Beyond Smith's rule: An optimal dynamic index, rule for single machine stochastic scheduling with convex holding costs

Most research on single machine scheduling has assumedthe linearity of job holding costs, which is arguablynot appropriate in some applications. This motivates ourstudy of a model for scheduling $n$ classes of stochasticjobs on a single machine, with the objective of minimizingthe total expected holding cost (discounted or undiscounted). We allow general holding cost rates that are separable,nondecreasing and convex on the number of jobs in eachclass. We formulate the problem as a linear program overa certain greedoid polytope, and establish that it issolved optimally by a dynamic (priority) index rule,whichextends the classical Smith's rule (1956) for the linearcase. Unlike Smith's indices, defined for each class, ournew indices are defined for each extended class, consistingof a class and a number of jobs in that class, and yieldan optimal dynamic index rule: work at each time on a jobwhose current extended class has larger index. We furthershow that the indices possess a decomposition property,as they are computed separately for each class, andinterpret them in economic terms as marginal expected cost rate reductions per unit of expected processing time.We establish the results by deploying a methodology recentlyintroduced by us [J. Niño-Mora (1999). "Restless bandits,partial conservation laws, and indexability. "Forthcomingin Advances in Applied Probability Vol. 33 No. 1, 2001],based on the satisfaction by performance measures of partialconservation laws (PCL) (which extend the generalizedconservation laws of Bertsimas and Niño-Mora (1996)):PCL provide a polyhedral framework for establishing theoptimality of index policies with special structure inscheduling problems under admissible objectives, which weapply to the model of concern.

The implicit theory of historical change in the work of Alan S. Milward

Alan S. Milward was an economic historian who developed an implicit theory ofhistorical change. His interpretation which was neither liberal nor Marxist positedthat social, political, and economic change, for it to be sustainable, had to be agradual process rather than one resulting from a sudden, cataclysmicrevolutionary event occurring in one sector of the economy or society. Benignchange depended much less on natural resource endowment or technologicaldevelopments than on the ability of state institutions to respond to changingpolitical demands from within each society. State bureaucracies were fundamentalto formulating those political demands and advising politicians of ways to meetthem. Since each society was different there was no single model of developmentto be adopted or which could be imposed successfully by one nation-state onothers, either through force or through foreign aid programs. Nor coulddevelopment be promoted simply by copying the model of a more successfuleconomy. Each nation-state had to find its own response to the political demandsarising from within its society. Integration occurred when a number of nation states shared similar political objectives which they could not meet individuallybut could meet collectively. It was not simply the result of their increasinginterdependence. It was how and whether nation-states responded to thesedomestic demands which determined the nature of historical change.

Patients's satisfaction concerning postoperative analgesia after caesarean section: a comparison of 4 alternative analgesic regimen

Introduction: Different routes of postoperative analgesia may be used after cesarean section: systemic, spinal or epidural [1]. Although the efficacy of these alternative analgesic regimen has already been studied [2, 3], very few studies have compared patients' satisfaction between them. Methodology: After ethical committee acceptation, 100 ASA 1 patients scheduled for an elective cesarean section were randomized in 4 groups. After a standardized spinal anesthesia (hyperbaric bupivacaine 10 mg and fentanyl 20 μg), each group had a different postoperative analgesic regimen: - Group 1: oral paracetamol 4x1 g/24 h, oral ibuprofene 3x600 mg/24 h and subcutaneous morphine on need (0.1 mg/kg 6x/24 h) - Group 2: intrathecal morphine (100 μg) and then same as Group 1 - Group 3: oral paracetamol 4x1 g/24 h, oral ibuprofene 3x600 mg/24 h and PCEA with fentanyl 5 μg/ml epidural solution - Group 4: oral paracetamol 4x1g/24 h, oral ibuprofene 3x600 mg/ 24 h and PCEA with bupivacaine 0.1% and fentanyl 2 μg/ml epidural solution After 48 hours, a specific satisfaction questionnaire was given to all patients which permitted to obtain 2 different scores concerning postoperative analgesia: a global satisfaction score (0-10) and a detailed satisfaction score (5 questions scored 0-10 with a summative score of 0-50). Both scores, expressed as mean ± SD, were compared between the 4 groups with a Kruskall-Wallis test and between each group with a Mann-Whitney test. A P-value <0.05 was considered significant. Results: Satisfaction scores Gr. 1 (n = 25) Gr. 2 (n = 25) Gr. 3 (n = 25) Gr. 4 (n = 25) P-value global (0-10) 8.2 ± 1.2 9.0 ± 1.0 7.8 ± 2.1 6.5 ± 2.5 0.0006 detailed (0-50) 40 ± 6 43 ± 5 38 ± 6 34 ± 8 0.0002 Conclusion: Satisfaction scores were significantly better in patients who received a systemic postoperative analgesia only (Groups 1 and 2) compared to patients who received systemic and epidural postoperative analgesia (Groups 3 and 4). The best scores were achieved with the combination of intrathecal morphine and multimodal systemic analgesia (Group 2) which allowed early ambulation without significant pain. Patients treated with postoperative epidural analgesia with combined local anesthetics and opioids (Group 4) obtained the worse scores (more restrictive nursing with less mobility, frequent asymmetrical block with insufficient analgesia on one side and motor block on the other)

Customer Focus Newsleter, October, November 2004, Vol. 1, no. 1

With this newsletter, we begin a bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, September-October, 2007, Vol. 4, no. 5

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, February-March 2005, Vol. 2, no. 3

With this newsletter, we begin a bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, May-June 2005, Vol. 2, no. 4

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, July-August 2005, Vol. 2, no. 5

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, September-October 2005, Vol. 2, no. 6

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, November-December 2005, Vol. 2, no. 7

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, May-June 2006, Vol. 3, no. 3

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, January-February 2006, Vol. 3, no. 1

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.

Customer Focus Newsleter, March-April 2006, Vol. 3, no. 2

A bi-monthly bulletin to keep the department/agency management teams of state government better informed. We hope to consolidate most of the service update messages we send throughout the month and keep you updated about the work of the Customer Councils. If yours is one of the many departments who participated in the second annual DAS customer satisfaction survey recently, we thank you for taking the time to give us this important feedback. We look forward to sharing survey results with you, and pledge to consider responses carefully as we work to determine benchmarks and set future priorities.