The inventory routing problem for the Mixed Car Model Assembly Line

Transcription

1 The inventory routing problem for the Mixed Car Model Assembly Line Raul Pulido, Alvaro García-Sánchez y Miguel Ortega-Mier CIO 2013 Valladolid 1

2 Relevance A high inventory level in the assembly line is a big cost contributor. Some of the car manufacturer s objectives are keeping low stock levels, performing the replenishment of the production line, and providing the required components at the right time (Monden, 1983). The contribution of this paper is a MIP for the feeding of the mixed model assembly line to minimize the cost of the replenishment, keeping a proper inventory level CIO 2013 Valladolid 2

3 Abstract A car assembly line usually produces hundreds of cars every day; each workstation in the assembly line needs car components to perform their task. The replenishment of the components is a critical issue for the proper operation of the assembly line. A lack of inventory could cause some problems in the production line; excess inventory could also create it. CIO 2013 Valladolid 3

4 Problem description The assembly line and the production plan already exists. Each model has a set of characteristics, such as types of wheels and tires, radio, sunroof, car seat, and so on. In every workstation, a kit of components is installed; these components can have different trim levels. The combination of components and trims give us the characteristics. Radio Tires GPS Sunroof Component n Model A Low(CD) High High High Low Model B High(MP3) High High High High Model 5 High(MP3) Low Low Low High CIO 2013 Valladolid 4

5 Problem description (cont) The early arrival of the component causes space problems with the buffers of the production lines; the late arrival causes several problems in the production line such as line stoppage. It is assumed that each component needed for the planning horizon is available in a single warehouse from where all the routes depart. A route is defined as the course taken by the transportation vehicles in order to get from the warehouse to the stations. Each route will be attended by one transportation vehicle. CIO 2013 Valladolid 5

6 Problem description (cont) The components needed for the operation of a station are delivered as a kit, where all the pieces needed for the operation are in the warehouse one day before the operation. Additionally, it is assumed that the model demand is known for the planning period and all the costs are known. The solution consists in finding routes for the transportation vehicles and the amount needed in order to replenish the components. The model contemplates safety stock to mitigate the risk due to any uncertainty; the level of the safety stock is determined by company policy and can be set to zero. CIO 2013 Valladolid 6

7 Problem description (cont) CIO 2013 Valladolid 7

8 Sets & Parameters Index Set R L M J τ Parameters ST js STo js A Y js TC Hj T CAP TDIS ll All routes operated by a transportation vehicle All locations All car model configurations Characteristic J all car components trim levels Discretized production time Safety stock corresponding to characteristic j J in the station s S Initial stock corresponding to characteristic j J in the station s S Amortization per transportation vehicle; it has to be paid if the transportation vehicle is used at least once 1 if component corresponding to characteristic j J is installed in station s S, 0 otherwise Traveling cost per distance unit Unitary holding cost of component corresponding to characteristic j J per time unit Number of cycles to be planned T= τ Maximum capacity of kits in a transportation vehicle Displacement time from l L to l ' L CIO 2013 Valladolid 8

9 Variables wr ł xr ł l' tr l demj τ s dem ac j τ s stj τ s αr qjl r fj l l' r p Variables 1 if route r R attends l L; 0 otherwise 1 if l L immediately precedes l' L, on route r R; 0 otherwise Discrete time in which the route r R arrives to the location l L Demand for component corresponding to characteristic j J in cycle τ τ, at station s S Accumulated demand for the component corresponding to characteristic j J at the beginning of cycle τ τ at station s S Stock of component corresponding to characteristic j J in the station s S at the beginning of cycle τ T 1 if the route r R is used for the replenishment; 0 otherwise Amount of component required with characteristic j J in station l L in route r R The flow of component corresponding to characteristic j J between l and l ' L in route r R CIO 2013 Valladolid 9

10 Objective function min. rrrrrrr TTTT TTTTTTTTTTTT x rrtttt + AA rr αα rr + jj HH jj ssss jjjjjj (3.1) The model is subject to the routing constraints equation (3.2) to (3.7) and to the inventory constraints equations (3.8) to (3.16). CIO 2013 Valladolid 10

11 Formulation Routing Inventory rrrr X rllr = 1 l ' L \{WH} (3.2) rrrr X rrrrrrr = 1 l L \{WH}' (3.3) rrrr X rllr - rrrr X rlrlrr = 0 l ' L, r R (3.4) lllll X rllr ' M α r r R (3.5) lllll X rllr ' L r R (3.6) α r α r+1 r R (3.7) jj ff jllr r CAP l, l ' L, r R (3.8) tr l' tr l + TDIS ll M(1 x r l l ) r R, l, l ' L (3.9) fjll' r fjl' l'' r qjl' r M(1 xr l l' ) M(1 xr l' l'' ) (3.10) aaaa aaaa cc jjjjjjjjj = cc jjll rrrr 1 + cc jjjjjjjjj j J, l ' L, r R, τ T \{1} (3.11) stj τ s = stj 0 s dem ac aaaa jτ s + rr cc jjjjjjjj j J, s S, τ T \{1} (3.12) dem ac jτ s = dem ac jτ 1 + demjτ s j J, s S, τ T \{1} (3.13) stj τ s ST js j J, τ T, s S (3.14) demj τ s = Rm jym js j J, s S, τ T (3.15) stj τ s = SToj s j J, s S CIO 2013 Valladolid 11

12 Results M3 M4 M5 M5 M4 M5 M5 M1 M5 M5 M3 M4 M2 M5 M5 M2 M5 M4 M3 M5 M5 M4 M5 M4 M5 M4 M5 M5 M4 M4 WH S1 S2 S3 S4 S5 Vehicle Vehicle CIO 2013 Valladolid 12

13 Practical conclusions In this work, the inventory and the routing problem have been solved jointly. The routing model should consider more factors than just the transportation cost. The main factor in the delivery of material should not only be the decrease of the transportation costs but also the decrease of the holding cost of the components. The cost of the space is an amplifier of the savings of the model. The replenishment is made before the inventory level reaches the safety stock. Following the Lean idea, it is possible to decrease the safety stock until it reaches zero safety stock, always keeping in mind the risk of any delay with the consequence of the stoppage of the assembly line. CIO 2013 Valladolid 13

14 Future research As this is a NP hard problem, many research directions come up. The first one is to try to change the sequence in order to create a joint algorithm to find a joint solution. Meta-heuristic has to be developed to solve bigger problems. CIO 2013 Valladolid 14

15 Thank you! CIO 2013 Valladolid 15