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Inventory control for multi-echelon systemCourse of Inventory Management

LOGOItems with Independent demand

Items with dependent demand

CharacteristicsISE - Inventory ManagementTruong Ba Huy2Example 1: A Series SystemItem NItem 1Item 2Customer demandExternal supplyISE - Inventory ManagementTruong Ba Huy3Example 2: An Assembly System1Customer demand234567External supplyISE - Inventory ManagementTruong Ba Huy4Example 3: A Disassembly SystemExternal supply 76 5 4 3 2 1Customer demandISE - Inventory ManagementTruong Ba Huy5Example 4: A Distribution SystemExternal supply 1Customer demand 2 5 43ISE - Inventory ManagementTruong Ba Huy6Disassembly/assembly systems

Assembly/distribution systems

Distribution systems with transshipments

Distribution systems with multiple supply sources

Other ExamplesISE - Inventory ManagementTruong Ba Huy7Lot Sizing with Multiple EchelonsISE - Inventory ManagementTruong Ba Huy89ExampleReactorReactor Feed AIntermediate EFeed CFeed BFeed DReactorReactorReactorProduct 1Product 2Intermediate FIntermediate GIntermediate HISE - Inventory ManagementTruong Ba Huy910The Item-Task Network RepresentationFeed AIntermediate EFeed CFeed BFeed DEnd Product 1End Product 2Intermediate FIntermediate GIntermediate H Task 2 Task 5Task1Task2Task5Task3Task4ISE - Inventory ManagementTruong Ba Huy10Items & TasksAn item can be a component purchased from an outside supplier or produced internally.An item can be a raw material (e.g., a component), a semi-finished (e.g., sub-assembly) or a finished product.A task can consume and produce multiple items (components/intermediates/products). An item can be consumed by more than one task; similarly, an item can be produced by more than one task. A finished product can be the result of several tasks done in series or in parallel.ISE - Inventory ManagementTruong Ba Huy11ExampleTask1Task2Task3Task41245367828556419551371ISE - Inventory ManagementTruong Ba Huy12System Descriptiont: a period (e.g., day, week, month); t = 1, ,T, where T represents the planning horizonDrt: demand for item r in period t (number of units), r=1,, R where R is the number of itemsrir: number of units of item r needed to carry out task i, i=1,, N where N is the number of taskssir: number of units of item r produced by task i

ISE - Inventory ManagementTruong Ba Huy13Given a demand profile over a set of T periods for each item (demand can be for either finished or semi-finished products), determine the quantity of each item to produce in each period in order to minimize the production, inventory and setup costs, while meeting demand and without exceeding production capacity.Problem StatementISE - Inventory ManagementTruong Ba Huy14FormulationsBig bucket formulation (the production planning problem)

Small bucket formulation (the production scheduling problem)

ISE - Inventory ManagementTruong Ba Huy15A Big-Bucket FormulationTasks initiated in a period are completed during the same periodThe same task can be carried out multiple times during a periodItems produced in a period can be used to satisfy demand during that periodDemand in each period must be satisfied in that period; no backorders allowedThere are no capacity limits (no limits on the number of times a task can be carried out in a given period)

ISE - Inventory ManagementTruong Ba Huy16NotationParameterscit: variable cost of carrying out task i in period t, i=1,, N where N is the number of tasksAit: fixed cost of carrying out task i in period t (incurred at most once during each period)hrt: cost of holding one unit of item r in inventory from period t to period t+1

ISE - Inventory ManagementTruong Ba Huy17Notation (Continued)Decision variablesIrt: inventory level of item r at the end of period t Qit: the number of times task i is carried out in period tYit = 1 if task i is initiated one or more times during period i and Yit = 0 otherwiseISE - Inventory ManagementTruong Ba Huy18Notation (Continued)Amount of item r produced in period t =

Amount of item r consumed in period t =

ISE - Inventory ManagementTruong Ba Huy19

FormulationISE - Inventory ManagementTruong Ba Huy20

FormulationM is a large numberISE - Inventory ManagementTruong Ba Huy21A Formulation with Capacity Constraints and Multiple MachinesA machine may correspond to a single processor, an assembly workstation, or a production line, among othersIt is possible for a task to be carried out on one or more machinesA machine could possibly carry out more than one taskA machine has a finite capacity and carrying out any task consumes some of this capacity ISE - Inventory ManagementTruong Ba Huy22NotationUtm: capacity of machine m in period t; m = 1, , M, where M is the number of machinesaim: units of capacity of machine m needed to carry out task ici,t,m: variable cost of carrying out task i on machine m in period tAi,t,m: fixed cost of carrying out task i on machine m in period t

ISE - Inventory ManagementTruong Ba Huy23 Note: A task that cannot be carried out on a machine is assigned a very large production cost.ISE - Inventory ManagementTruong Ba Huy24Notation (Continued)Qi,t,m: the number of times task i is carried out in period t on machine m

Yi,t,m = 1 if task i is initiated one or more times during period i on machine m and Yi,t,m = 0 otherwise

ISE - Inventory ManagementTruong Ba Huy25Formulation

ISE - Inventory ManagementTruong Ba Huy26A Formulation with Setup TimesA setup time si,m is incurred if task i is carried out on machine m, one or more times in any given period.

The capacity constraint is modified as follows

ISE - Inventory ManagementTruong Ba Huy27A Small-Bucket FormulationTime periods are chosen to be small enough so that only one task on a particular machine can be either initiated or completedThe processing time of each task consists of one or more periods A setup cost is incurred when the task initiated on a machine is different from the task that was just completed on that machine.ISE - Inventory ManagementTruong Ba Huy28Notationai,m: processing time (in number of time periods) of task i on machine m

Qi,t,m = 1 if task i is initiated on machine m at time t, and Qi,t.m = 0 otherwise

Zi,t,m = 1 if machine m at time t is set up for task i ISE - Inventory ManagementTruong Ba Huy29

FormulationISE - Inventory ManagementTruong Ba Huy30Solution MethodsSmall to medium problems can be solved exactly (to optimality)Large problems may not solve within a reasonable amount of time (the problem belongs to a class of combinatorial optimization problems called NP-hard)Large problems can be solved approximately using a heuristic approach

ISE - Inventory ManagementTruong Ba Huy31Example HeuristicsDecompose the problem into a series of sub-problems with smaller planning horizonsDecompose the problem into a series of subproblems, each pertaining to a single itemDecompose the problem into one big bucket problem (the production planning problem)Solve the problem by relaxing one or more sets of constraintsSolve the problem on a rolling horizon basis

ISE - Inventory ManagementTruong Ba Huy32