aretemq , {Vitem

q} andX = {pitem

q}, Vtem

q= ∑n

i=1Vitemq

, V fi (tem

q ), p, andDtemq

, 1 ≤ i ≤ n.

Note that Step 5 is applied both in the course of optimizing vectorEX at Step 4 and in the course ofrealon-line control, within the interval [tem

q , temq+1]. The main difference is that to determine new optimized

valuespit at any emergency pointt = temq one has to simulate the manufacturing process until the end,

i.e. until pointF, and to undertake such a simulation run many times for each search pointEX. Moreover,future emergency pointstem

q+1, . . . , temNem

have also to be implemented in such a simulation run, togetherwith the corresponding optimization problems to be solved at those points. Thus, Problem A, i.e. problems(1)–(5), is predictive in its nature.

Problem A includes Problem B, i.e. includes problems (6)–(12) at the manufacturing level, which dealsonly with an on-line control for a pregiven production plan.

Note that after each emergency momenttemq in the course of a simulation run, valuesVt andDt have to

be updated


Vtemq+1

= Vtemq

−n∑

i=1

V fi (tem

q+1), (13)

Dtemq+1

= Dtemq

− (temq+1 − tem

q ). (14)

The enlarged stages of Step 5 are as follows [7]:Step5.1.Start withi=1, k=0, D = Dtem

q, Vitem

q, tem

q , pitemq

, V fi (ti0 = 0) andti0=0.

Step5.2.For each speedvij , 1≤j≤m, determine via simulation valuesWpitem

q[V f

i (tik), Vitemq

, j ] (see Section 8).Step5.3.Determinez = min1≤v≤mv{v : Wpitem

q[V f

i (tik), Vitemq

, j ]} ≤ Dtemq

. If z cannot be determined, go to Step6. Otherwise apply Step 5.4.

Step5.4.Consider couples as follows:

(1, z), (2, z), . . . , (z − 1, z),

(1, z + 1), (2, z + 1), . . . , (z − 1, z + 1),

(1, z + 2), (2, z + 2), . . . , (z − 1, z + 2),...

(1, m), (2, m), . . . , (z − 1, m),

(z, z), (z, z − 1), . . . , (z, 2), (z, 1).

(15)

Step5.5.For each combination of couples (j1, j2) from Eq. (15), we calculate the value ofti ,k+1 (the next

inspection point) as the abscissa of the intersection point of two straight lines

V = V fi (tik) + vij1(t − tik) and

V = Vitemq

− V fi (tik)

Wpitemq

[V fi (tik), Vitem

q, j2] − tik

+ Vitemq

− Dtemq

Vitemq

− V fi (tik)

Wptemq

[V fi (tik), Vitem

q, j2] − tik

. (16)

Step5.6.For each combination of couples (15) check ifti ,k+1−tik≥d holds. If not, calculateti ,k+1=tik+d.Step5.7.For each combination of couples (15) check ifti,k+1 > Dipem

qor Dipem

q− ti,k+1 < 1. If one of these

relations holds, setti,k+1 = Dipemq

.Step5.8.For each combination of couples (15), simulate the random forecasted output product

V [V fi (tik), j1, j2] = vij1(ti,k+1 − tik) + vij2(Ditem

q− ti,k+1) + V f

i (tik) (17)

in order to obtain representative statistics, i.e. calculate the empirical distribution of Eq. (17) via simulation.Step5.9.Exclude from the list of couples all couples which satisfy Pr{V [V f

i (tik), j1, j2] ≥ Vitemq

}, i.e. which donot honor chance constraint (2). If the list of couples becomes empty, go to Step 6. Otherwise apply thenext step.


Step5.10.Calculate for all remaining couples (j1, j2) the forecasted average costsC(i,k, j1, j2) as follows:

C(i, k, j1, j2) = Cij1(ti,k+1 − tik) + Cij2(Ditem

q− ti,k+1) + Cins,

if j1 6= j2 and ti,k+1 6= Ditemq

,

C(i, k, j1, j2) = Cij1(Ditemq

− tik), if j1 6= j2 and ti,k+1 = D, andC(i, k, j1, j2) = Cij1(Ditem

q− tik), if j1 = j2.

Step5.11.Consider the optimal couple (j1, j2) which delivers the minimal valueC(i,k, j1, j2). Note that the latter

fully determines the next control pointti ,k+1.Step5.12.Transfer the determined next control point to a special array of future control points (AFCP).Step5.13.Repeat Steps 5.1–5.12 for alli, 1≤i≤n.Step5.14.Find from AFCP valuet∗ = tiξ kξ

= mini{ti,k+1}, iξ being the index of the unit to be inspected first.Step5.15.Cancel Step 5.13 and simulate speedviξ j1.Step5.16.ObserveV f

iξ(tiξ kξ

) at inspection pointtiξ kξ.

Step5.17.Accumulate the operating and inspection costs for unitUξ in a special arrayC{Uξ}. Note that at Step

5.17 the costs are not forecasted, as at Step 5.10, but calculated according to the simulated speed, i.e. byusing(tiξ kξ

− tiξ kξ −1).Step5.18.If at momenttiξ kξ

production planViξ temq

has not been accomplished, seti=iξ , k=kξ and go to Step 5.2.If the production plan for unitViξ is accomplished, an emergency call is declared, since the remainingtarget has to be rescheduled anew among all production units. Go to Step 5.19.

Step5.19.Inspectall production unitsUi at point tiξ kξ

and, thus, obtainVtiξ kξ. If Vtiξ kξ

=V the target amount iscompleted. Go to Step 11. Otherwise apply the next step.

Step5.20.Sett = tem

q+1 = temq + tiξ kξ

. Go to Step 5.21.Step5.21.Restore Step 5.13 and go to Step 3 at the section level.Step6.Applying this step means, that for one of production unitsUiη at the routine control pointtiηkη

problems(6)–(12) cannot be solved. Thus, pointtiηkη

is an emergency point. Setξ=η, and go to Step 5.19.Step7.This step is applied in the case when problems (1)–(5) at the section’s level cannot be solved, i.e. even

a feasible solution{pit} honoring Eq. (2) cannot be obtained.Introduce at the corresponding emergency pointtem

q maximal speeds for all production units untilsimulate the latter and proceed manufacturing until the next control pointtem

q+1 = temq + d for all units.


Step8.Inspect at pointtem

q+1 all the actual outputsV fi (tem

q+1) and the accumulated costs.Step9.DetermineVtem

q+1= ∑

iVfi (tem

q+1).Step10.If Vtem

q+1=V, apply the next step. Otherwise sett = tem

q+1 and go to Step 3.Step11.Using arraysC{Uξ}, 1≤ξ≤n, calculate valueC via Eq. (3). Thus,C determines the total expenses

within a routine simulation run.Step12.The two-level control algorithm terminates.Thus, the algorithm enables proceeding the manufacturing process until production planV will be

accomplished.The algorithm is performed in real time, although it is based on numerous forecasting optimization

problems. Each iteration of the algorithm can be done only after valueVf (tik) is actually observed.However, the efficiency of the algorithm can be examined by simulating speedsvij in each interval[tik, ti ,k+1]. This means that to justify the algorithm one has to simulate Steps 1–11 many times in orderto obtain representative statistics and the system’s outcome parameters.

Note, in conclusion, that in the case the section’s target amount is completedbeforethe due date itis not easy to determine the actual momentF exactly. This, in turn, makes it difficult to determine thestorage expenses. We suggest a simple approximate model as follows.

Consider theearliestinspection moment when the section’s observed output exceeds production planV. Call such a momentF∗, together with the correspondent observed outputV∗≥V. ValueF can be easilycalculated via linear interpolation

F = V

V ∗ F ∗. (18)

Thus, the storage charges can be calculated by[D − V

V ∗ F ∗]

C∗∗∗. (19)

5. Numerical example

A numerical example is outlined below to illustrate a simulation run for a two-level cost-optimizationproduction control model.

A section comprises three different production units with five possible speeds, which are subject todisturbances with uniform density probability functions. The speeds’ parameters are as follows:

Unit 1:

v11 = U(0.5, 1.2) C11 = 10v12 = U(2, 2.8) C12 = 20v13 = U(2.9, 3.8) C13 = 40v14 = U(3.5, 4.5) C14 = 60v15 = U(4, 5) C15 = 70


Unit 2:

v21 = U(0.1, 1.2) C21 = 15v22 = U(1, 2.5) C22 = 25v23 = U(2.4, 3.5) C23 = 40v24 = U(2.5, 3.9) C24 = 50v25 = U(3, 5) C25 = 60

Unit 3:

v31 = U(0.5, 1.2) C31 = 10v32 = U(1, 2) C32 = 20v33 = U(1.8, 2.8) C33 = 30v34 = U(2, 4) C34 = 45v35 = U(3.5, 5.5) C35 = 60

The system’s time parameters are: the due dateD=30,d=3 and1=3.The cost parameters areC∗=1000,C∗∗=100,C∗∗∗=10,Cem=20 andCins=10.Other parameters are as follows: target amount (production plan)V=300 and chance constraintp=0.75.The solution according to the heuristic algorithm outlined above (for one simulation run) is as follows:Stage1.t=0, V0=300.After rescheduling the target amountV0 along the units via Eq. (5) we obtain:V10=103.846,V20=92.308

andV30=103.846.Using a cyclic coordinate descent search at Step 4 (see Section 4) results in obtaining in a three-

dimensional space the optimized variablespi0 as follows:p10=0.88,p20=0.87 andp30=0.88.Afterwards all units work independently, and production control centers on determining productions

speedvit and control pointstik (at momentt=0, valuek is set equal 0). The corresponding subalgorithmis outlined in Section 4 (see Steps 5.1–5.13). For each unitUi we obtain the corresponding valuez, theoptimal couple (j1, j2) and the next inspection pointti1, as follows:

Unit 1: z=4, (j1, j2)=(2, 5),t11=11.695.Unit 2: z=5, (j1, j2)=(2, 5),t21=3.49.Unit 3: z=5, (j1, j2)=(3, 5),t31=5.75.Thus, the next inspection point to be observed is the minimum of 11.695, 3.49 and 5.75, i.e. is equal to

3.49.Stage2.t21=3.49. The simulated product of Unit 2 within the period [0, 3.49], taking into account the simulated

value of speedv22 = 1.85 at pointt20, is V f2 (3.49) = 3.49× 1.85 = 6.45. The accumulated costs within

the period [0, 3.49] for Unit 2 are 3.49×C22+Cins=97.25.After inspecting Unit 2 at pointt21 we determine the next control pointt22 and the speed to proceed

with up to that point:z=5, (j1, j2),=(4, 5),t22=9.3. Thus, the next inspection pointt31 is calculated as theminimum of valuest11=11.695,t22=9.3 andt31=5.75.

Stage3.t31=5.75. The simulated product of Unit 3 within the period [0, 5.75], taking into account the simulated

value of speedv33 = 2.55 at pointt30, isV f3 (5.75) = 5.75×2.55 = 14.66. The corresponding accumulated

costs within the period [0, 5.75] for Unit 3 are 5.75×C33+Cins=5.75×30+10=182.5.


Using Steps 5.1–5.13 of the two-level algorithm results in determining the next control pointt32 andthe speed to proceed with up to that point is as follows:z=5, (j1, j2)=(5, 5),t32=D=30.

Thus, the next inspection point ist22=9.3, since 9.3=min(11.695, 9.3, 30)Stage4.t22=9.3. The simulated product of Unit 2 within the period [0, 9.3], taking into account valueV f

2 (3.49)=6.45and the simulated value of speedv24 = 2.6 at pointt21=3.49, is equal.

V f2 (9.3) = 6.45+ (9.3 − 3.49) × 2.6 + Cins = 21.556.

The accumulated costs for Unit 2 at pointt=9.3 are 97.25+(9.3−3.49)×50+Cins=397.75.Unfortunately, the algorithm fails to obtain the corresponding valuez, as well as the next control point

t23 and the speed to proceed with up to that point. Thus,t23 is an emergency point (see Step 5.9), and Step6 has to be applied.

Stage5.At point t=9.3 the observed, i.e. simulated production output values for all production units are as

follows:V f

1 (9.3)=0+9.3×2.68=24.92, where 2.68 is the simulated value of speedv12, which has been intro-duced at pointt10.

V f2 (9.3) = 21.556, V f

3 (9.3) = 14.66+ (9.3 − 5.75) × 4.12 = 29.29,

where 4.12 is the simulated value of speedv35, which has been introduced at pointt31=5.75.The corresponding accumulated costs are as follows:

C f1(9.3) = C12 × 9.3 + Cins = 9.3 × 20+ 10 = 196, C f

2(9.3) = 397.75,

C f3(9.3) = 182.5 + (9.3 − 5.75) × C35 + Cins = 182.5 + 3.55× 60+ 10 = 405.5.

Stage6.After inspecting all production units at the emergency pointt=9.3 the remaining target amount has to

be rescheduled among the units via Eq. (5). Thus, we obtain fort=9.3

V1t = 77.6, V2t = 68.9, V3t = 77.6.

Stage7.At this stage, Step 4 of the two-level algorithm has to be applied in order to determine quasi-optimal

valuespit , 1≤i≤3, t=9.3. However, the algorithm fails to obtain a quasi-optimum solution of problems(1)–(5), i.e. a feasible solution honoring chance constraint (2) cannot be found. According to the algorithmStep 7 has to applied.

Stage8.At point t=t12=t22=t32=9.3 we introducemaximalspeedsv15, v25 andv35 for all production units

until the next control pointt=9.3+d=12.3. Pointt=12.3 is an emergency point too. The correspondingsimulated speeds are as follows: 4.55 (for Unit 1), 3.54 (for Unit 2) and 5.3 (for Unit 3). Thus, at pointt=12.3 the system has to be inspected anew.

Stage9.At point t=12.3, which is an inspection point for all units (12.3=t12=t23=t33), we inspect the accu-

mulated outputs for all production units, together with the corresponding accumulated costs. Thus, weobtain


V f1 (12.3) = 24.92+ 3 × 4.55 = 38.574, V f

2 (12.3) = 21.556+ 3 × 3.54 = 32.176,

V f3 (12.3) = 29.29+ 3 × 5.3 = 45.19.

C f1(12.3) = 196+ 10+ 3 × C15 = 416, C f

2(12.3) = 397.75+ 10+ 3 × C25 = 587.75,

C f3(12.3) = 405.5 + 10+ 3 × C35 = 595.5.

Stage10.After inspecting the system at pointt=12.3 we reschedule the remaining target amount among all

production units via Eq. (5) and, thus, obtain

V1t = 63.71, V2t = 56.64, V3t = 63.71.

Stage11.After rescheduling the remaining target amount the algorithm centers on realizing Step 4, i.e. determin-

ing the quasi-optimal valuespit , t=12.3, 1≤i≤3, honoring chance constraint (2). The algorithm succeedsto obtain the following values:

p1t = 0.87, p2t = 0.88, p3t = 0.87.

Thus, each production unitUi starts working independently in accordance with the assigned input pa-rametersVit andpit .

The corresponding indices of the speedz, optimal couples (j1, j2) and next inspection pointti ,k+1, foreach unit are as follows:

Unit 1: z=4, (j1, j2)=(2, 5),t13=17.5.Unit 2: z=5, (j1, j2)=(2, 5),t24=30.Unit 3: z=5, (j1, j2)=(3, 5),t34=30.Thus, the closest inspection point ist13=17.5.Stage12.At point t=17.5 the observed, i.e. the simulated accumulated production output valueV f

2 (17.5), is asfollows:

V f1 (17.5) = V f

1 (12.3) + (17.5 − 12.3) × 2.37 = 50.9.

Here 2.37 is the simulated value of speedv12, which has been introduced at pointt=12.3. The accu-mulated cost valueC f

1(17.5) isC f1(12.3)+(17.5−12.3)×C12+Cins=416+5.2×20+10=530.

The next inspection pointt14=20 is determined forz=4, (j1, j2)=(4, 5). Thus, speedv14 has to beintroduced until the next inspection point 20.5. It can be well-recognized thatt14 is the control point ofthe unit to be inspected first.

Stage13.At the inspection pointt=20.5 the results of the inspection are as follows: since the simulated value of

speedv14 is 4.22, the accumulated outputV f1 (20.5) is 50.9+4.22×3=63.55. The accumulated costs are

C f1(20.5) = C f

1(17.5) + C14 × 3 + Cins = 530+ 60× 3 + 10 = 720.

The index of the speedz=5, while the quasi-optimal couple (j1, j2) is (5, 5). Thus, the next (and thelast) inspection point for all units ist15=t24=t34=30.


Stage14.Note that at the pointt=20.5 the simulated value of speedv15 is 4.74, while at the pointt=12.3 the

simulated valuesv25 andv35 are 3.48 and 4.91, respectively. Thus, the observed accumulated outputs atthe due date are as follows:

V f1 (D) = V f

1 (20.5) + (30− 20.5) × 4.74 = 108.6,

V f2 (D) = V f

2 (12.3) + (30− 12.3) × 3.48 = 93.77,

V f3 (D) = V f

3 (12.3) + (30− 12.3) × 4.91 = 132.1.

The corresponding accumulated costs are as follows:

C f1(D) = C f

1(20.5) + (30− 20.5) × C15 + Cins = 720+ 9.5 × 70+ 10 = 1395,

C f2(D) = C f

2(12.3) + (30− 12.3) × C25 + Cins = 587.75+ 17.7 × 60+ 10 = 1659.75,

C f3(D) = C f

3(12.3) + (30− 12.3) × C35 + Cins = 595.5 + 1072= 1667.5.

Thus, the total outputVf (D) is 108.6+93.77+132.1=334.47.The total costs for our example are as follows:

C =3∑

i=1

C fi (D) + CemNem + (D − F)C∗∗∗.

HereNem=3 since target amount rescheduling to determine optimal valuespit has been undertaken at threeemergency points within the simulation run:t=0,t=9.3 andt=12.3. SinceF∗=D with V∗=334.47>V=300,valueF can be determined via Eq. (18)

F = D 300334 ≈ 26.95.

Thus, using Eq. (3), we finally obtain

C = 1395+ 1659.75+ 1667.5 + 20× 3 + 10× (30− 26.95) = 4812.75.

The section completes its production plan with three target amount rescheduling and 13 inspections ofproduction units.

6. Experimentation

In order to evaluate the performance of the algorithm, various examples were run. The experimentaldesign is given in Table 1. Other parameters are similar to those outlined in Section 5. Four parametersare varied, namely,p, d (and1), V and the distribution ofvij . Thus, the controlled section comprisesthree production units with five possible speeds. Three distributions ofvij are considered:1. vij is a random value uniformly distributed in the interval [aij ,bij ].2. vij is a random value normally distributed with the mean valueµij=0.5(aij ,bij ) and with the variance

σ 2=(1/36)(bij−aij )2 (a truncated distribution).


Table 1The experimental design

Variable Values given in the experiment Number of values

The least permissibilityp in the chance constraint 0.75, 0.85 2Distribution ofvij Uniform, normal, beta 3The target amountV 350, 365 2The minimal time span valuedd=1 3, 5 2

3. vij has a beta distribution density function

pij (v) = 12

(bij − aij )4(v − aij )(bij − v)2, (20)

which has been recently used in many publications on production control under random disturbances,e.g. in [1,7,9].

It can be well-recognized that using distribution (20) results in lower production speeds than in the case ofa normal or an uniform distribution. This is because the average value of the asymmetric beta distribution(20) is by 0.1(bij−aij ) lower than the corresponding average values 0.5(aij+bij ) for symmetric normaland uniform density functions.

Note that all pregiven average processing costsCij , 1≤i≤n, 1≤j≤m, depend only on the couple ofvalues (aij ,bij ), but not on the type of distribution in the interval [aij ,bij ].

A total of 24 combinations (2×3×2×2) were considered. For each combination 1000 runs were carriedout. Several output measures were considered as follows:C the optimal average value of total expenses within one simulation run;p the average actual probability of meeting the due date on time;Nins the average number of inspection points for all units within one simulation run;Nem the average number of emergency points (withoutt=0);sik the average index of the speed introduced by the decision-maker at a routine control pointtik

(within one simulation run);pi the average quasi-optimal chance constraint values for each production unit determined at a

routine emergency momentt, 0≤t≤T (within one simulation run).

Note that valuesik for one simulation run is calculated as follows:

sik = 1

nF

n∑i=1

Ni−1∑k=0

[(tik+1 − tik)sik].

The summary of the experimentation is shown in Table 2. The following five conclusions can be drawnfrom the summary:1. Increasing valuep results in increasing the optimal average valueC, together with valuessik, Nins and

Nem.2. The average actual probabilityp of meeting the due date on time for all types of distributions exceeds

its corresponding pregiven chance constraint valuep. Thus, the control algorithm minimizes objectiveC with respect to Eq. (2).


Table 2The summary of the experimentation

Distribution p d V p1 p2 p3 p C sik Nins Nem

Uniform 0.75 3 350 0.61 0.62 0.60 0.82 5660 4.32 14.57 2.87365 0.62 0.62 0.60 0.76 5991 4.59 14.89 3.32

5 350 0.57 0.56 0.55 0.80 5688 4.35 8.5 1.14365 0.62 0.61 0.62 0.83 5943 4.68 9.47 1.83

0.85 3 350 0.68 0.66 0.69 0.88 5682 4.43 15.08 3.06365 0.67 0.68 0.67 0.92 5959 4.69 21.47 5.35

5 350 0.63 0.63 0.62 0.90 5626 4.46 8.89 1.31365 0.65 0.66 0.66 0.85 5991 4.75 12.5 2.79

Normal 0.75 3 350 0.48 0.49 0.49 0.81 5451 4.11 9.77 1.34365 0.63 0.63 0.63 0.81 5842 4.49 8.99 1.45

5 350 0.47 0.47 0.46 0.77 5570 4.21 8.23 1365 0.47 0.47 0.47 0.78 5849 4.48 7.29 0.79

0.85 3 350 0.64 0.63 0.64 0.85 5564 4.18 10.02 1.7365 0.66 0.67 0.65 0.9 5777 4.49 8.16 1.2

5 350 0.56 0.55 0.57 0.86 5538 4.31 7.57 0.84365 0.61 0.60 0.62 0.90 5781 4.57 6.57 0.73

Beta 0.75 3 350 0.58 0.58 0.56 0.78 5859 4.47 10.75 1.93365 0.64 0.65 0.64 0.81 6283 4.93 20.81 5.47

5 350 0.51 0.50 0.50 0.77 5888 4.50 7.46 0.92365 0.64 0.62 0.63 0.77 6299 4.96 13.97 3.41

0.85 3 350 0.67 0.68 0.66 0.89 5857 4.59 12.79 2.73365 0.73 0.73 0.71 0.87 6376 4.94 28.11 7.65

5 350 0.60 0.61 0.63 0.88 5814 4.60 7.49 1.12365 0.91 0.92 0.92 0.87 6297 4.98 18.11 4.95

3. Using the normal distribution yields lower total cost expensesC than by using the uniform or the betadistributions. Using the beta distribution results in the highest total cost expenses. Thus, practicallyspeaking, the normal distribution enables the cheapest output’s manufacturing while beta distribution(20) is the least effective one.

4. Increasing valuesd and1 results usually in decreasing the efficiency of the production control, e.g.in increasing valuesC, Nem, Nins andsik.

5. With the exception of one combination (beta distribution,p=0.85,V=365,d=5) average valuespi oflocal chance constraints are usually smaller than the pregiven probability valuep. This phenomenonoccurs thanks to the decision-making outlined at Step 7 of the two-level algorithm (see Section 4).The decision-making enables using non-exhausting speeds at non-emergency inspection points incombination with introducing maximal speeds for all production units in case of critical emergencysituations.

7. Conclusions and future research

The following six conclusions can be drawn from the study:


1. The two-level control model under consideration is a further development in the area of productioncontrol. The suggested control algorithm enables meeting the target amount on time subject to a chanceconstraint for two-level man-machine production systems under random disturbances.

2. The developed control model can be used for various semi-automated production systems [1,9] underrandom disturbances where the outputs (target amounts) can be measured only at pregiven controlpoints and are gauged by a single measure (money terms, cubic meters, percentage completion ofthe project, etc.). For example, when a building construction company performs a building project,units are teams, which can work with several alternative speeds, while the output of a team is usuallymeasured in percentage completion of the project.

3. The system’s target amounts are transferable and may be rescheduled among the production units.4. For all previously developed multilevel semi-automated production systems under random distur-

bances the risk average principle has been implemented in the control model at the lower level[1,3–6,8–13]. Those models do not deal with chance constraints. The model under considerationis based on another principle, namely, the chance constraint principle [2,7] which is very effective forcost objectives.

5. The two-level cost-optimization control algorithm minimizes the system’s total expenses subject to achance constraint, while the pregiven probability value of the chance constraint is the input value ofthe algorithm. Similar results for multilevel production systems have not been published as yet.

6. Future research may be undertaken to develop three-level production control models, e.g. factories,etc., with chance constraints at each hierarchical level.

8. Nomenclature

aij lower bound of random speedvij (pregiven)bij upper bound of random speedvij (pregiven)C the total operational costs, penalties and charges accumulated for the system

in the course of accomplishing the production plan (a random value)Cem the average cost of rescheduling the remaining target amountVt among units

Ui by the section at a routine emergency momentt≥0Cij the average processing cost per time unit of speedvij , 1≤i≤n, 1≤j≤m

(pregiven); note that for a fixedi relationj1≤j2 results inCij1 < Cij2

Cins the average cost of performing a single inspection of a production unit(pregiven, equal for all units)

C fi (t) the actual accumulated processing and inspection costs calculated at momentt

for unit Ui , 0≤t≤D, 1≤i≤n, C fi (0) = 0

C∗ the penalty paid to the customer by the section for not accomplishing thetarget amount on time, i.e. whenF>D (a single payment, pregiven)

C∗∗ the penalty cost for each time unit of delayF−D (pregiven)C∗∗∗ storage charges per time unit for the target amount’s completion before the

due date (pregiven)d the minimal given time span between two consecutive control pointstik

andti ,k+1 (pregiven and equal for all units)D the due date (pregiven)


Dt the length of the remaining planning horizon at momentt, Dt=D−tF the actual moment the target amount is completed (a random value)hi the search step (increment) for determining optimal valuespit

m number of possible speeds (common to all units)n number of production unitsNem the number of emergency moments (a random value)Ni the number of control points of unitUi

p the chance constraint, i.e. the minimal permissible confidenceprobability of accomplishing the production plan on time (pregiven)

pit the chance constraint value for production unitUi determined at theemergency momentt≥0, 1≤i≤n (to be determined as an optimized variable)

sik the index of the speed chosen by the decision-maker at the control pointtiktik thekth inspection moment (control point) of unitUi , k=0, 1,. . . ,Ni

temq theqth emergency moment at the section level, 1≤q≤Nem (random value)

Ui the ith production unit entering the section, 1≤i≤nvij the jth speed of unitUi to manufacture the product, 1≤j≤m (a random value

with pregiven density functionfij (v))vij the average of speedvij . It is assumed that for each production unitUi speeds

vi1, vi2, . . . , vim are sorted in ascending order of their average values andare independent oft. Thus, valuevim is the maximal average value of unitUi

V the pregiven section’s production plan gauged by a single measure(target amount)

V f (t) = ∑ni=1V

fi (t) the actual section’s output observed at momentt (a random value)

Vit the production plan assigned to unitUi at the emergency pointt (to bedetermined); note that

∑iVit = Vt

V fi (t) the actual output of unitUi observed at momentt, 0≤t≤D; V f

i (0) = 0(a random value)

Vt the section’s remaining production plan at momentt, V0=VWp[V f

i (t), Vit , j ] thep-quantile of the moment production planVit will be completed onconditions that: (a) speedvij is introduced for unitUi at momentt and will beused throughout, and (b) the actual observed output of unitUi at momenttis V f

i (t)

1 the minimal value of the closeness of the inspection momenttik to the due date(pregiven and equal for all units)

Acknowledgements

This research has been partially supported by the Paul Ivanier Center on Robotics and ProductionManagement, Ben-Gurion University of the Negev, Beer-Sheva, Israel.

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