Valid Inequalities for the Proportional Lotsizing and Scheduling Problem with Fictitious Microperiods. Waldemar Kaczmarczyk

Transcription

1 Valid Inequalities for the Proportional Lotsizing and Scheduling Problem with Fictitious Microperiods Waldemar Kaczmarczyk Department of Operations Research AGH University of Science and Technology Kraków, Poland 29th European Conference On Operational Research Valencia, 8-11 July 2018 Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 1 / 17

2 Contents 1 Basic model 2 Motivation 3 Valid inequalities One lot per product per week at most Predefined sequence of products 4 Experiments 5 Conclusions Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 2 / 17

3 Basic model Lot sizing and scheduling discrete time scale, i.e., planning horizon split in periods, several products with deterministic dynamic (variable) demand, single machine with limited capacity, minimize machine set-up costs and inventory holding costs. Small time buckets one set-up per time period at most, PLSP one product before and one after each set-up, to enable high-quality solutions real periods (macroperiods, here: weeks) are often split into shorter fictitious periods (microperiods, here: shifts), Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 3 / 17

4 Motivation I Multi-level environments dependent demand, multi-stage (flow) lines, Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 4 / 17

5 Motivation II Lead time integer multiple of period (shift) length Zero components and finished products processed during the same period, for basic models solution may be not feasible, for advanced models additional variables and constraints (Stadtler, 2011, Stadtler and Sahling, 2013), One batch of components completed in a period before processing of finished products, long periods large transfer batches: long cycle times, high work-in-process, i.e., high inventory holding cost and may infeasible because of storage area upper limit, short periods many binary variables. Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 5 / 17

6 Motivation III Models with fictitious microperiods the whole demand of a week is cumulated at the end of its last shift, inventory holding costs are accounted only at the end of each week, approximation of a large bucket model, Weeks 1 2 Shifts Demand Holding cost 5 5 Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 6 / 17

7 Motivation IV Properties of models with fictitious periods If demand may be non-zero only at the end of week, then: 1 One lot per product per macroperiod at most (Tempelmeier and Buschkühl, 2008). 2 Different sequences of lots during of a week give indistinguishable symmetrical solutions, if set-up costs are not sequence dependent. The sequence of lots may be fixed in advance: except weekly set-up carry-over i.e., the first and last product during a week. may lead to non-optimal solution, minimize holding cost if products sorted in increasing order of the ratio of the holding cost to processing time. Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 7 / 17

8 Notation N set of products, W set of weeks (macroperiods), T set of shifts (microperiods), T w set of shifts during week w, L(w) last shift in week w, I jt inventory of product j at the end of period t, x jt production volume of product j during period t, y jt = 1, if at the end of period t machine is set up for j; 0 otherwise, z jt = 1, if during period t machine starts up to process j; 0 otherwise. Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 8 / 17

9 VI: One lot per product per week at most I 1 permit any start-up if the machine was set-up already at end of the previous week: y j,l(w 1) + z jt 1, j N, w W, t T w, (y0z) 2 permit all start-ups if the machine was set-up already at the end of the previous week: y j,l(w 1) + s T w z js 1, j N, w W, (y0w) 3 permit start-up if the machine was already set-up in any previous period of the same week, y j,t 1 + z js 1, j N, w W, t T w (yw) s T w : s t Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 9 / 17

10 VI: One lot per product per week at most II 4 new binary variable weekly set-up state Y jw : limit the numbers of start-ups, variables describing aggregated decisions, create nodes in the search tree (Willimas, 1999), Y jw {0, 1}, j N, w W, Y jw = y j,l(w 1) + z jt, j N, w W, (y0wb) t T w 5 inventory lower bounds only for the last shift of each week: w+p u I j,l(w 1) d ju [1 Y jr ], u=w r=w j N, w W, p = 0,..., W w (ilbw) Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 10 / 17

11 VI: Predefined sequence of products I Sequence of lots consistent with increasing order of product indices. 1 Permit start-up for product j after set-up for product k if j < k: z j,s, 1 y kt + y k,l(w 1) + y j,l(w) s T w :s>t j, k N : j < k, (syy) 2 product with higher index ca to precede product with smaller j y jt k y k,t+1 + n(2 b t e t ), j N k N w W, t T w \ {L(w) 1, L(w)}, (sjy) Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 11 / 17

12 VI: Predefined sequence of products II 3 with flow and set-up variables f kjt S ( ) y k,l(w 1) + y j,l(w) t T w 4 with flow and set-up or start up variables: j, k N : j < k, w W (sf) f kjt y k,l(w 1) + y j,l(w) j, k N : j < k, w W, t T w (sff, sffz) Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 12 / 17

13 Data sets 36 different data sets, 5 or 10 products, 6 weeks, numbers of shifts: 3, 5 or 10 for 5 products, 5, 10 or 15 for 10 products, 3 random demand instances for 5 and 10 products, average replenishment cycles in range [1, 2] or [3, 4] weeks, machine utilization 80%, Cplex , Intel Core i7, 2.8 MHz, 8 GB RAM. Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 13 / 17

14 Results 5 products Valid Obj. MIP Time Nodes Iterations It./ It. time ineq. [%] gap[%] [s] node [µs] None y0z y0w yw y0wb ilb ilbc ilbw syy sjy sf sff sffz means that all values are equal to zero, Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 14 / 17

15 Results 10 products Valid Obj. MIP Time Nodes Iterations It./ It. time ineq. [%] gap [%] [s] node [µs] None y0wb ilb ilbw ilbw, ilb ilbw, syy ilbw, sjy ilbw, sf ilbw, sff ilbw, sffz means that all values are equal to zero, 0.0 means that average value is smaller than 0.05 Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 15 / 17

16 Impact of microperiod number on results for ten products a) Time [s] b) MIP gap [%] Number of shifts Number of shifts Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 16 / 17

17 Conclusions A. Weekly binary set-up variable and weekly inventory lower bounds reduce number of nodes, iterations and computational effort, make the small bucket models more competitive against the large bucket models, B. Predefined sequence reduces number of nodes, but increases iterations and computational effort; increases total cost in average by 0.1%, i.e., could be used in heuristics. Waldemar Kaczmarczyk (AGH) Valid inequalities for fictitious microperiods EURO 2018, Valencia 17 / 17