Valeria Romanelli

Julian Nevado

Mario Fraga

Alex Martín Trujillo

Maria Ángeles Mori

Luis Fernández

Guiomar Pérez De Nanclares

Víctor Martínez-Glez

Guillermo Pita

Heloisa Meneses

Julián Nevado

Ricardo Gracia

Sixto García-Miñaur

Purificación García De Miguel

Beatriz Lecumberri

José Ignacio Rodríguez

Anna González Neira

David Monk

M.D Pablo Lapunzina
email: plapunzina.hulp@salud.madrid.org

Constitutional Mosaic Genome-Wide Uniparental Disomy due to Diploidization: an Unusual Cancer-Predisposing Mechanism

Keywords: Beckwith-Wiedemann syndrome, paternal uniparental disomy, diploidization, imprinting

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Beckwith-Wiedemann syndrome (BWS; [MIM 130650]), one of the commonest overgrowth syndromes, is characterized by pre and postnatal overgrowth, hemihyperplasia, macroglossia and an increased frequency of embryonic tumors [START_REF] Beckwith | Extreme cytomegaly of the adrenal fetal cortex, omphalocele, hyperplasia of kidneys and pancreas, and Leydig-cell hyperplasia: Another syndrome? Western Society for Pediatric Research[END_REF][START_REF] Lapunzina | Risk of tumorigenesis in overgrowth syndromes: a comprehensive review[END_REF][START_REF] Wiedemann | Familal malformation complex with umbilical hernia and macroglossia: A "new syndrome[END_REF]. It is caused, in up to 90% of patients, by a variety of genetic or epigenetic alterations within two imprinting sub domains located on human chromosome 11p15 [4;5]. The largest molecular subgroup (~60-70% of BWS cases) is comprised of patients carrying an epigenetic error at either the H19DMR (imprinting control region, ICR1) or KvDMR (ICR2) [START_REF] Weksberg | Beckwith-Wiedemann syndrome demonstrates a role for epigenetic control of normal development[END_REF], the regions of differential DNA methylation that regulate the appropriate allelic expression of the paternal IGF2 and maternal CDKN1C genes respectively. Patients may also have paternal uniparental disomy (patUPD11p15) (~10-15%), i.e. two paternally derived copies of chromosome 11p15 and no maternal contribution, that results in over-expression of the potent mitogen IGF2, and absence of the cell cycle regulator CDKN1C [START_REF] Henry | Uniparental paternal disomy in a genetic cancer-predisposing syndrome[END_REF]. Chromosome rearrangements comprising maternal translocations/inversions or paternal duplications are rare (~2-3% of cases) as are CDKN1C mutations (~5-7 %).

Loss of heterozygosity (LOH) may be due either to haploinsufficiency (deletion leading to the lack of contribution of one allele) or uniparental disomy (UPD) when both copies of a chromosome pair originate from one parent. LOH is a common mechanism of tumorigenesis where it indicates the absence of a functional tumor suppressor gene in the lost region. LOH has been described in almost all cancer types [START_REF] Tuna | Uniparental disomy in cancer[END_REF]. Localized loss of imprinting (LOI) is a common consequence of LOH due to the lack of balanced parental contribution in specific region with imprinting either by deletion/duplications or UPD. As stated above, LOH/LOI due to patUPD11p15 is relatively common in BWS and this molecular subgroup of individuals displays one of the highest risks of cancer [START_REF] Cooper | Molecular subtypes and phenotypic expression of Beckwith-Wiedemann syndrome[END_REF].

Diploidization means the recovery of the normal diploid human chromosome number (2n=46) from an initially haploid cell (n=23) or triploid zygote (3n=69). Diploidization is a relatively common mechanism in plants and non-mammalian species but it has been unfrequently observed in humans [START_REF] Golubovsky | Postzygotic diploidization of triploids as a source of unusual cases of mosaicism, chimerism and twinning[END_REF][START_REF] Morales | Reproductive consequences of genome-wide paternal uniparental disomy mosaicism: description of two cases with different mechanisms of origin and pregnancy outcomes[END_REF][START_REF] Reed | Androgenetic/biparental mosaicism in an infant with hepatic mesenchymal hamartoma and placental mesenchymal dysplasia[END_REF][START_REF] Wilson | The clinical phenotype of mosaicism for genome-wide paternal uniparental disomy: two new reports[END_REF].

We here report on a woman with BWS phenotype and tumors with almost complete LOH in all chromosomes, showing a near-complete contribution of the paternal genome for the diploid status. Postzygotic diploidization due to failure of maternal DNA replication followed by paternal genome endoreduplication would be the most plausible explanation for the observed findings. Thus, present patient would represent a further example of diplodization in humans.

The diagnosis of Beckwith-Wiedemann syndrome was established at birth in a girl of healthy and non-consanguineous parents because of macrosomia, overgrowth, hypoglycaemia and hemihypertrophy. The patient is included as part of a Research Program in the Spanish Overgrowth Syndrome Registry (IRB-HULP PI446). At birth she weighed 3750 g, and showed capillary malformations in legs, hands and thorax, contracture of fingers, facial asymmetry and hemihypertrophy of left hand and leg. She continued with hypoglycaemia and needed treatment with diazoxide for years. At 16 months she had episodes of convulsions needing valproic acid treatment. She started to walk at 19 months. Language was delayed and her school performance was lower than average for age. At age 9 years she underwent pancreatic resection due to intractable, symptomatic hypoglycaemias. At age 4 years 5 months a grade II mesenchymal Wilms tumor was diagnosed and received a combination of complete nephrectomy (including right adrenalectomy) and 9 months of chemotherapy. She also had renal stones (calcium) needing close follow-up for her unique kidney. Calcium, phosphate, alkaline phosphatase, and PTH were among normal ranges. She has an average height for Spanish standards (163.5 cm; 50 th centile). Skeletal surveys were normal except for the slight asymmetry due to hemihyperplasia. Her clinical outcome was then unremarkable until 20 years when an ectopic adrenocortical virilizing adenoma was suspected (and confirmed) due to hirsutism, secondary amenorrhea and the finding at MRI of an abdominal retroperitoneal mass, independent from the left adrenal gland. An excision surgery was followed by complete recovery. Six years later she remains asymptomatic, with normal adrenal function and regular menses. The patient is now 26 years old and works as administrative assistant.

Standard karyotyping along with single nucleotide polymorphism (SNP) and HumanMethylation 27 Infinium arrays, microsatellites, Methylation Specific-Multiplex Ligation dependent Probe Amplification (MS-MLPA), Methylation Sensitive-High Resolution Melting Assay (MS-HRMA), pyrosequencing, and array-CGH were performed on patient's samples (see Supplementary methods).

The diagnosis of BWS due to patUPD11p15 was suspected upon microsatellite analysis in the patient and her parents and confirmed through MS-MLPA (Fig. 1A,B). As expected, MS-HRMA and pyrosequencing of specific CpG dinucleotides in the KvDMR and H19DMR, further delineated the degree of aberrant methylation reflecting the percentage of the patUPD11p15 (Fig. 1C-F).

SNP-arrays showed an abnormal, unforeseen, unusual pattern not only at chromosome 11 but also in all chromosomes. The Log R ratio (which evaluates the genomic dosage, e.g. deletions/duplications) was between normal ranges in all chromosomes but the allele frequency (evaluation of homozygosity/heterozygosity of each SNP) disclosed an almost complete drift (∼85% LOH) to homozygosity in all chromosomes (Fig. 2A and Supplementary Fig. 1).

Subsequent microsatellite analyses of several chromosomes confirmed in blood, skin, urine, buccal cells and some tissues (Wilms tumor, adrenal, ceccal appendix) that almost all of the genomic contribution was paternal with a minimum, low mosaic maternal contribution (Supplementary table 1). All tissue showed a similar pattern with different degree of mosaicism.

Calculation of the percentage of mosaicism was applied to some tissues and showed an average contribution of ∼85% from the paternal genome and ∼15% of maternal genome (Supplementary table 1). Analysis of microsatellites of X chromosome in the patient indicated that in most cells both X chromosomes came from the only paternal X chromosome with only a minimum fluorescence of maternal markers (∼15%) (Supplementary Fig. 2). SRY and amelogenin amplifications were indicative of absence of Y chromosome in the patient (data not shown).

Using MLPA targeting the Prader-Willi and Angelman syndrome (PW/AS) and Pseudohypoparathyroidism (PHP) loci, and the Illumina Infinium DNA-methylation BeadArray, we were able to clearly show the lack of appropriate allelic methylation in the patient at 14 DMRs located on 7 different autosomes. These included the maternally methylated SNPRN, NDN, GNAS XL, GNAS EX1A, GRB10, MEST, PEG10, PEG3, ZAC/PLAGL1, L3MBTL, KvDMR/KCNQ1 and DIRAS3, as well as the paternally methylated NESP and H19 DMRs (Fig. 2B). These results confirm the methylation profiles determined by both the MS-MLPA and MS-HRM assays.

There have been scant reports of genome-wide paternal UPD in the literature [START_REF] Kotzot | Complex and segmental uniparental disomy updated[END_REF] and as far as we know our case is the only adult patient reported to date. Three patients with proved genome-wide paternal UPD developed pediatric tumors (pheochromocytoma, hepatoblastoma and pancreatic tumor) [13;15] and another two individuals without tumors have been recently reported [11;16]. These cases presented with combined features consistent in part with known imprinting disorders associated with paternal UPD of specific chromosomes, but the most pronounced phenotypic features in all cases resembled patUPD11/BWS. Other authors have described genome-wide paternal UPD confined to specific tissues but not present in blood [12;17] or exclusively confined to placenta and not to fetal tissues [START_REF] Kaiser-Rogers | Androgenetic/biparental mosaicism causes placental mesenchymal dysplasia[END_REF][START_REF] Makrydimas | Complete hydatidiform mole and normal live birth: a novel case of confined placental mosaicism: case report[END_REF][START_REF] Robinson | Origin and outcome of pregnancies affected by androgenetic/biparental chimerism[END_REF][START_REF] Surti | Twin pregnancy with a chimeric androgenetic and biparental placenta in one twin displaying placental mesenchymal dysplasia phenotype[END_REF]. Although, the percentage of uniparental cells has not been calculated in all reports the patient presented herein showed one of the highest mosaicism (∼85% in blood, 65-87% in other tissues).

In fitting with these previous reports, our patient showed clinical features of some classical paternal UPD syndromes (Supplementary Fig. 4). She showed capillary malformation, hemihyperplasia, macroglosia, Wilms tumor and adrenocortical tumor (patUPD11), linear hypopigmentation, seizures and low-average mental development (patUPD15) and hypotonia (patUPD14). The observation of BWS phenotype in this woman is in agreement with our laboratory findings since patUPD11p15 is observed in about 15-20% of individuals with BWS. Furthermore, both tumors observed in the patient, Wilms tumor at 4 years and extra-adrenal adrenocortical tumor at age 20, are also prevalent in BWS. Somatic pat11pUPD15 has been encountered in about one third of Wilms tumors [START_REF] Scott | Constitutional 11p15 abnormalities, including heritable imprinting center mutations, cause nonsyndromic Wilms tumor[END_REF] and such a tumor is the commonest neoplasia observed in BWS patients encompassing about half of tumors reported in this disorder [START_REF] Lapunzina | Risk of tumorigenesis in overgrowth syndromes: a comprehensive review[END_REF].

Regional or partial LOH is observed in deletions or UPDs. In the present patient, deletion was ruled-out due to normal log R ratios in the CGH-array and SNP-array. In addition, almost complete LOH in all chromosomes observed by means of SNP-arrays, microsatellite analyses, MS-MLPA and MS-HRM is consistent with a mechanism of genome-wide UPD.

Potential mechanisms of genome-wide UPD are diploidization of an initial triploid foetus or a normal fertilization followed by a failure of replication and chromosome condensation of the maternal pronuclei with normal replication and segregation of the paternal chromosome (Supplementary Fig. 5). We discarded diploidization after dispermic triploidy as a possible mechanism because the microsatellite analysis in our patient revealed the involvement of a unique X-bearing sperm. We believe that the most probable and simplest explanation for our case is normal fertilization followed by failure of maternal DNA replication and paternal genome endoreduplication , resulting in androgenetic/biparental mosaicism, with over-representation of the genome-wide paternal UPD cell lines (Supplementary Fig. 5E). This diploidization, the recovery of the normal diploid human chromosome number (2n= 46) is common in plants and is in some ways similar to "trisomy rescue" observed in UPD, when trisomic cells (47 chromosomes) recover to normality loosing the extra chromosome in the next division. The pathological consequences of UPD are identical for a single chromosome as a results of trisomy rescue or for the entire genome due diploidization.

Differences in gene expression between the allele inherited from the mother and that inherited from the father are the result of genomic imprinting, and it is the most frequent molecular aberration in BWS. Thus, BWS patients carrying a mosaic paternal UPD of chromosome 11p15 have increased risk of cancer, mainly Wilms tumor, hepatoblastoma and others embryonal tumors [2;23]. Further, the frequency of UPD detected in neoplastic tissues varies from 3 to 100% [START_REF] Tuna | Uniparental disomy in cancer[END_REF]. The patient reported herein developed two different neoplasias: Wilms tumor and adrenocortical neoplasia. Consistent with this observation, chromosome 11p15 UPD is observed in 30-40% and 32% of Wilms and adrenocortical tumors, respectively [22;24].

Aberrant allelic methylation at all loci analyzed was demonstrated in the patient by means of MS-MLPA, MS-HRMA, pyrosequencing and methylation array (Fig. 1B-F and2B).

Analysis of the BWS, PW/AS and PHP regions revealed a methylation level partially abnormal due to the mosaic status; such an aberrant methylation led to abnormal imprinting of these loci. Thus, our patient showed typical clinical findings of BWS but not typical features of AS, PHP or transient neonatal diabetes mellitus (TNDM), demonstrating that partial loss of imprinting is sufficient for developing BWS but not sufficient for others disorders which usually requires total LOI to manifest the clinical features of the diseases. In addition, the lack of atypical BWS phenotypic features also suggests that inappropriate methylation at the MEST, L3MBTL, GRB10, PEG3, and DIRAS3, as revealed by the methylation array, is not associated with obvious dysmorphologies. However, the resultant increase in expression of the paternally expressed imprinted genes, due to biallelic expression, and the loss of maternally expressed transcripts maybe involved in tumorigenesis in this patient.

Because of the patient had two different cancers and she carried genome-wide paternal UPD, she may theoretically have increased risks for additional UPD-related tumors [START_REF] Tuna | Uniparental disomy in cancer[END_REF]. Thus, her prospective follow-up raised some important questions because there are no published recommendations for clinical evaluation schedule and survey for these individuals. Classic recommendations for cancer surveillance in general population may or may not apply for her, since the genetic background is quite different and any cancer surveillance schedule in this patient should involve the whole body. Finally we empirically opted to evaluate this woman with regular (twice a year) clinical evaluation together with annual body MRI (including breasts), colonoscopy and a routine laboratory work-up including WBC and serological markers for cancer.

The demonstration of genome-wide paternal UPD has important implications for medical genetics, counseling and clinical oncology. Genomic technologies such as array-CGH, highdensity methylation and SNP-arrays should be incorporated in diagnostic workflows of patients with suspected imprinting/UPD disorders presenting with dysmorphic features and/or cancer.

Tumor surveillance in these patients is empiric since neither guidelines are available yet nor enough experience has been accumulated for guiding the follow-up. Finally, the mechanism of diploidization should be taken into account in patients with genetic disorders and cancer because diploidization of triploid or haploid fetuses is now a recognizable, emerging mechanism with potential risk of UPD and consequently, high risk of neoplasia.

Figure 1 .

 1 Figure 1. Microsatellites, MS-MLPA, MS-HRM and CpG pyrograms of patient's blood sample.

Figure 2 .

 2 Figure 2. SNP-array of chromosome 11 and methylation profile using Illumina Bead

Summary

Molecular studies in a patient with Beckwith-Wiedemann syndrome phenotype who developed two different tumors revealed an unexpected observation of almost complete loss of heterozygosity of all chromosomes. We demonstrate, by means of numerous molecular methods that the absence of maternal contribution in somatic cells is due to high-degree (∼85%) genomewide paternal uniparental disomy. Our observations indicate the genome-wide UPD results from diploidization, and have important implications for genetic counseling and tumor surveillance for the growing number of UPD-associated imprinting disorders.

Competing Interests: None to declare


Mahmoud R Khoshnoud
email: mahmoud.khoshnoud@karolinska.se

Britta Löfdahl

Helena Fohlin

Tommy Fornander

Olle Stål

Lambert Skoog

Jonas Bergh

Bo Nordenskjöld

Immunohistochemistry compared to cytosol assays for determination of Estrogen receptor and prediction of the long term effect of adjuvant tamoxifen

Keywords: breast cancer, estrogen receptor, tamoxifen, cytosol, Immunohistochemical

Immunohistochemistry compared to cytosol assays for determination of estrogen receptor and prediction of the long-term effect of adjuvant tamoxifen

Introduction

Estrogen (ER) and progesterone (PGR) receptors are well-established predictors for the effect of endocrine treatment in both primary and metastatic breast cancer [START_REF] Goldhirsch | Meeting highlights: updated international expert consensus on the primary therapy of early breast cancer[END_REF][START_REF] Mcguire | An update on estrogen and progestrone receptors in prognosis for primary and advanced breast cancer[END_REF][START_REF] Group | Tamoxifen for early breast cancer: an overview of the randomised trials[END_REF]. The prognostic value for these receptors still is a matter of debate. Earlier studies reported a more favourable prognosis for patients with ER-positive tumors [START_REF] Crowe | Estrogen receptor determination and long term survival of patients with carcinoma of the breast[END_REF][START_REF] Maynard | Estrogen receptor assay in primary breast cancer and early recurrence of the disease[END_REF][START_REF] Knight | Estrogen receptor as an independent prognostic factor for early recurrence in breast cancer[END_REF],

while later studies showed that the early advantage for ER-positive breast cancers was not sustained over time [START_REF] Aamdal | Estrogen receptors and long-term prognosis in breast cancer[END_REF][START_REF] Donegan | Prognostic factors. Stage and receptor status in breast cancer[END_REF][9]. Furthermore, we previously reported a frail prognostic value for ER [START_REF] Khoshnoud | Long-term pattern of disease recurrence among patients with early-stage breast cancer according to estrogen receptor status and use of adjuvant tamoxifen[END_REF] and that the better outcome for patients with ER-positive tumors is obvious only for patients who received endocrine therapy.

Before approximately 1990 determination of ER and PGR was based on biochemical methods, which measured the level of receptors in cytosol initially by a ligand binding assay, later replaced by monoclonal antibody based assays. Cytosol assays were time consuming, expensive and required fresh tissue and measured receptor content in both cancer cells and normal cells in the tumor tissue. However an advantage of the cytosol method was that it offered quantitation of the receptor levels. On the other hand IHC has many advantages compared with cytosol methods. It is cheaper, faster and does not require fresh material. Furthermore, it is possible to perform on small amounts tumor cells, and ER and PGR are measured in cancer cells only.

A high concordance between biochemical and immunochemical methods was reported in several studies [START_REF] Pertschuk | Immunocytochemical estrogen and progestin receptor assays in breast cancer with monoclonal antibodies. Histopathologic, demographic, and biochemical correlations and relationship to endocrine response and survival[END_REF][START_REF] Stierer | Comparison of immunohistochemical and biochemical measurement of steroid receptors in primary breast cancer: evaluation of discordant findings[END_REF][START_REF] Zafrani | High sensitivity and specificity of immunohistochemistry for the detection of hormone receptors in breast carcinoma: comparison with biochemical determination in a prospective study of 793 cases[END_REF][START_REF] Chebil | Comparison of immunohistochemical and biochemical assay of steroid receptors in primary breast cancer--clinical associations and reasons for discrepancies[END_REF].

Immunohistochemistry and cytosol assays were shown to yield similar predictive and prognostic information [START_REF] Beck | Immunohistochemical detection of hormone receptors in breast carcinomas (ER-ICA, PgR-ICA): prognostic usefulness and comparison with the biochemical radioactive-ligand-binding assay (DCC)[END_REF][START_REF] Molino | Prognostic significance of estrogen receptors in 405 primary breast cancers: a comparison of immunohistochemical and biochemical methods[END_REF][START_REF] Fisher | Solving the dilemma of the immunohistochemical and other methods used for scoring estrogen receptor and progesterone receptor in patients with invasive breast carcinoma[END_REF], but the majority of these studies had a short follow-up and many patients received a range of therapies.

To our knowledge there are only few studies comparing these methods in an adjuvant randomized setting [START_REF] Fisher | Solving the dilemma of the immunohistochemical and other methods used for scoring estrogen receptor and progesterone receptor in patients with invasive breast carcinoma[END_REF][START_REF] Alberts | Comparison of estrogen receptor determinations by a biochemical ligandbinding assay and immunohistochemical staining with monoclonal antibody ER1D5 in females with lymph node positive breast carcinoma entered on two prospective clinical trials[END_REF][START_REF] Regan | Re-evaluating adjuvant breast cancer trials: assessing hormone receptor status by immunohistochemical versus extraction assays[END_REF].

The aim of this study was to compare the predictive value of ER measured in the cytosol of the unfixed tumor with ER status obtained by an immunohistochemical assay performed on archival tissue. For this, we used tumors from a well-defined randomized clinical adjuvant trial conducted by Stockholm Breast cancer Group where high quality data on long-term follow-up is available. We furthermore investigated the extent to which the methods of assessing the hormone receptors may affect trial conclusions.

Material and methods

Study design:

Patients with an operable invasive breast cancer entered in a controlled clinical study of adjuvant endocrine therapy conducted by the Stockholm Breast Cancer Study Group, the Stockholm Adjuvant Tamoxifen Trial (STO-3). The trial was approved by the local ethical committee and the details of this study were described previously [START_REF] Rutqvist | The Stockholm trial on adjuvant tamoxifen in early breast cancer. Correlation between estrogen receptor level and treatment effect[END_REF]. Postmenopausal women at the age of 70 or less with invasive breast cancer were randomly given tamoxifen postoperatively at a dose of 40 mg per day compared with no adjuvant endocrine therapy. A woman was considered postmenopausal if more than 6 months had emerged since her last menstrual period.

During November 1976 through June 1990, a total of 2,738 patients entered the trial. The trial included patients irrespective of hormone receptor content but prospectively collected data on ER status were available in 2012 (85%) patients. ER-status was absent mainly in patients with small tumors from which all tissue was used to verify the histophatological diagnosis of breast cancer.

Before randomization the patients were stratified according to stage of disease and primary therapy. 1780 patients (65%) with no lymph node metastases and a tumor diameter of 30 mm or less (established by histological examination) were classified as "low risk". In this group, 432 patients were treated with breast conserving surgery including axillary dissection plus radiation to the breast (50 Gy/ 5 weeks). The remaining 1,348 patients had a modified radical mastectomy.

Present study population:

For the purpose of this study, we selected the 1,780 "low risk" patients because they received no adjuvant chemotherapy known to affect recurrence free and overall survival. These patients were randomized between no systemic therapies versus tamoxifen in contrast to a high-risk group also given chemotherapy. The data on the estrogen receptor status was known for 1183 (66.5%) which was originally determined by cytosol assays. Tumor material was available for 1,001 of these 1,780 patients. From 912 of these 1001 tumors, tissue microarray (TMA) could be created and sections stained for estrogen receptor by IHC, for the 89 remaining patients the archival material was of poor quality or not representative ( e.g cancer in situ, normal cells, connective tissue ). 790 of 912 tumors were technically evaluable by IHC and the remaining was difficult to interpret because of too few tumor cells. Data on ER receptor by cytosol for 777 of 912 patients were available. Re-evaluation was performed on 683 patients for whom the ER status was known by both cytosol and IHC (Figure1).

Follow -up strategies:

Clinical visits took place once every 3 months during the first 2 years, every 6 months during 2-5 years and yearly thereafter. These visits routinely included a physical examination and an annual mammogram. Chest Xray, blood sample, bone scans, etc were performed if clinical symptoms and signs indicated a probable relapse. Disease recurrence was confirmed when possible by biopsy. However, bone and visceral metastases were sometime established on unequivocal radiological evidence. Recurrence was dated from the first evidence of relapse based on physical, histological, or imaging data. After recurrence, treatment was decided individually.

The current results were based on follow-up until December 31, 2005. The proportion of patients lost to followup before this date was < 1%.

ER determination:

The ER cytosol assays were performed in one laboratory in Stockholm, a participant in the Swedish national wide quality control programme for hormone receptor laboratories sponsored by the Swedish Cancer Society.

Representative tumor sample was collected immediately after surgery and transported to the laboratory in saline according to the guidelines in the study protocol. The samples were frozen and stored at -80 C for 2 weeks or less. After thawing, the content of ER was determined using isoelectric focusing (IF) on polyacrylamide gel as previously described [START_REF] Wrange | Isoelectric focusing of estradiol receptor protein from human mammary carcinoma -a comparison to sucrose gradient analysis[END_REF][START_REF] Wrange | Cytosol estradiol receptor in human mammary carcinoma: an assay based on isoelectric focusing in polyacrylamide gel[END_REF]. This technique was used for samples assayed before 1988. After 1988 an enzyme immunoassay (ElA) was used. Studies have showed that results with these techniques correlate well with those obtained using methods based on ultra centrifugation, dextran-coated charcoal, and monoclonal antibodies [START_REF] Wrange | Isoelectric focusing of estradiol receptor protein from human mammary carcinoma -a comparison to sucrose gradient analysis[END_REF][START_REF] Wrange | Cytosol estradiol receptor in human mammary carcinoma: an assay based on isoelectric focusing in polyacrylamide gel[END_REF][START_REF] Ferno | A comparison of two steroid receptor assays in breast cancer: dextran coated charcoal and isoelectric focusing[END_REF][START_REF] Ferno | Enzyme immuno assay of the estrogen receptor in breast cancer biopsy samples. A comparison with isoelectric focusing[END_REF]. The receptor values in this study were normalized to DNA content as measured by Burton to avoid the influence of the large variations in extracellular protein [START_REF] Burton | A study of the conditions and mechanism of the diphenylamine reaction for the colorimetric estimation of deoxyribonucleic acid[END_REF]. A tumor with a receptor content of ≥0.05 fmol/µg DNA was classified as ER positive. This cut-off point was chosen since a previous analysis indicated that it distinguished between patients who benefited from adjuvant tamoxifen and those who did not.

Immunohistochemistry:

The immunohistochemical staining was performed using the Ventana automatic BenchMark machine (Ventana Medical Systems, S.A, lllkirch, Cedex, France). A rabbit monoclonal ER antibody (Sp1 250) was used. Antigen retrieval and staining procedure were performed according to the Ventana instructions.

Statistics:

Overall (OS) and recurrence-free survival (RFS) was estimated by the Kaplan-Meier method. The end-point in calculations of RFS was loco-regional recurrence, distant recurrence or death due to breast cancer. Hazard ratios (HR) were estimated using the Cox's proportional hazards model. The pattern of the treatment effect of tamoxifen for different ER values of both cytosol and IHC was studied. For this purpose, Subpopulation Treatment Effect Pattern Plots (STEPP) analysis [START_REF] Bonetti | A graphical method to assess treatment-covariate interactions using the Cox model on subsets of the data[END_REF] was performed. The method used was the tail-oriented version with g = 7, generating thirteen overlapping subgroups (six for decreasing ER values, one for all patients and six for increasing ER values). The STEPP plots show the tamoxifen effect estimated in each of these created subgroups.

The sensitivity and specificity corresponding to different choices of cut-off for ER cytosol values compared to ER determined by IHC were explored using a Receiver Operating Characteristic curve (ROC). For this analysis the cut-off for ER positivity by IHC was set to 10% stained cells. The software used for the statistical analysis was Stata/SE 10.0

Results

Clinical characteristics:

Among 1780 postmenopausal patients with "low-risk" invasive breast cancer randomly assigned to treatment in the STO-3 trial, 1183 patients (66. 5%) had ER-positive disease, 296 (16.6%) had ER-negative disease and the remaining 301 patients (17%) had unknown estrogen receptor status. Median follow-up was 17 years. The tamoxifen and control groups were well balanced regarding clinical characteristics (Table 1).

Among 912 women with tumor material available in tissue microarrays, ER had been determined in the cytosol in 777 cases (85%) and by IHC in 790 cases (86.5%). ER was determined by both methods in 683 tumors (75%) described below. The clinical characteristics of patients in this study and patients in "low-risk group" of the STO-3 trial were comparable (table 1), However in this study, there were fewer patients with tumor size less than 10 mm than the low-risk group (22% and 31% respectively) and more patients with tumor size 11-30 mm (76% and 65% respectively). The other dissimilarity was that all patients in this cohort have a known ER status but in the "low-risk" population 17% had unknown ER status.

Concordance between the methods: Six hundred eighty three patients had tumors with known ER status by both methods, 536 (78.5%) were ERpositive by cytosol assays (ER≥0.05 fmol/µg DNA) and 539 ER-positive (79%) by IHC (ER ≥ 10%). Thirty nine (5.7%) had ER-positive cytosol assays but negative IHC, whereas the opposite pattern was found in 42 cases (6.1%) (Table 2). ER status from cytosol assays indicated that 147 (21.5%) of 683 tumours were ER-negative.

When the ER status of these ER-negative tumors by the cytosol assays was determined by IHC, 42 (28.5%) were ER positive. Of 536 tumors classified as ER-positive by the cytosol assays, 39(7%) were classified as ER negative by IHC. The concordance between the two assays was high (88%). The kappa statistic for concordance was 0.65, (CI 0.58 to 0.72) when using two categories (ER-positive and ER-negative) for each method. Only seven tumors were classified into the low category by IHC (1%-9% of cells stained).

ER status and clinical outcomes:

Recurrence free survival, overall survival and breast cancer specific survival in the tamoxifen and control groups were correlated to hormone receptor levels. With cytosol data we compared ER-negative cases (ER-negative<0.05 fmol/µg DNA) and ER-positive ones (≥0.05 fmol/µg DNA) and also compared ER-negative cases with cases with intermediate levels (ER+ 0.05≤ ER<1 fmol/µg DNA), and high levels (ER++, ER≥1 fmol/µg DNA). Similarly, we compared patients grouped by IHC data, ER-negative (<10% stained cells) or ER-positive (≥10% cells stained). We also compared the cases with no staining and cases with 10-85% cells stained and a group with more than 85% stained cells. Both assays yielded ER-data with strong correlation to treatment benefit. Patients with ER-negative tumors showed no benefit from treatment with tamoxifen neither in term of recurrence free survival nor overall survival regardless with which assay the ER status was determined (e.g., for the comparison tamoxifen versus no tamoxifen: (ER-negative cytosol HR 0.87, 95%Cl=0.48 to 1.60; ERnegative IHC HR=0.74, 95%Cl=0, 43 to 1.28; ER-absent ( 0% stained cells) by IHC, HR= 0.73, 95% Cl 0.42 to 1.28) (figure2). Patients with ER-positive breast cancer allocated to tamoxifen showed statistically significant better recurrence free survival. Both assay methods defined ER positive groups with similar estimated hazard ratios and 95% confidence intervals for tamoxifen benefit ( ER cytosol -positive HR 0.51, 95% Cl 0.36 to 0.73, P<0.001; ER IHC>0% present HR 0.54, 95%Cl 0.38 to0.75, P<0.001; ER IHC>10% -positive HR 0.53, 95% Cl 0.37 to 0.74, P<0.001) (figure 3). Overall mortality was slightly reduced by tamoxifen treatment but the differences were not statistically significant, likely due to a relatively low power. However the treatment benefit with tamoxifen appeared to be marginally greater among those classified as ER-positive by cytosol assays compared to those classified with IHC assay (HR 0.80 and 0.87 respectively)(Figure 3). For more detailed correlation of treatment benefit to ER levels measured by IHC and cytosol assays we divided the patients in three groups, (ER<0.05 fmol/µg DNA/147 tumors (21.5%), 0.05 fmol/µg DNA≤ER<1.0 fmol/µg DNA/264 tumours (38.7%) and ER≥1.0 fmol/µg DNA/ 272 tumours (39.8%) measured by cytosol assays) and (ER<10%/144 tumours (21%), 10%≤ER<85%/ 260 tumours (38%) and ER≥ 85%/279 tumors (41%) by IHC). The analyses revealed that there was no benefit of tamoxifen among patients with ER-negative by any assays. Among patients with ERpositivity defined by IHC, the benefit from tamoxifen appeared to be unrelated to the percentage stained cells, but in the group with ER-positivity measured by cytosol assays the benefit from tamoxifen treatment appeared to be related to the ER concentration and an increased benefit with ER (≥1.0 fmol/µg DNA (data not shown).

To investigate the trends in tamoxifen treatment effect differences across the continuum of ER status, we shows an obvious trend in tamoxifen effect with greater effect in subpopulation with higher median ER content.

However, based on these analyses we were unable to identify an abrupt changing in tamoxifen effect separating the subpopulations without effect from the subpopulations with effect from tamoxifen. Further statistical analyses by ROC curves and testing different cutoff levels for cytosol (0.01 fmol/µDNA and 0.05fmol/µDNA) and IHC (1%, 10% and 20%), to identify an optimal cutoff level for both assays were performed. However, the ROC curve which was created by choosing different cutoff level for ER cytosol values, compared to ER determined by IHC and if the cutoff level for considering ER to be positive by IHC was set to 10%, the majority of tumors were correctly classified with a cutoff level for cytosol from 0.05-0.08 fmol/µDNA (between 88.1%-88.4% with sensitivity 92.2%-90.7% and specificity 72.9%-79.9% respectively) (Fig 5a). The sensitivity and specificity corresponding to cutoff of 10% for ER determined by IHC compared to ER cytosol values and if the cutoff level for considering ER to be positive by cytosol was chosen 0.05 fmol/µDNA, was 92.7 and 71.4 respectively and 88.1% of the tumors could be classified correctly (Fig 5b). It means that these two cutoff levels (10% by IHC and 0.05 fmol/µDNA) are likely to be corresponding. We also conducted STEPP analysis for the two discordant groups: ER-positive by IHC but negative by cytosol (42 cases) and ER -negative by IHC but positive by cytosol (39 cases). The STEPP analyses of these two subgroups were not conclusive because of the small number of cases in each group (data not shown). On the other hand, the Kaplan-Meier plots of recurrencefree survival for tamoxifen effect in the discordant subgroup of ER-negative by cytosol(ER<0.05 fmol/µg DNA) but positive by IHC (ER≥10%) revealed similar effect for both tamoxifen and control group (Fig 6a). In contrast, the RFS in the discordant subgroup that was defined as ER-positive by cytosol (ER≥0.05fmol/µg DNA) but negative by IHC (ER<10%), was greater in the tamoxifen group compared with the control group (Fig 6b).

These results may indicate some more predictive ability of cytosol assays compared to IHC assay. However, this difference was not statistically significant and should be interpreted with caution because of the small number of patients in these groups. An analysis of events by period of follow-up showed that disease recurrences and total deaths were reduced among the tamoxifen treated patients mainly during the first 5 years (table 3).

Discussion

The results from randomized studies in the 1970s changed the treatment of early stage breast cancer and established adjuvant tamoxifen therapy for ER-positive breast cancer [START_REF] Rutqvist | The Stockholm trial on adjuvant tamoxifen in early breast cancer. Correlation between estrogen receptor level and treatment effect[END_REF][START_REF] Baum | Improved survival among patients treated with adjuvant tamoxifen after mastectomy for early breast cancer[END_REF][START_REF] Stewart | Adjuvant tamoxifen therapy and receptor levels[END_REF]. However the importance of ER status as a predictor was not completely established until 1990 [START_REF]Systemic treatment of early breast cancer by hormonal, cytotoxic, or immune therapy. 133 randomised trials involving 31,000 recurrences and 24,000 deaths among 75,000 women. Early Breast Cancer Trialists' Collaborative Group[END_REF]. The Stockholm adjuvant trial (STO-3) conducted by the Stockholm Breast Cancer Study Group was initiated in 1976. This study is one of the largest of adjuvant systemic therapy versus no adjuvant treatment. Inclusion was not based on hormone receptor status, although prospective data on receptor status was available for 85% of the patients. This has permitted evaluations of the treatment predictive ability of estrogen receptor. This report is based on the "low-risk" subpopulation of STO-3. The follow-up is long and few patients were lost to follow up.

Our data confirm that ER status measured by IHC is similar to that determined by the cytosol assays. The result showed no benefit from tamoxifen therapy for ER-negative patients regardless of assays performed. The pattern of the tamoxifen effect was similar for patients with absent staining cells (ER=0%) compared to ER-negative (ER<10%). However only 7 tumors were stained between 1-9%. Patients with ER-positivity defined by IHC or cytosol assay obtained similar benefit from tamoxifen treatment. In discordant groups the effect of tamoxifen in term of recurrence free survival showed dissimilarity. There were no differences between the tamoxifen group and the no tamoxifen group in ER cyt -negative/ER IHC -positive, despite the majority of cytosol negative tumors showing high percent staining cells by IHC. In contrast, the tamoxifen group showed greater RFS compared to the no tamoxifen group in ER cyt -positive/ER IHC -negative discordant group. Whether this indicates a more sensitive predictive ability for cytosol assay or is a result by chance due to very small number of cases in each group, needs further validation from other studies. There may also be the possibility that the larger amount of fresh tissue used in the cytosol method positively impact the accuracy of this method. Although adjuvant tamoxifen was established based on ER measured by cytosol assays, we conclude that immunohistochemistry yields similar correlation to benefit of therapy. Two retrospective adjuvant studies conducted on archival tissue, one by Ferno et al [START_REF] Ferno | Oestrogen receptor analysis of paraffin sections and cytosol samples of primary breast cancer in relation to outcome after adjuvant tamoxifen treatment. The South Sweden Breast Cancer Group[END_REF] on 98 patients which received adjuvant tamoxifen, the other reported by Harvey et al [START_REF] Harvey | Estrogen receptor status by immunohistochemistry is superior to the ligand-binding assay for predicting response to adjuvant endocrine therapy in breast cancer[END_REF]showed that estrogen receptor positivity determined by IHC significantly predicted improved disease free survival in patients receiving adjuvant tamoxifen therapy. To our knowledge only three other studies [START_REF] Fisher | Solving the dilemma of the immunohistochemical and other methods used for scoring estrogen receptor and progesterone receptor in patients with invasive breast carcinoma[END_REF][START_REF] Alberts | Comparison of estrogen receptor determinations by a biochemical ligandbinding assay and immunohistochemical staining with monoclonal antibody ER1D5 in females with lymph node positive breast carcinoma entered on two prospective clinical trials[END_REF][START_REF] Regan | Re-evaluating adjuvant breast cancer trials: assessing hormone receptor status by immunohistochemical versus extraction assays[END_REF] used randomized clinical trial populations receiving adjuvant therapy but the cohort of patients were smaller, had shorter duration of follow up and mixed treatment arms [START_REF] Fisher | Solving the dilemma of the immunohistochemical and other methods used for scoring estrogen receptor and progesterone receptor in patients with invasive breast carcinoma[END_REF][START_REF] Alberts | Comparison of estrogen receptor determinations by a biochemical ligandbinding assay and immunohistochemical staining with monoclonal antibody ER1D5 in females with lymph node positive breast carcinoma entered on two prospective clinical trials[END_REF]. Regan at al re-evaluated and reported results from International Breast Cancer Study Group (IBCSG) trial VIII (pre-and perimenopausal patients) and trial IX (postmenopausal patients). The subset of patients in trial IX is similar to our cohort but randomization was between chemotherapy followed by tamoxifen versus tamoxifen. The median duration of follow-up was shorter and biochemical assays were performed at different local pathology laboratories [START_REF] Regan | Re-evaluating adjuvant breast cancer trials: assessing hormone receptor status by immunohistochemical versus extraction assays[END_REF]. However our results confirm results from these cohorts.

There is no general consensus concerning the estrogen receptor cut-off level separating endocrine responsive tumors from non-responsive one. In STO 3 trial a cut-off level of 0.05 fmol/µg DNA was used to distinguish receptor positive tumors from receptor-poor/receptor-negative tumors. Analysis of treatment benefit versus ER levels did not show any benefit for ER levels below 0.05 fmol/µg DNA [START_REF] Rutqvist | Long-term follow-up of the randomized Stockholm trial on adjuvant tamoxifen among postmenopausal patients with early stage breast cancer[END_REF]. In this study we used this cut-off level for cytosol assays. On the other hand, the IHC method has increasingly been used for determination of ER during recent decades and has been established as the routine assay for ER determination. Concerns associated with IHC assay which reduce its accuracy include variation in pre-analytical, analytical and post-analytical procedures, using different antibodies and diversities in interpretation and reporting results from IHC assays for determination of ER in breast cancer. To overcome this issue standardization of handling tissue, procedures, use of antibodies, interpretation and reporting result has been discussed and recently the American Society of Clinical Oncology (ASCO) and the college of American Pathologists (CAP) presented a guideline recommendations for IHC testing of ER and PR to improve accuracy of the method [START_REF] Hammond | American Society of Clinical Oncology/College Of American Pathologists guideline recommendations for immunohistochemical testing of estrogen and progesterone receptors in breast cancer[END_REF]. The second issue is to find a optimal cutoff level for IHC. For the immunohistochemical assay we used 10% stained cells as cut-off level that has been commonly used worldwide. Some recently data provide support for the use of cutoff level of 1% staining cells for ER as showing a better prognosis or some extent of endocrine effect [START_REF] Harvey | Estrogen receptor status by immunohistochemistry is superior to the ligand-binding assay for predicting response to adjuvant endocrine therapy in breast cancer[END_REF][START_REF] Hammond | American Society of Clinical Oncology/College Of American Pathologists guideline recommendations for immunohistochemical testing of estrogen and progesterone receptors in breast cancer[END_REF][START_REF] Viale | Prognostic and predictive value of centrally reviewed expression of estrogen and progesterone receptors in a randomized trial comparing letrozole and tamoxifen adjuvant therapy for postmenopausal early breast cancer: BIG 1-98[END_REF]. However the 11 th St Gallen conference, defined endocrine responsiveness as the presence of any detectable estrogen receptor [START_REF] Goldhirsch | Thresholds for therapies: highlights of the St Gallen International Expert Consensus on the primary therapy of early breast cancer 2009[END_REF]. Our study could neither support nor reject these statements since only a small number of tumors had stained cells between zero and 25%. Some limitations of this cohort must be taken into account when interpreting the results. This study included only 38% of patients randomly allocated in the "low-risk" group. Although the clinical characteristics of patients in this study and those of the overall trial were comparable, the statistical power is reduced. The other limitation is the issue of cut point for Immunohistochemical assays. Only seven tumors had between 0 and 9 percent stained cells. We are thus unable to suggest an optimal cut point for the immunohistochemistry assay. Finally, a direct comparison of outcomes (e.g., recurrence free and overall survival) between patients whose tumours were estrogens receptor absent or negative by IHC and positive by cytosol assay with patients whose tumors were estrogen positive by IHC but negative by cytosol assay was not meaningful because of the small number of tumors with discordant data.

Conclusion

We conclude that determination of estrogen receptor status by IHC could substitute cytosol assays with preserved ability to predict response to adjuvant tamoxifen treatment of early stage breast cancer. However, we observed 12% discordance in estrogen receptor status determined by IHC as compared to cytosol assays. In spite of this, estrogen receptor status determined by IHC confirmed randomized clinical trial results that were based on predicted response to tamoxifen therapy as well as estrogen receptor status measured by cytosol assay.

  performed STEPP (Subpopulation Treatment Effect Pattern Plot) analyses (fig 4). The STEPP curves show the effect of tamoxifen for overlapping subgroups with different median ER values by cytosol (Fig 4a) and by IHC (Fig 4b). The STEPP curve for ER determined by IHC does not start with a zero point on log relative hazard indicating some effect of tamoxifen in patients with low ER value by IHC. The Kaplan-Meier curve of recurrence free survival for tamoxifen effect for ER IHC -negative(ER<10%) shows as well the same trend (FIG 2b). Likewise, the STEPP curve of IHC has a shallow slip in comparison to the STEPP curve of cytosol that

"Figure 1 ;

 1 Figure 1; Summary of trial design

Figure 2 ;Figure 3 ;Figure 4 ;

 234 Figure 2; Recurrence-free survival (a,b) and overall survival (c,d) of tamoxifen treated and control groups patients with ER negative tumors analysed with cytosol assays and IHC

Figure 5a ;

 5a Figure 5a; The ROC curve shows the sensitivity and specificity corresponding to different choices of cut-off for ER cytosol values compared to ER determined by IHC. The cut-off for considering ER to be positive by IHC was set to 10%.

Figure 6 ,

 6 Figure 6, Influence of tamoxifen therapy on recurrence-free survival of patient with ER cyt -negative/ER IHC -positive (a) and ER cyt positive/ER IHC -negative (b)

Table 1 ;

 1 Distribution of patient characteristics by risk strata and allocated treatment in present study and entire "low-risk" population

	Patient characteristics	This study	Low-risk group
		TAM+ 1	TAM-2	TAM+	TAM-
		(n=350)	(n=333)	(n=886)	(n=894)
	RT
	Yes	19.7	19.8	24.0	24.5
	No	80.3	80.2	76.0	75.5
	Age (years)
	40-49	0.9	1.8	0.9	1.7
	50-59	29.7	33.9	33.4	35.9
	60-69	65.1	60.4	62.0	58.7
	70-	4.3	3.9	3.7	3.7
	Histopathological tumor size (mm)
	<10	21.1	23.1	31.6	30.5
	11-20	54.6	52.9	46.5	47.9
	21-30	22.9	23.1	18.5	17.1
	>30	0.0	0.3	0.1	0.6
	Unavailable	1.4	0.6	3.3	3.9
	Histopathological nodal involvement
	N0	99.7	99.7	99.1	98.2
	N1-3	0.0	0.3	0.1	0.8
	N4+	0.0	0.0	0.1	0.0
	Unavailable	0.3	0.0	0.7	1.0
	Estrogen receptor status (IHC)
	Positive	79.4	78.4	n.a 5	n.a 5
	Negative	20.6	21.6	n.a 5	n.a 5
	Estrogen receptor status (cytosol)
	Positive 3	80.0	76.9	67.2	65.8
	Negative 4	20.0	23.1	16.0	17.2
	Unavailable			16.8	17.0
	1-adjuvant tamoxifen. 2-No adjuvant tamoxifen.3-≥0.05 fmol/µg DNA.4-<0.05 fmol/µg DNA
	5-in original cohort ER status was not performed by IHC.

Table 2 ;

 2 Distribution of ER status determined by immunohistochemistry with different percentage of stained cells in relation to cytosol assays

		ER fmol/µg DNA (Cytosol)
	ER % (IHC)	<0.05	≥0.05	Total
	0	101	36	137
	1-9	4	3	7
	10-24	0	6	6
	25-49	4	18	22
	50-74	8	77	85
	≥75	30	396	426
	Total	147	536	683

Table 3 ;

 3 time-dependent effect of tamoxifen during the first five years, five to 20 years and all 20 years after start of tamoxifen treatment

		TAM+ vs. TAM-		TAM+ vs. TAM-		TAM+ vs. TAM-
		0-20 years		0-5 years		5-20 years
		HR (95% CI)	P	HR (95% CI)	P	HR (95% CI)	P
	Overall survival
	ER+ (IHC 10%)	0.87 (0.68-1.11)	0.26	0.84 (0.45-1.56)	0.58	0.87 (0.67-1.14)	0.33
	ER-(IHC 10%)	1.01 (0.63-1.62)	0.96	0.98 (0.45-2.11)	0.95	1.04 (0.57-1.87)	0.91
	ER+ (Cytosol 10%)	0.80 (0.63-1.02)	0.08	0.64 (0.35-1.17)	0.15	0.84 (0.64-1.10)	0.20
	ER-(Cytosol 10%)	1.33 (0.81-2.19)	0.26	1.76 (0.76-4.06)	0.19	1.14 (0.61-2.12)	0.68
	Recurrence free survival
	ER+ (IHC 10%)	0.53 (0.37-0.74) <0.001 0.29 (0.16-0.53) <0.001 0.78 (0.50-1.22)	0.28
	ER-(IHC 10%)	0.75 (0.44-1.29)	0.30	0.61 (0.31-1.24)	0.17	1.03 (0.43-2.48)	0.95
	ER+ (Cytosol 10%)	0.51 (0.37-0.71) <0.001 0.26 (0.15-0.46) <0.001 0.82 (0.53-1.26)	0.36
	ER-(Cytosol 10%)	0.88 (0.48-1.60)	0.68	0.96 (0.44-2.07)	0.91	0.77 (0.29-2.03)	0.60

Acknowledgment This research was supported by grants from the Swedish Cancer Society and the Swedish Research Council to Olle Stål and Jonas Bergh and funds of Jonas Bergh´s research group at Radiumhemmet and the Stockholm County Council.


Angelos A Economopoulos

Vassilis S Kouikoglou

Lead-time analysis and a control policy for production lines

Keywords: Production lines, admission control, inventory control, balking, quoted lead time, closed queueing networks, simulation

We address a control problem for a production line which produces one product to stock and faces random demand. During stockouts, the system quotes a fixed response time for each arriving order, and the customers place their orders only if the response time promised meets their deadlines. Customer orders are filled on a first come, first served basis. A penalty cost is incurred whenever a customer is served later than promised. A two-parameter admission/inventory control policy is implemented that maintains a bounded backlog and a constant inventory position (total inventory minus backlog) in the system. For production lines with exponential processing times and Poisson demand, the mean profit rate of the system is computed analytically using closed queueing network formulas. For systems with general processing or interarrival time distributions, the profit rate is estimated via simulation.

Simple properties are established which ensure that the profit maximizing control parameters can be determined in finite time using exhaustive search.

Numerical results show that the proposed policy performs better than the make-to-order/zero-inventory and the lost-sales/make-to-stock policies.

Introduction

Production control involves decisions concerning when and how much to produce and when to accept incoming orders. Such decisions have significant effects on a number of operational indices such as throughput, mean inventory and backlog levels, and quality of service. These operational indices affect the mean profit rate of the system, which comprises profit from sales and the costs associated with inventory, backlog, delays in filling orders, possible loss of sales during stockout periods and so on (see, e.g., Porteus 1990, Section 3.7, p. 615). In the literature of production control there is a rich collection of policies that minimize the inventory and backlog costs.

A simple and often effective inventory control policy is one that specifies a target value for the stock level, called base stock (Zipkin 2000, Buzacott and[START_REF] Buzacott | Stochastic Models of Manufacturing Systems[END_REF]; the manufacturing facility produces as long as the number of finished items is smaller than the base stock and idles otherwise.

In order to handle incoming orders during stockout periods, many systems apply a lost sales (LS -all incoming orders rejected) or a complete backordering (CB -all incoming orders accepted) policy. When applying LS, there is no cost of delay in filling orders but the loss of sales to competition during stockouts makes it mandatory to keep a high stock level. On the contrary, facilities under CB appear to have vast backlog queues and long delays in filling orders which, apart from high backlogging costs, could also lead to potential loss of sales due to customer impatience.

An admission control policy which is analogous to the base stock policy and generalizes LS and CB, balancing between loss of sales and long delays, is the base backlog policy [START_REF] Naor | The regulation of queue size by levying tolls[END_REF]. This policy admits incoming orders during stockouts as long as the backlog level is less than a certain threshold called base backlog and rejects them otherwise. When a base backlog is applied to systems with zero stock level, the policy is called make-to-order (MTO).

In this paper, we consider a production line producing one product to meet random demand from impatient customers. We adopt a joint inventory/admission control policy called base stock/base backlog (BSBB) to maximize the net profit rate of the system, which includes profit from sales less the costs of inventory, 3 backordering and failure to meet the delivery deadlines promised. According to BSBB, the inventory levels are handled as in base stock and, during stockouts, a base backlog admission control policy is implemented. Joint inventory and admission control policies have been studied in the past years both for single stage systems [START_REF] Caldentey | Analyzing the make-to-stock queue in the supply chain and e-business settings[END_REF][START_REF] Kouikoglou | Design of product specifications and control policies in a single-stage production system[END_REF], Ioannidis and Kouikoglou 2008) and production networks (Ioannidis et al. 2008), yet without considering customer impatience and delivery due dates.

Impatience of customers is an important factor that must be taken into consideration when controlling inventory and backlog in production systems. This subject has drawn some attention in the past decades, mainly for its applicability to telecommunication systems rather than manufacturing systems. When we refer to customer's impatience we use the terms balking and reneging (see e.g. [START_REF] Haight | Queueing with balking[END_REF][START_REF] Haight | Queueing with reneging[END_REF][START_REF] Haight | Queueing with balking II[END_REF].

Balking occurs when a customer arrives during a stockout and decides not to place an order, rather than join the backlog queue, if he believes that he will be served later than desired. Balking causes loss of sales. It can be modeled by assigning a balking probability to the arrival process, which depends either on the state of the system (Subba Rao 1967) or on some response time estimate [START_REF] Liu | Balking and reneging in M/G/s systems exact analysis and approximations[END_REF]. In the former case, an arriving customer enters the backlog queue with some probability which depends on the number of outstanding orders already pending. In the latter case, each customer has a due time (deadline measured from the time of his arrival) and joins the queue only if he estimates that the facility's response time to his order is less than the due time.

Reneging, also referred as abandonment, is the process where outstanding customers decide to withdraw their orders if they have waited longer than their deadlines. Reneging not only causes loss of sales but also loss of customer goodwill. The case of queue abandonment has been studied in the past decades, but analytical results for performance measures such as mean backlog queue sizes, waiting times or abandonment rates, are limited to single-stage systems with one (Movaghar 2006, Economopoulos andKouikoglou 2008) or several servers (Bacceli and Hebuterne 1981[START_REF] Ward | Improving service by informing customers about anticipated delays[END_REF][START_REF] Brandt | On the M(n)/M(m)/s queue with impatient calls[END_REF]. For more general production systems such as production lines or networks with reneging, mathematical difficulties arise in deriving analytical results and determining In this paper, we only consider balking of customers but we include a penalty cost for the system when it fails to meet a quoted lead time (QLT). Specifically, a customer arriving during a stockout is promised a fixed response time QLT to his order. If QLT is greater than the customer's due time, then the customer balks.

Otherwise, the customer places his order, but if his waiting time exceeds QLT, then, though it is not possible to renege and abandon the backlog queue, the production facility incurs an extra delay cost as a penalty or discount offered to the customer.

Our aim is to determine optimal base stock and base backlog levels, denoted s and c respectively, for a production line so as to maximize its mean profit rate. In section 2, we describe the production line and model the balking process. In section 3, we use an equivalent closed queueing system to derive expressions for the steady-state probabilities of the system and the various components of the profit rate. In section 4, we show that the optimal values for s and c are bounded from above. This allows us to develop a simple and fast algorithm to track down the optimal solution. Numerical results are presented in section 5 and conclusions in section 6. Early results of this work are reported in [START_REF] Economopoulos | Lead-time analysis and a control policy for production lines with exponential machines[END_REF].

System description

Consider a production line with N machines that produce one product to meet random demand. Machine M i along with its input buffer will be referred as node i, i = 1, 2, …, N, and n i will denote the number of workpieces in node i, including the one that may be processed by M i . Raw items are first processed at M N .

Workpieces leaving M i are sent to node i -1. Machine M 1 performs the last operation and sends finished items to the output buffer. The processing times of M i are independent exponential random variables with mean 1/µ i . For simplicity, we assume that all production rates µ i are different. Customers arrive according to a Poisson process with mean rate λ.

Each customer requests one unit of product and, if stock is available, a sale 5 occurs immediately and a new raw item is sent to the buffer of node N to replace the item sold. During stockouts, orders cannot be fulfilled immediately and customers are reluctant to place orders. To cope with order cancelations and attract reluctant customers, the firm announces a standard time frame QLT from the moment an order is placed, after which if the order is still pending, the customer would benefit from a discount. We assume that each arriving customer has a maximum waiting time tolerance upon arrival until his order is shipped. If QLT is greater than the customer's maximum waiting time he will not place the order. Otherwise, he places the order but cannot abandon the backlog queue thereafter.

Let q denote the probability that an arriving customer will place his order during a stockout period. The maximum waiting tolerance of customers is a random variable θ with known distribution function G(t) = P(θ ≤ t) and the firm's QLT is constant. The probability that an arriving customer facing stockout will place his order is

q = P(QLT ≤ θ) = 1 -P(θ ≤ QLT) = 1 -G(QLT).
(1) Therefore, when no stock is available, the rate at which products are requested is qλ. Every time a customer requests a product and enters the backlog queue, a new raw item is released to node N. Orders are shipped on a FCFS basis.

The number of finished items in the output buffer is denoted n F , the number of backorders is denoted n B and the total number of items (raw items, workpieces, products) n H . The inventory and backlog levels of the system are controlled by implementing the joint inventory/admission control policy BSBB described in Section 1. Accordingly, the facility produces until the inventory level n F of the output buffer reaches the base stock s and then stops. During stockouts, all incoming orders are accepted as long as the number of backorders n B is lower than the base backlog c. A new raw item is released into the production line each time an order is accepted. It then follows that, at any time instant, the total number of items in the system equals s plus the number of backorders, i.e., The net profit of the system comprises four types of profit or cost. Each time a product is sold, the system earns a profit p; penalties due to lost sales can be taken into account by adding a unit lost sale cost to p. The system incurs a holding cost rate h for every item held in the facility, a backlog cost rate b for every pending order and a penalty cost d for every order pending longer than QLT. Different holding costs h 1 , …, h N for the inventories can easily taken into account into the model. Yet, a uniform holding cost h is assumed for simplicity in the presentation.

n H = s + n B . (2)
The problem is to determine the optimal values of s and c so as to maximize the average net profit rate of the system. This quantity is given by

J(s, c) = pΤΗ s,c -hH s,c -bB s,c -dD s,c , (3)
where TH s,c is the throughput (mean production rate), H s,c is the average inventory of the total number of items in the system (raw items, workpieces, products), B s,c

is the average backlog of pending orders and D s,c is mean rate of delayed orders beyond the QLT.

Performance evaluation

In this section, we derive analytical expressions for the performance measures involved in equation (3).

An equivalent cyclic Jackson network

Consider the cyclic network of queues shown in Fig. 1, in which there are N + 1 service nodes, i = 0, 1, …, N, and a total population of s + c jobs. Suppose that server M 0 at node 0 has a service rate µ 0 equal to the rate at which orders are placed in the production system. The servers of the queueing system have the same rates µ i and the same contents n i as the nodes the production facility for i = 1, …, N.

Insert Figure 1 Under the above assumptions, both systems evolve in the same manner. We assume that when n 0 = 0 in the queueing system, the production system has n F = 0 finished items and n B = c orders pending, and when n 0 = s + c the production system has n F = s and n B = 0 (for a detailed discussion of the equivalence of the two systems see e.g. Section II.B of Ioannidis et al. 2008). Table 1 summarizes the relationships among the number of items in node 0 of the closed queueing system and the state variables n F , n B and n H of the production system. Since the service rate of node 0 is equal to the demand rate of the production system, µ 0 is state-dependent, hence,

µ 0 (n 0 ) =    ≤ > . c n q c n 0 0 , , λ λ (4)
Insert Table 1 about here Next, with the use of Table 1, we derive expressions for the throughput TH s,c , average inventory H s,c and backlog B s,c , and the mean rate of delayed orders D s,c .

The throughput of the system equals the rate of accepted orders. We see from equation (4) that the latter drops from λ to λq during stockouts, i.e., when n F = 0 or, in the equivalent queueing system, n 0 ≤ c. Therefore,

ΤΗ s,c = λP(n 0 ≥ c + 1) + qλP(n 0 ≤ c).
(5)

The average backlog of the production line by definition is B s,c = E(n B ), where E is the expectation operator. By inspection of Table 1 we have n B = 0 for n 0 ≤ c

and n 0 = c -n B for n B ≥ 1; thus, B s,c = ∑ = - = c k k c n kP 1 0 ) ( . (6)
The average overall inventory of raw, semi-finished and finished items is

H s,c = E(n H ); it follows from equation (2) that H s,c = s + B s,c . (7)
Finally, the mean arrival rate D s,c of orders delayed longer than QLT is given by

F o r P e e r R e v i e w O n l y 8 D s,c = ∑ - = - = 1 0 0 ) ( c m m m c n P q Π λ , ( 8
)
where qλP(n 0 = cm) is rate at which a new order is placed when n B = m orders are already pending, and Π m denotes the probability that the lead time (time-to- fill) of this order will exceed QLT.

Equations for the computation of Π m will be established in the next section. All probabilities of the state variable n 0 involved in equations ( 5)-( 8) can be calculated using standard formulas for cyclic Jackson networks (see, e.g., Section II.C of Ioannidis et al. 2008[START_REF] Buzen | Computational algorithms for closed queuing networks with exponential servers[END_REF]):

(a) Normalization constant of a cyclic Jackson network with nodes k = i, i -1, …, j having constant service rates µ k , node j feeding back to i, and a total population of n jobs,

G i,j (n) = ∑ ∏ = + + = - n n ... n j i k n k j i k µ , j ≤ i.
(9) (b) Normalization constant of the original cyclic queueing system (i.e., with a total population s + c and node 0 having a state-dependent service rate given by equation ( 4)),

G(s + c) = ∑ + = - - - + c s n N n c n n c s G q 0 0 1 , ) , min( 0 0 0 ) ( λ . ( 10
)
(c) Product-form equilibrium probabilities of the original cyclic queueing system,

P(n 0 , …, n N ) = ∏ = - - - + N k n k n c n k q c s G 1 ) , min( 0 0 ) ( 1 µ λ . ( 11
) (d) Marginal probabilities for n 0 , P(n 0 = k) = ) ( ) ( 1 1 , ) , min( k c s G q c s G N k c k - + + - - λ , ( 12
)
P(n 0 ≥ k) = ∑ + = - - - + + c s k n N n c n n c s G q c s G ) ( ) ( 1 1 , ) , min( λ . ( 13

Lead-time analysis

We now derive equations for the computation of Π m . Assume that s ≥ 1. The case s = 0 will be analyzed later as a special case. We consider the system in equilibrium. Suppose that a customer arrives during a stockout interval (t, t + dt)

and decides to place an order while m, m < c, orders are already pending in the system. At time t, right before the customer places his order, there are a total of

Location of item m + 1

Proposition 1: Suppose that a new order arrives when m others are already pending. Then, for s ≥ 1,

P(item m + 1 is at M i | n B = m at
+ + = = + t t m n P t t m n M m P B B i (14) = ) ( ) 1 ( ) ( 1 , , 1 ,1 m s G s G m G N i N i i + - - µ . ( 15
)
Proof: First we calculate the denominator of ( 14). The probability that a customer will arrive during a stockout interval (t, t + dt) and decide to place an order while m orders are already pending in the system, m < c, equals the product of probabilities of the following events:

(i) at time t, in steady state, the system is out of stock with m orders pending or, equivalently, a total of s + m workpieces are in nodes N, N -1, …, 1;

(ii) a new customer arrives in the interval (t, t + dt) and decides to place an order.

Thus, using ( 4) and ( 12), we can calculate the denominator of ( 14) as follows:

P(n B = m at time t and a new customer arrives at t

+ ) = = P(n 0 = c -m) × (λqdt) = (qλdt) (
)

) ( ) ( 1 1 , m s G q c s G N m c + + + - λ . ( 16
)
The numerator of ( 14), is the probability of intersection of events (ii) and a subset of (i) satisfying the following conditions:

(iii) the first m workpieces are anywhere between machines M i , M i-1 , …, M 1 ;

(iv) one more (item m + 1) is in M i ;

(v) the remaining s -1 workpieces are anywhere between machines M N and M i .

The probability of intersection of (iii)-(v) is computed as follows:

P(events (iii) and (iv) and (v)) = = P(in equilibrium, item m + 1 is at M i and

n 0 = c -m) = = + ′ ′ + ′ = - = ∑ - = ′ ′ + + + = + + + ′ + - 1 (iii) (i) 1 0 1 1 1 ) , , 1 , , , ( s n n ... n m n n n i i i N i i N i i n n n n n m c n P K K K = ( )                 + ∑ ∑ = + + + ′ - - - ′ - - - = ′ ′ + + + ′ ′ - - + - - - - - + + m n n n n n i n i i s n n n n i n i n N m c i i i i i N i i N q c s G 1 1 1 1 1 1 1 1 1 1 1 ) ( ) ( 1 K K K K µ µ µ µ µ µ µ λ = ( ) ) 1 ( ) ( ) ( 1 , 1 ,1 - + - + - s G m G q c s G i N i i m c µ λ .
Multiplying the above by qλdt and dividing by ( 16) yields (15). that at time t item m + 1 is located at machine M i awaiting to be processed by M i , M i-1 , … and M 1 . The distribution of its sojourn time depends on the locations of the remaining m workpieces ahead of it. The enumeration of all possibilities is fraught with combinatorial difficulties. We circumvent this using the following result.

Remaining sojourn time distribution of item m + 1 from M i on. Suppose

Proposition 2: The time to produce item m + 1 has the same distribution as the cycle time of a job in a closed Jackson network with nodes i, i -1, …, 1 and a total population of m + 1 jobs. The probability that this time is greater than QLT is given by

DP i (m) = ( ) ∑ ∑ = = -       i k j k m j i , k ! j e m k 1 0 QLT QLT ) ( µ β µ ; ( 17
)
where, for distinct service rates,

β k,i (m) = ∑ = - - i j m j i j m k i k 1 , , µ α µ α and α k,i = ∏ ≠ = - i l k l k l l 1 µ µ µ . ( 18
)
Proof: The first part of the proof is similar to the proof of Proposition 1. Consider an order arriving when the system is out of stock with m orders already pending.

Suppose further that this order "sees" the system in the following state:

(iii)' the first m workpieces are at specific locations between machines M i , M i-1 , …, M 1 ;

(iv) one more (item m + 1) is in M i ;

(v) the remaining s -1 workpieces are anywhere between machines M N and M i .

Event (iii)' is a subset of (iii), which was defined in the proof of Proposition 1.

The conditional probability of (iii)' and (iv) given (iii)-(iv) is given by 5)-( 8) and, finally, the average profit rate of the system given by (3).

)
i i i i i i i M m m P m n n n n n n n n m c n P + = ′ + + + + ′ ′ + ′ = - = - - K K F o r P e e r R e v i e w O n l y ( ) ( )( ) ( ) ( ) dt q q c s G dt q q c s G m n n n n n i n i i s n n n n i n i n N m c n n i n i i s n n n n i n i n N m c i i i i i N i i N i i i N i F o r P e e r

Optimization

We now consider the problem of determining optimal values for s and c, denoted s opt and c opt respectively, so as to maximize the average profit rate of the system.

We have the following propositions.

Proposition 3: The optimal base stock level is bounded from above as follows Proof: Consider an arriving customer who finds c -1 pending orders and decides to place one more. This customer must wait until c items are produced. The bestcase scenario would be that these c items are in the last machine M 1 and require only one operation. The mean time to fill this customer's order is thus greater than c/µ 1 and the associated mean holding and backlog cost is c(h + b)/µ 1 . The latter must be less than p to ensure that the mean profit to be earned from this customer is positive. This yields a bound on c opt above which it is not profitable for the system to accept an incoming order. The following algorithm performs an exhaustive search to find the optimal base stock and base backlog.

Algorithm 1:

1. Set J(0, 0) = 0, J opt = 0, s opt = 0 and c opt = 0. Initialize the base stock level and the base backlog; thus, s = 1 and c = 0.

2. Compute the profit rate J(s, c). Keep the optimal control parameters and the maximum profit rate in memory; thus, if J(s, c) > J opt , then set J opt = J(s, c), s opt = s and s opt = c.

3. Apply Proposition 3: If (s + 1) < pλ/h, then set s := s + 1 and go to step 2; else go to step 4. 4. Apply Proposition 4: If (c + 1) < pµ 1 /(h + b), then set c := c + 1, s = 0, and go to step 2; otherwise stop. It follows from Propositions 3 and 4, that the above algorithm requires a finite number of iterations to find the optimal control parameters.

Numerical results and discussion

We now use Algorithm 1 of the previous section to illustrate the advantages of the proposed BSBB policy over two simpler, yet commonly used policies.

Analytical results for a production line with exponential machines

Consider a production line similar to the one described in Section 2 consisting of six machines M i , i = 1, 2, …, 6, with exponentially distributed service and customer interarrival times. Standard values for the system parameters are: mean demand rate λ = 4.0, mean production rates µ 1 = 5.25, µ 2 = 6.5, µ 3 = 5.5, µ 4 = 5.0, µ 5 = 7.0, µ 6 = 6.0, unit profit p = 100.0, unit costs h = 8.0, b = 8.0 and d = 10.0, and QLT = 1.0. The maximum waiting tolerance of customers is a random variable θ with a uniform distribution function in [0, 10], i.e., G(t) = t/10. Upon substitution of G(t) into equation (1), we obtain q = 0.9. This means that 90% of the customers arriving during stockouts are willing to wait one time unit and place their orders; the rest balk.

We compare BSBB, denoted (s, c), with the lost sales (LS) and make to order (MTO) policies. Because these policies are special cases of BSBB (LS is the same as (s, 0) and MTO coincides with (0, c)), their performance will not, in general, be as good as that of BSBB. We examine the effects of varying λ, h, b, d and QLT on the net profit rate of each policy. In each experiment, we modify one of these parameters fixing the others to their standard values and compute the optimal parameters under BSBB, LS and MTO. Tables 2 and3 show the profit rate for various values of λ and QLT. Figure 2 shows the profit rate versus the various values of the cost parameters h, b and d.

Insert Table 2 about here

From Table 2 we see that when λ is greater than minµ i = 5, LS and BSBB perform similarly. This happens because BSBB tends to reject customers during stockouts and keeps a high stock when customers arrive frequently. However, as the demand rate decreases the probability of a stockout decreases as well, and the system does not need to maintain high stock levels.

An interesting result can be drawn from Table 3, where we observe that BSBB always performs better than but close to LS for all values of QLT we tested. When QLT is zero, all customers who arrive during stockouts place their orders with probability one. In this case, BSBB reacts to protect the system against delay penalties, so it rejects all customers beyond c = 1 and, therefore, it is similar to an LS policy. As QLT increases, customers become more reluctant to place orders and so a relaxed admission control policy benefits the firm more.

Simulation results for production lines with non-exponential machines

So far we have assumed that the processing times of all machines of the production line are exponentially distributed. This assumption permits modeling the system as a Jackson network and expressing analytically all important performance indices. However, this assumption is often violated in practice. For example, [START_REF] Inman | Empirical evaluation of exponential and independence assumptions in queueing models of manufacturing systems[END_REF] studied the machine processing times in two automotive assembly plants and observed that the Erlang distribution provides more reasonable approximations of the empirical processing time distributions. To investigate the performance of the proposed joint inventory/admission control policy for more realistic systems, we examine six-machine production lines whose standard parameter values are as in the previous section. The processing times have Erlang distributions with various stages but the mean processing time of each machine M i is equal to 1/µ i , as in the exponential case. Optimal control parameters for each case are found by applying Algorithm 1 and using simulation, rather than analysis, in step 2 to compute the profit rate J(s, c). Note that Propositions 3 and 4 hold regardless of the type of distribution assumed for the processing or interarrival times.

Four distributions of processing times are tested: exponential (Erlang-1), Erlang-2, Erlang-4 and deterministic (Erlang-∞). To perform a fair comparison, we used common random numbers to generate the processing times for each machine. In all experiments, the errors in estimating the mean profit rate of the system were less than 0.5% with a 95% confidence level. On an Intel Core 2 Duo 2.33GHz personal computer, the computational time for an exhaustive search of the control parameters varies from about half to three hours, depending on the parameters of the system and especially the processing time distributions. Table 4 gives simulation results for the maximum profit rates under BSBB, LS and MTO and the corresponding optimal values for s and c. The column labeled "QLT = 1" of Table 2 and column "Exponential" of Table 4 correspond to the analytical and simulation results of the same system. We see that the simulation model gives the exact optimal control parameters, except for the MTO policy where the optimal c is overshot by 1 (yet the profit estimates J ≈ 177 of the two models deviate by less than 0.002%). As the number of Erlang stages increases 4 we see that decreasing the variance results in an increase of the profit rate, a reduction of the safety stock s under BSBB and LS and an increase of the maximum number c of backorders allowed under BSBB and MTO.

Insert Table 4 about here

We also examine the second-order properties of the mean profit rate with respect to s and c under the BSBB policy. For this experiment we use an Erlang-2 distribution for the machine processing times and standard values for the remaining system parameters. Figure 4 shows that the profit rate is a unimodal and submodular function as in the exponential case. Establishing these properties for non-exponential production lines would substantially reduce the number of simulation runs needed to optimize the system. Because the optimal parameters for non-exponential systems are not too far from those of exponential systems, one could track down the optimal control parameters of the latter, using the fast analytical model of Section 3, and set them as initial values in a local search thereafter using simulation. Some guidelines for performing local search are given in Section 5.1.

Insert Figure 4 about here

Conclusion

In this paper, we have studied a control policy for production lines with exponential machines and impatient customers. During stockouts, the system informs customers about anticipated delays by quoting a fixed lead time, QLT, to each arriving order, and offers a discount per unit of delay beyond QLT. An arriving customer balks if QLT exceeds his due time, which is a random variable with known distribution. A joint threshold-type inventory and admission control policy is proposed to maximize the net profit rate of the system. The system accepts orders as long as the backlog level is under c and produces until the number of finished products reaches s. We derive analytical expressions for the performance indices involved in the mean profit rate of the system and develop a simple but efficient algorithm that uses bounds on the optimal parameters and performs an exhaustive search to determine their values. These bounds hold regardless of the distributions of machine processing times.

We have compared the proposed policy, BSBB, with the LS (lost sales) and MTO (make-to-order) policies. In general BSBB manages the values s and c in an optimal fashion and shows a better adaptability than LS or MTO to seasonal demand variations and changes in the cost parameters. Systems with exponentially distributed processing times are analytically tractable and the computational time required to find the optimal control parameters is very short.

Systems with more general distributions are analyzed with exceptional accuracy and optimized using simulation. In all cases, BSBB performed much better than LS and MTO.

From the numerical results it appears that the mean profit rate of the system is a unimodal and submodular function with respect to the control parameters. As these properties could significantly reduce the search effort for finding the optimal control parameters, they are subjects of future work. Also, because QLT has a significant impact on system performance, considering it as an additional control parameter to be optimized is another challenging subject of research.

F

  n H = s + m workpieces in the production line (a fixed inventory s plus one item per outstanding order). Since the firm adopts the FCFS policy, the lead time (time-to-fill) of the new order will be the time needed by the facility to produce a total of m + 1 items. Various cases have to be considered depending on the location of item m + 1 in the production facility. In summary, the probability that the lead time of the order exceeds QLT given that n B = m is computed in three steps: (a) we calculate the probabilities of various locations of item m + 1 in the production facility; (b) for each location i = N, N -1, …, 1, we compute the tail of the probability distribution of the total time item m + 1 spends in M i , M i-1 , … and M 1 ; and (c) we add the products of (a) and (b) for all i.

  Τhe throughput of the system cannot exceed the demand rate λ. Also, because n H = s + n B , we have H s,c = E(n H ) ≥ s. It follows from equation (3) that J(s, c) ≤ pλsh. Since J(s, c) must not be negative, we obtain the bound on s opt .Proposition 4:The optimal base backlog level c opt is bounded from above as follows

  exponential case) to infinity (deterministic case), the variance of processing times decreases to zero. From Table

Figure 2 .

 2 Figure 2. Profit rate versus various values of system parameters.

Figure 3 .

 3 Figure 3. Profit rate versus control parameters.

Figure 4 .

 4 Figure 4. Profit rate versus control parameters (Erlang-2 distribution).

  time t and a new order arrives at t + ) =

	=	(	item	1	( is	at	at time and	new at time a and	at order arrives new a order and	) arrives	at	)

  Figure 1. Equivalent cyclic network of queues.

	FOUR FIGURES IN SEPARATE PAGES
					node i
	raw
	items,	M N	n N-1	M N-1	n i M i	n 1 M 1
	o r F o r n N F o r	virtual machine M 0 node 0	virtual buffer, n 0
	P P P
	e e e
		e r e r e r
			R R R
				e e e
				v i e v i e v i e
					w w w
						O n l O n l O n l
						y y y
				22

Table 1 .

 1 Relationship among state variables.

	Production system Cyclic network of queues
	n F	n B	n H	n 0
	s	0	s	s + c
	s -1 0	s	s + c -1
	…	…	…	…
	1	0	s	c + 1
	0	0	s	c
	0	1 s + 1	c -1
	…	…	…	…
	0	c s + c	0

Table 2 .

 2 Optimal control parameters and profit rate versus demand rate.

	Page 27 of 29
					Demand rate
		λ = 3		λ = 4		λ = 4.95	λ = 6.95
	Policy	s c	J	s c	J	s c	J	s c	J
	BSBB	7 6 215.97	11 3 259.21	13 2 280.39	13 1 294.17
	MTO	0 13 160.29	0 10 177.06	0 9 182.81	0 8 186.51
	LS	9 0 202.10	12 0 249.91	14 0 275.48	14 0 293.18

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uk International Journal of Production Research

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	28

Table 3 .

 3 Optimal control parameters and profit rate versus QLT.

		Quoted lead time
	QLT = 0	QLT = 1	QLT = 2	QLT = 5
	Policy

Table 4 .

 4 Optimal control parameters and profit rates for various distributions of machine processing times (simulation results).

	Page 29 of 29
				Distribution
		Exponential	Erlang-2		Erlang-4		Deterministic
	Policy	s c	J	s c	J	s c	J	s c	J
	BSBB	11 3 259.26	9 4 287.65	8 5 304.86	7 5 327.47
	MTO	0 11 177.06	0 11 207.27	0 12 226.60	0 12 249.75
	LS	12 0 249.96	11 0 275.73	10 0 291.62	9 0 313.46

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Acknowledgements

A. A. Economopoulos was supported in part by a doctoral scholarship from the Alexandros S. Onassis Public Benefit Foundation, Greece.

The above expression is the product-form equilibrium probability of a queueing network with m jobs and nodes i, i -1, …, and 1 in a cycle. Now the well-known arrival theorem for Jackson networks (see, e.g., [START_REF] Sevcik | The distribution of queuing network states as input and output instants[END_REF] says that this probability is the same as the probability of the state seen by a job arriving at node i from node 1 in the same cyclic network but with a total population m + 1. Therefore the distribution of the time until item m + 1 is produced is the same as the distribution of the cycle time in the network with nodes i, i -1, …, and 1 and population m + 1. Moreover, the probability that this time is greater than QLT is given by equation ( 17), as shown in [START_REF] Zazanis | Cycle times in single server cyclic Jackson networks[END_REF].

When all the node rates are different, β k,i (m) and α k,i are given by equation ( 18).

When this is not the case, the coefficients can be computed using a partialfractions expansion of the z-transform of the normalizing constant G i,1 (m + 1) or, equivalently, the method of [START_REF] Lam | An extension of Moore's result for closed queueing networks[END_REF].

Estimation of performance indices.

Combining Propositions 1 and 2 we see that the probability Π m that the waiting time of an arriving customer will exceed QLT when m orders are already pending is given by

In the special case s = 0 we have a make-to-order policy, and each workpiece in the system is associated with an order. Thus, when order m + 1 is placed, the corresponding item is released into the buffer of raw items upstream of M N . In this case, equation ( 16) simplifies to Π m = DP N (m), for MTO systems.

Using equations ( 9)-( 10), ( 12)-( 13), (15), and ( 17)-( 19) we can compute the Insert Table 3 about here

Insert Figure 2 about here Finally we investigate the second-order properties of the mean profit rate for the BSBB policy with respect to s and c. Figure 3 shows that the profit rate appears to be a unimodal function in each parameter s and c when the other is fixed. Unimodality with respect to s implies that the optimal base stock for a system with fixed base backlog c is the value s c such that J(s c , c) > J(s c + 1, c).

Unimodality with respect to c can be defined in a dual fashion. Establishing this property could reduce the computational effort for tracking down the optimal parameters for BSBB since, at step 3 or 4 of Algorithm 1, a jump to step 2 would occur whenever a local maximum for s or c, respectively, is attained regardless of validity of the condition of Proposition 3 or 4.

Insert Figure 3 about here

The profit rate also appears to be a submodular function. By submodularity, if c s is the largest value of c at which J(s, c) is maximized for some fixed s, then c s is decreasing in s (here we take decreasing to mean nonincreasing). For example, from the second graph of Figure 3 we see that the maxima of the profit rate for s = 0, 6, 11, and 16 are respectively at c 0 = 10, c 6 = 4 and c 11 = c 16 = 3. Similarly, submodularity of J implies that s c ≤ s c-1 . Use of submodularity can potentially reduce the search effort. For example, if s c ≤ s c-1 holds then, once we have found the optimal base stock s c-1 for base baclog c -1, the optimal base stock s c for c is in the subset {0, 1, …, s c-1 }; thus in step 3 of Algorithm 1 we do not need to try values for s larger than s c-1 even though they may satisfy the inequality of Proposition 3.