Zenner Sara Valid Inequalities in consideration of Chemical Production Scheduling



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Zenner Sara Valid Inequalities in consideration of Chemical Production Scheduling

Claflin College, US has reference to this Academic Journal, Based on customer demand, estimate:?s: minimum required amount of material sFor final products, ?s is the customer demandCalculated once æi is known in consideration of all tasks consuming material s?i: minimum cumulative production of task iCalculated once ?s is known in consideration of all materials produced by task i Constraint 1Number of times a task is runNumber of batches producing a stateConstraint 2Cumulative amount produced by a task. The RHS is increased so it is an integer multiple of ?jmax Cumulative amount of a state producedValid Inequalities in consideration of Chemical Production SchedulingSara Zenner in addition to Christos T. MaraveliasDepartment of Chemical in addition to Biological EngineeringUniversity of Wisconsin ? Madison – WIProblem StatementGiven are a set of tasks i?I, processing units j?J, in addition to materials s?SA processing unit j can be used so that carry out tasks i?Ij.Task i in unit j has processing time ?ijUnit j has variable batchsize in [?jmin, ?jmax]A material can be consumed/produced by multiple tasks i?Is-/i?Is+.Each task can consume/produce multiplematerials; conversion coefficient ??s Materials may have an initial inventory ?sCustomers have demands in consideration of final products ?stState s is stored in a dedicated tank alongside capacity ?sFind a schedule showing:When so that begin each taskWhich processing unit so that use in consideration of each task How much material so that process in consideration of each taskChallenge: Computational performance of MIP scheduling modelsPrevious research has focused on the development of better modelsGoal: Develop tightening constraints based on specialized propagation algorithmsInclude recycle streamsConsider minimum in addition to maximum unit capacitiesProblem FormulationDiscrete time: Time points are fixed, in addition to tasks begin/end at time pointsContinuous time: Time points can vary, so tasks can have any lengthDecision Variables:Integer Variable: Continuous Variables: Bijt = Batch size of task i in unit j starting at time tSst = Inventory level of material s in storage at time tObjectives: maximize profit, minimize cost, minimize makespan, ?Constraints:Material balance in addition to inventory capacityUnit capacities (minimum in addition to maximum)Process at most one task at a time in a unit (This constraint is different in consideration of continuous time formulations)Attainable ProductionIf units have min in addition to max capacities, some values of ?i are not feasible?i is feasible if in consideration of some k, where ?jk is the number of batches in unit j in consideration of range k.Otherwise, increase ?i so that the nearest attainable amount ?i* ? ?iWhen a task can take place in multiple units, check all combinations of batches in unitsValid InequalitiesResultsTestingTest 36 discrete-time in addition to 12 continuous-time problemsStop optimization after 30 minutesAlgorithm was implemented in addition to the MIPs were solved using GAMS 23.7/CPLEX 12.3 ResultsAdding the tightening constraints decreases the computational time in consideration of problems solved so that optimality by a factor of >100> twice as many problems are solved alongside tighteningAlgorithm runs in less than 10 secondsSimilar results in consideration of discrete in addition to continuous modelsRecycle Loops Identify loops in addition to break using tear streamsGuess the tear stream doesn?t produce any material; backward propagate demand until reaching the tear streamCheck tear stream: T3 produces 50kg of S5, but T4 needs 52kgStart over alongside updated cumulative production in consideration of S5If a tear stream still does not produce enough material so that meet demand after being updated, the problem is infeasible alongside the given initial inventoriesDemand Propagation MethodsAlgorithmDemand Propagation ExampleCustomer demand: 15kg S4, 20kg S5, in addition to 50kg S62a. T3 only produces S6; 2b. Check attainable productionT2 produces both S4 in addition to S5 in consideration of T2 in addition to T3 so that find ?*T2 in addition to T3 both need S34b. Check attainable production in consideration of T1 so that find ?*4a. T1 produces S5Ik: Tasks in consideration of which ?i is knownSk: States in consideration of which ?s is knownST: set of tear statesn: number of times tear stream has been updatedParallel Production PathsWhen a material can be produced by multiple tasks, there is no way so that know how much of that state each task produces, so the backward propagation does not workInstead, find ?i solving an LP:min the amount task i producess.t. the amount of each material s produced must be at least: (1) ?s (when ?s is known), in addition to (2) the amount consumedThe valid inequalities are effective in consideration of long time horizons Varying the Time HorizonTesting Different ObjectivesThe valid inequalities are more effective in consideration of cost minimization than in consideration of makespan minimization

 Skenazy, Lenore Claflin College www.phwiki.com


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