THIS ARTICLE INVESTIGATES THE PERFORMANCE OF THE WEIGHTED SHORTEST PROCESSING TIME (WSPT) RULE AS A LOCAL SEQUENCING POLICY IN A SCHEDULING GAME FOR UNIFORMLY RELATED PARALLEL MACHINES, WHERE THE SOCIAL OBJECTIVE IS THE TOTAL WEIGHTED COMPLETION TIME. OUR RESEARCH AIMS TO ESTABLISH IMPROVED UPPER BOUNDS FOR THE PRICE OF ANARCHY IN THIS GAME. WE DETERMINE TWO BOUNDS, INCORPORATING PARAMETERS THAT CHARACTERIZE THE INSTANCE FAMILY, SUCH AS THE SPEED OF THE FASTEST MACHINE (??) AND THE NUMBER OF MACHINES (M). ONE BOUND ESTABLISHES A FIXED UPPER BOUND FOR THE PRICE OF ANARCHY, WHILE THE OTHER OUTPERFORMS THE PARAMETRIC UPPER BOUND FOUND IN THE EXISTING LITERATURE. THESE NEWLY DERIVED BOUNDS PROVIDE BETTER INSIGHTS INTO THE PERFORMANCE OF THE SCHEDULING GAME UNDER STUDY, PROVING THAT THE PRICE OF ANARCHY IS UPPER BOUNDED BY MIN{??(1+1/2???1/2?),?,4}.

THIS STUDY INVESTIGATES THE WORST-CASE PERFORMANCE GUARANTEE OF LOCALLY OPTIMAL SOLUTIONS TO MINIMIZE THE TOTAL WEIGHTED COMPLETION TIME ON UNIFORMLY RELATED PARALLEL MACHINES. THE INVESTIGATED NEIGHBORHOOD STRUCTURE IS JUMP, ALSO CALLED INSERTION OR MOVE. THIS RESEARCH FOCUSED ON ESTABLISHING THE LOCAL OPTIMALITY CONDITION EXPRESSED AS AN INEQUALITY AND MAPPING THAT MAPS A SCHEDULE INTO AN INNER PRODUCT SPACE SO THAT THE NORM OF THE MAPPING IS CLOSELY RELATED TO THE TOTAL WEIGHTED COMPLETION TIME OF THE SCHEDULE. WE DETERMINE TWO NEW UPPER BOUNDS FOR THE PERFORMANCE GUARANTEE, WHICH TAKE THE FORM OF AN EXPRESSION BASED ON PARAMETERS THAT DESCRIBE THE FAMILY OF INSTANCES: THE SPEED OF THE FASTEST MACHINE, THE SPEED OF THE SLOWEST MACHINE, AND THE NUMBER OF MACHINES. THESE NEW BOUNDS OUTPERFORM THE PARAMETRIC UPPER BOUND PREVIOUSLY ESTABLISHED IN THE EXISTING LITERATURE AND ENABLE A BETTER UNDERSTANDING OF THE PERFORMANCE OF THE SOLUTIONS OBTAINED FOR THE JUMP NEIGHBORHOOD IN THIS SCHEDULING PROBLEM, ACCORDING TO PARAMETERS THAT DESCRIBE THE FAMILY OF INSTANCES.

THE PRODUCTION PLANNING PROBLEM IN THE PINUS RADIATA SAWMILL INDUSTRY REVOLVES AROUND DETERMINING HOW TO CUT A SET OF LOGS OF DIFFERENT DIAMETERS TO OBTAIN PIECES WITH A RECTANGULAR BASE, TYPICALLY OF THE SAME LENGTH AS THE ORIGINAL LOG. THE PRIMARY OBJECTIVE IS OFTEN TO MAXIMIZE THE VOLUMETRIC YIELD OR THE RATIO BETWEEN THE TOTAL VOLUME OF THE PRODUCED PIECES AND THE AVAILABLE VOLUME OF THE LOGS. GIVEN THE SCARCITY OF TIMBER FORESTS, SMALL AND MEDIUM-SIZED (SME) SAWMILLS MUST OPTIMIZE THEIR OPERATIONS TO MAXIMIZE THE VOLUMETRIC YIELD OF LOGS, USUALLY PROCURED FROM THIRD PARTIES AND MAY VARY IN QUALITY AND DIMENSIONS. IN THIS CONTEXT, THE ABSENCE OF DECISION SUPPORT TOOLS DIRECTLY CONTRIBUTES TO INEFFICIENT RAW MATERIAL UTILIZATION, CONSEQUENTLY IMPACTING THE BUSINESS?S PROFIT. THIS STUDY INTRODUCES A MULTI-OBJECTIVE MIXED-INTEGER LINEAR PROGRAMMING MODEL INCORPORATING LOG AVAILABILITY AND PRODUCT DEMAND AS INPUT PARAMETERS. THE OBJECTIVE FUNCTIONS AIM TO MINIMIZE THE TOTAL VOLUMETRIC LOSS OF UTILIZED LOGS AND THE SURPLUS QUANTITY ASSOCIATED WITH PRODUCTS EXCEEDING DEMAND. THE MODEL INTEGRATES CUTTING PATTERNS PRE-DETERMINED FOR EACH LOG DIAMETER AS AN ADDITIONAL INPUT. THE CUTLOG SOFTWARE WAS USED TO IDENTIFY ALL THE OPTIMAL CUTTING PATTERNS. THE ?-CONSTRAINT METHOD, IMPLEMENTED IN THE CPLEX SOLVER, WAS EMPLOYED TO SOLVE THE MODEL. THE MODEL WAS VALIDATED USING REPRESENTATIVE INSTANCES DESIGNED TO EMULATE THE CHALLENGES FACED BY SME SAWMILLS. REAL INDUSTRY DATA, SURVEYS, AND COMMERCIAL RECORDS FROM SME SAWMILLS IN SOUTHERN CHILE WERE UTILIZED. THE RESULTS CONFIRM THE EFFECTIVENESS OF THE PROPOSED MODEL IN ADDRESSING THE MULTI-OBJECTIVE CHALLENGES ENCOUNTERED BY THESE BUSINESSES. THE MODEL SUCCESSFULLY IDENTIFIES MULTIPLE SOLUTIONS ON THE PARETO FRONTIER, OFFERING VALUABLE INSIGHTS FOR DECISION-MAKING.

WE STUDY THE WORST CASE PERFORMANCE GUARANTEE OF LOCALLY OPTIMAL SOLUTIONS FOR THE PROBLEM OF SCHEDULING JOBS ON UNIFORMLY RELATED PARALLEL MACHINES WITH THE OBJECTIVE OF MINIMIZING THE TOTAL WEIGHTED COMPLETION TIME. THE QUALITY OF LOCALLY OPTIMAL SOLUTIONS UNDER THE JUMP NEIGHBORHOOD IS ANALYZED, WHICH CONSISTS OF ITERATIVELY MOVING A SINGLE JOB FROM ONE MACHINE TO ANOTHER, IMPROVING THE TOTAL WEIGHTED COMPLETION TIME IN EACH ITERATION AND STOPPING ONCE IMPROVEMENT IS NO LONGER POSSIBLE. WE PROPOSE AN UPPER BOUND FOR THE TOTAL WEIGHTED COMPLETION TIME FOR THE SOLUTIONS OBTAINED BY THIS LOCAL SEARCH, AND UPPER AND LOWER BOUNDS FOR THE PERFORMANCE GUARANTEE OF THE OBTAINED LOCALLY OPTIMAL SOLUTIONS. ADDITIONALLY, THE CASE OF IDENTICAL PARALLEL MACHINES IS ANALYZED.

WE STUDY THE WORST-CASE PERFORMANCE GUARANTEE OF LOCALLY OPTIMAL SOLUTIONS FOR THE PROBLEM OF MINIMIZING THE TOTAL WEIGHTED AND UNWEIGHTED COMPLETION TIME ON PARALLEL MACHINE ENVIRONMENTS. OUR METHOD MAKES USE OF A MAPPING THAT MAPS A SCHEDULE INTO AN INNER PRODUCT SPACE SO THAT THE NORM OF THE MAPPING IS CLOSELY RELATED TO THE COST OF THE SCHEDULE. WE APPLY THE METHOD TO STUDY THE MOST BASIC LOCAL SEARCH HEURISTICS FOR SCHEDULING, NAMELY JUMP AND SWAP, AND ESTABLISH THEIR WORST-CASE PERFORMANCE IN THE CASE OF UNRELATED, RESTRICTED RELATED AND RESTRICTED IDENTICAL MACHINES.

THIS ARTICLE INVESTIGATES THE EFFICIENCY OF NASH EQUILIBRIA IN A UTILITARIAN SCHEDULING GAME, WHERE EACH JOB, ACTING AS AN AGENT, STRATEGICALLY SELECTS A MACHINE FOR ITS PROCESSING. THE STUDY FOCUSES ON A UNIFORM PARALLEL MACHINE ENVIRONMENT, EMPLOYING THE WEIGHTED SHORTEST PROCESSING TIME RULE AS THE LOCAL POLICY AND THE TOTAL WEIGHTED COMPLETION TIME AS THE SOCIAL GOAL. WE ESTABLISH UPPER AND LOWER BOUNDS FOR THE PRICE OF ANARCHY, OFFERING VALUABLE INSIGHTS FOR THIS SCHEDULING GAME.