A Lot Sizing Model with Integrated Tour Planning

Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009

A Lot Sizing Model with Integrated Tour Planning Carsten B¨ ohle ∗ Wilhelm Dangelmaier ∗∗ Bernd Hellingrath ∗∗∗ ∗

Heinz Nixdorf Institute, University of Paderborn, Germany (e-mail: [email protected]) ∗∗ Heinz Nixdorf Institute, University of Paderborn, Germany (e-mail: [email protected]) ∗∗∗ University of M¨ unster, Germany (e-mail: [email protected]) Abstract: In spite of the importance of efficient supply chain cooperation, planning in production and transportation are still conducted separately today. This paper presents a model that simultaneously creates lot-based plans for both production and transportation. It is assumed that transports are organized as milk runs, i.e. one truck stops at several suppliers before heading to the buyer. The integration of production planning and inbound logistics tour planning into a dynamic model has not been studied before. After a brief review of literature on integrated planning models, requirements for a model that is able to solve the outlined problem are given. A mathematical model is formulated along with a heuristic which is necessary because the model combines production planning and vehicle routing which are both computationally intensive problems and thus not mathematically solvable in acceptable time. Results show that supply chain-wide costs can be reduced with the help of the presented algorithm. Some comments on the applicability in practice and the need for further research are given in an outlook. Keywords: Production systems, Transportation control, Enterprise integration, Intelligent manufacturing systems 1. INTRODUCTION In the last years, many companies have outsourced a significant share of their production activities. As a result, efficient cooperation in the supply chain has become even more important. This reflects in the extensive research devoted to the subject matter which is documented by the increasing amount of publications. However, very few methods and concepts pay attention to the integration of logistics. In most cases, the cooperation occurs between buyer and supplier, neglecting the carrier. The assumption is that transportation capacities are available at any time in any quantity which disregards potential savings that might stem from a more considerate transportation planning. The carrier is the last actor in the supply chain that is informed about plans. It can be assumed that the successive propagation of plans from buyer to supplier and ultimatively to the carrier results in suboptimal solutions, because small changes for the worse on the buyer’s side may result in bigger overall savings but cannot be realized. In fact, there is almost no leeway left for the carrier, as the optimization space has been narrowed by both the buyer’s and the supplier’s selfish optimization algorithms (cf. fig. 1). It might be argued that opening this space in order to enhance transportation planning will result in higher inventories and is in consequence opposed to the primary objective of lean manufacturing. In fact, this is ? The work on this paper was partially funded by the Fraunhofer Institute for Material Flow and Logistics, Dortmund, Germany.

978-3-902661-43-2/09/$20.00 © 2009 IFAC

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not true, because lean does not mean zero inventory but rather the elimination of waste. Waste may also be present in transportation operations, but as logistics are commonly not part of the analyzed production system, improvements can be made on the expense of transportation. This moves waste out of sight, but does not eliminate it. For holistically better results, the system boundaries have to be extended. The aim of this paper is to present a model that is able to simultaneously create production and transportation plans for procurement in a supply network. The resulting production and transportation plans minimize supply chain wide costs and can also help to reduce CO2 emissions. The transports in this work are organized as milk runs. Those are pickup and delivery tours with multiple pickups and a single delivery. This concept has become popular as it provides for low inventories in combination with low transportation costs, because small quantities are transported more frequently. Direct deliveries are a special case of milk runs and thus inherently included in this work. There are certain drawbacks to the presented approach which will be discussed in Section 7. 2. STATE OF THE ART IN INTEGRATED PLANNING MODELS Both production planning in the form of lot sizing problems and vehicle routing have been studied extensively in numerous variants. In spite of this, the combination of these problems has received relatively little attention.

10.3182/20090603-3-RU-2001.0383

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

for procurement has so far not been formulated. Yung (2006) presents a tactical model that determines annual procurement and supply quantities, thus spanning even one more level in the supply chain. Savings ranging from 5-20% are commonly reported for integrated models. 3. PROBLEM OUTLINE

Fig. 1. Plan propagation in successive planning At the same time, transportation costs have an increasing share in the total costs. Studies suggest that distribution structures will change once a certain oil price is reached, because running more warehouses closer to customers is cheaper compared to shipping on long routes (cf. Gosier (2008)). However, this is a strategic issue in the field of Supply Chain Management. The need for integrated operational models has been identified in the past, though. Voß (2003) briefly addresses the problem of simultaneously planning production and transport and names it a ”rather new research area”. There has been some research in the related field of simultaneous production and distribution planning which is different from the approach pursued here as there is only a single production system. However, these models can add some ideas. Early contributions are made by Blumenfeld (1985) and Blumenfeld (1991) where the effects of synchronizing production and distribution schedules are examined statically. Chandra (1994) presents a dynamic model. Fumero (1999) presents a mathematical formulation and a decomposition procedure for solving. Boudia (2006) deals with a similar model, but devises a tabu search algorithm. Most often the routing problem is omitted when analyzing the integration of supplier’s planning and transportation. Further reviews of the state of the art are given by Sarmiento (1999) and Ereng¨ u¸c (1999). Concerning inbound logistics, Hofmann (1995) presents a static lot sizing model that includes transport lot sizes. Here, transports are organized as direct deliveries. V¨olkl (2006) develops a similar solution. This work also names two assumptions of conventional lot sizing models, namely that variable unit costs are independent of transportation costs and that there are no dependencies between products regarding procurement. Both do not hold in a supply chain environment, because the total price of a unit is less if it is transported efficiently. The dependencies stem from the tours that are based on order and production dates. Kuhn (2008) presents a static model based on economic order quantity (EOQ) that incorporates tour planning. Besides the general sparseness of literature on this topic, a dynamic lot sizing model that includes tour planning

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The integrated planning in this case will occur between a manufacturing company as the buyer and its suppliers as well as one carrier. This is a reasonable scenario derived from reality where also one carrier operates all transports for a factory within a certain region. Each truck delivers to a single destination only. This is based on an observation from practice. Unloading times are unpredictable and can last several hours. Regarding drivers’ working hours, unloading more than once becomes impossible. Plans have to be made for all the actors in the supply chain which determine when a product is produced, when it is transported, and where and for how long it is kept on stock. The combined costs of these plans have to be minimal. This is where the complexity of the problem stems from, because instead of being able to make isolated decisions, now every decision is interrelated to those of other participants, e.g. it could be the case that producing a lot two days before its due date does not yield savings for the supplier, but enables the carrier to plan shorter routes because its pickup can be combined with another one close-by. Finding those globally optimal plans is the aim of this paper. 4. MATHEMATICAL MODEL The model is based on a slightly modified Capacitated Lot Sizing Problem (CLSP, cf. Tempelmeier (2006)) and a Vehicle Routing Problem (VRP, cf. Toth (2002)) with multiple use of vehicles and time windows for delivery. The CLSP was chosen as a starting point because of its simplicity. Later research will have to deal with the problem of how to integrate different lot sizing problems into the integrated model. This is necessary if special restrictions of suppliers have to be included. The CLSP is extended with the additional index z which adds the dimension of supplier to production related variables. At the same time it omits restrictions commonly found in literature as it assumes the same product related production costs over all periods and includes only a single resource per supplier. The VRP includes the multiple use of vehicles, is capacitated in terms of transportation capacity and distance, and adds the edge from buyer to depot to every tour. The changeover from production to transportation is made by the variable tp which subtracts the transported goods from the suppliers’ stock and adds them to the customer’s stock. It is also used for the trucks’ capacity restrictions. tp is thus the connecting link from the production models to the tour planning model. This technique has also been used by Fumero (1999). A list of the symbols used in this model is given in table 1.

min

Z X K X T X (sk ∗ γz,k,t + hk ∗ yz,k,t + pk ∗ qz,k,t ) z

k

t

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

+

Z X L X T N X N X L X T X X (xDepot,z,l,t ∗ f ) + (ci,j ∗ xi,j,l,t ) z

l

t

i

+

T X K X t

j

t

l

(hk ∗ uk,t )

yz,k,t + qz,k,t − yz,k,t =

tpn,k,l,t ≤ M ∗

N X

xn,m,l,t

∀n ∈ Z, l ∈ L, t ∈ T (18)

m

k

(1)

k L X

K X

uk,t +

Z X

tpz,k,t − vk,t = uk,t

∀k ∈ K, t ∈ T

(19)

z

tpz,k,t,l

l

∀z ∈ Z, k ∈ K, t ∈ T

Table 1. Model symbols

(2) Symbol γz,k,t

qz,k,t − M ∗ γz,k,t ≤ 0 K X

∀z ∈ Z, k ∈ K, t ∈ T

(tbk ∗ qz,k,t + trk ∗ γz,k,t ) ≤ bz,t

(3)

∀z ∈ Z, t ∈ T (4)

k

qz,k,t ≥ 0

∀z ∈ Z, k ∈ K, t ∈ T

(5)

∀k ∈ K, t ∈ T

(6)

uk,t ≥ 0 yz,k,t ≥ 0

∀z ∈ Z, k ∈ K, t ∈ T

(7)

∀z ∈ Z, k ∈ K

(8)

∀k ∈ K

(9)

yz,k,0 = 0 uk,0 = 0 γz,k,t ∈ {0, 1}

∀z ∈ Z, k ∈ K, t ∈ T

qz,k,t = qz,k,t ∗ capz,k N X

xi,n,l,t −

i

N X

∀z ∈ Z, k ∈ K, t ∈ T

xn,j,l,t = 0

(10) (11)

∀l ∈ L, t ∈ T, n ∈ N (12)

j

N X N X i

xi,j,l,t ≤ xBuyer,Depot,l,t ∗ M

∀l ∈ L, t ∈ T (13)

j

xDepot,Buyer,l,t = 0 Z X Z X K X i

j

∀l ∈ L, t ∈ T

(14)

(15)

k

N X N X i

(tpi,k,t ∗ xi,j,l,t ) ≤ C

∀l ∈ L, t ∈ T

(ci,j ∗ xi,j,l,t ) ≤ D

∀l ∈ L, t ∈ T

(16)

j

XX

xi,j,l,t ≤ |S| − 1

i∈S j∈S

∀S ⊆ N \{Depot, Buyer}, |S| 6= ∅, l ∈ L, t ∈ T

(17)

590

bz,t ci,j capz,k f hk pk qz,k,t sk tbk tpz,k,l,t trk uk,t vk,t xi,j,l,t yz,k,t C D K L M N S T Z

Definition Binary setup variable for product k at supplier z at time t Capacity of supplier z at time t Distance from node i to node j Ability to produce product k at supplier z Fixed costs per truck Warehousing costs for product k Production costs for product k Lot size of product k at supplier z at time t Setup costs for product k Unit processing time for product k Transported products k from supplier z at time t in truck l Setup time for product k Product k in stock at time t at the buyer Consumption of product k at time t at the buyer Binary indicator for truck l traveling from node i to node j at time t Products k in stock in stock at supplier z at time t Truck Capacity Truck maximum traveling distance Products Trucks Big number All nodes, N = Z ∪ Buyer ∪ Depot Node subset Length of time period Suppliers

The target function aims at minimizing the costs for producing, storing, and transporting the requested products. The costs for production consist of setup costs and unit production costs. The costs for transportation consist of fixed costs per truck and variable costs per km. Restriction 2 updates the suppliers’ stocks with regard to newly produced products and those shipped. Restriction 3 forces setup costs if a product is produced in a period. Restriction 4 assures that the suppliers’ capacity restrictions are held. Restrictions 5,6,7 are the non-negativity restrictions for lot sizes and stock levels. Restriction 8 and 9 set the initial inventories. Restriction 10 defines the setup indicator as binary. Restriction 11 assures that each supplier can only produce certain products. Restriction 12 assures that each truck leaves the supplier it has visited. Restrictions 13 and 14 force the edge from buyer to depot to be included in each tour and the edge from depot to buyer not to be included in any tour thus rendering each tour into a milk run. Restriction 15 sets the capacity limit for transported goods. Restriction 16 set the distance limit for trucks. Restriction 17 accounts for subtour elimination. Restriction 18 forces a stop at every node that delivers product in a period. Restriction 19 updates the buyer’s stock with regard to all incoming tours.

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

5. MODEL SOLVING Although the mathematical model would allow for exact solving, it cannot be used for reasonably sized instances due to its complexity. The VRP alone is known to be NPhard and the CLSP is also computationally intensive. That is why a heuristic has been developed. In the following, the algorithms of the heuristic and the data set used for tests are described. 5.1 Heuristic Basically, the heuristic consists of two nested tabu searches (cf. Reeves (1995)). The first one alters the production dates of orders and is similar to known CLSP heuristics. Even if transportation is not considered to be part of the target function, orders are brought forward in case of capacity shortages or saved setup costs. The result is a time window between the actual production date and the latest delivery date/due date. This can be used for transportation planning. Each order is assigned a transportation date which can move inside this window. The determination of the transportation date is made by the second tabu search. This randomly assigns orders to a feasible transportation day and then calculates the savings matrix for finding tours. Orders are then shifted to other dates and the savings algorithm starts over again until no better solution can be found. In order to get the correct savings, the calculation for two orders i and j is: s[i,j] = d[i,Buyer] + d[Depot,j] + d[Buyer,Depot] - d[i,j]. A more detailed description of the VRP heuristic is given in B¨oohle (2009). The assessment of shifting the production date is therefore twofold. The impact on production costs is directly calculable whereas the impact on transportation costs has to be found by a second heuristic. To simplify the problem, maximum sizes for bringing forward orders are introduced, called an order’s shift window. It is not necessary to allow big shift windows as the possibility that bringing forward an order reduces costs shrinks with every additional day because of stock keeping costs. Still, checking all possible shifts is hard even with the use of shift windows, e.g. with a shift window of 3 and a problem size of 50 orders, there are (3 + 1)50 = 12 ∗ 1029 possible shift combinations, of which each necessitates a run of the tour planning heuristic. The tour planning tabu search works in the same way as the main tabu search. The difference is that it shifts another parameter for demands, i.e. the transportation period instead of the production period with respect to the inequality production period ≤ transportation period ≤ due date and calculation of transportation costs only. 5.2 Test Data Set There is no standard data set available for this problem as for example for different variants of the VRP. That is why a new data set had to be designed. It can be obtained from the authors upon request. It contains six suppliers that are clustered in two groups both within a 70 km radius around the customer. The carrier’s depot is also closely located to the customer. Each supplier produces a single product. The period length is 50 days. Orders are issued daily and according to constant patterns (cf. fig.

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Algorithm 1. Find solution: 1: f easible ← true 2: for all Suppliers do 3: shortage ← 0 4: for all P eriods(inreverseorder) do 5: addT oStock(shortage+demand broughtf orward 6: shortage ← planLots 7: if nrOf ItemsOnStock > stockCapacity then 8: f easible = f alse 9: end if 10: end for 11: if shortage > 0 then 12: f easible ← f alse 13: end if 14: end for 15: planT ransports 16: if numberOf T rucks ≤ f leetSize then 17: return f easible 18: else 19: return f alse 20: end if Algorithm 2. Main tabu search: 1: f indSolution 2: bestCosts ← getP rodAndT ransCosts 3: while i < maxIter do 4: shif tRandomDemandT oRandomDateN otOnT abuList 5: f indSolution(cf.Alg.1) 6: if Solution is not f easible OR newCosts > 7: 8: 9: 10: 11: 12: 13: 14: 15:

bestCosts then undo demand shif t set on tabu list i←i+1 else bestCosts ← getP rodAndT ransCosts set previous shif t on tabu list i←0 end if end while

2). The order quantities fluctuate between 100 and 300 units. The supplier’s production capacities are assumed as unlimited, likewise stock capacities at the suppliers and at the customer. A truck can transport 1,000 units over a distance of 500 km. The fleet consists of three trucks. Production costs range from 2 to 10, and setup costs from 12 to 21. Production and setup time are negligible in this case. Stocking costs go from 0.01 to 0.05 per unit depending on the product. In every period, each operating truck generates fixed costs of 250 and a variable costs of 1 per km. CO2 emissions depend on the loaded number of units and range from 10 kg per km for 100 units and increase by 5 kg per 100 units. 6. RESULTS The scenario described above was solved with and without consideration of transportation costs. The figures given in table 2 are the mean values of ten respective runs. In this case, transportation costs could be reduced by 16%, resulting in 2% overall savings. Due to better truck utilization, emissions have been reduced by 10%. Emissions are not taxed so this does not reflect in monetary savings.

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

This is currently subject to research and a prototype is currently being developed. The contribution of this paper is first the formalization of a transportation method called milk run which currently gains more importance in practice and its integration with lot sizing problems at multiple suppliers and second the theoretical impulse to study the alignemnt of production and transportation for procurement. The potential benefits of the described integration of transportation issues seem promising, the more so as energy prices are expected to rise steadily. Efforts are currently being made to collect figures from real industrial cases to prove the benefits that can be achieved using the presented model. Fig. 2. Supplier locations and ordering patterns

REFERENCES

Table 3 shows results with transportation costs increased by 20%. The savings accordingly rise slightly. It has to be kept in mind that results depend heavily on structural parameters and ordering patterns. Still, refining the model and approximating real world conditions is justified by these results. Table 2. Results Indicator Total Costs Production Costs Transportation Costs CO2 Emissions Stock Keeping Costs

Succ. Value 139,421 117,011 22,410 166,900 139

Integr. Value 136,103 117,261 18,842 148,750 194

Diff. -2% +0.2% -16% -11% +40%

Table 3. Results with 20% increased transportation costs Indicator Total Costs Production Costs Transportation Costs CO2 Emissions Stock Keeping Costs

Succ. Value 141,369 116,999 24,370 166,900 144

Integr. Value 137,491 117,420 20,071 149,170 294

Diff. -3% +0.4% -18% -11% +104%

7. OUTLOOK Concerning the mathematical models, some additions can be thought of that are worth implementing. First of all, variable transportation costs should be calculated on the basis of a tariff graded by distance and weight. Furthermore, carbon emissions could be weighted with costs to create a probable future scenario. Regarding the heuristic, it became apparent that the tabu search algorithms lock in to inferior results too early. Also, lot sizes currently depend on order sizes which deviates from the mathematical model where there are no restrictions on the lot size. In general, the solution given here has two major drawbacks. The first is that the target function does not exist in reality. No supply chain member will attempt to reduce supply chain wide costs, but rather aims at minimizing its own expenditures. Second, this is a centralized approach which will not work in inherently decentralized supply chains consisting of legally independent companies which neither share confidential information nor give up their planning authority. For these reasons, an approach which takes these non-negotiable restrictions into account is required. A possible solution is a multi-agent system capable of conducting negotiations between buyer, supplier, and carrier.

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Blumenfeld, D. E., Burns, L. D. (1991). Synchronizing Production and Transportation Schedules Transportation Research Part B: Methodological, Vol. 25, No. 1, pages 23-37, 1991 Blumenfeld, D. E., Burns, L. D., Diltz, J. D., Daganzo, D. F. (1985). Analyzing Trade-offs between Transportation, Inventory and Production Costs on Freight Networks Transportation Research Part B: Methodological, Vol. 19B, No. 5, pages 361-380, 1985 B¨ohle, C., Dangelmaier, W. (2009). Milk run optimization with delivery windows and hedging against uncertainty Operations Research Proceedings 2008, Springer-Verlag, Berlin, 2009. Boudia, M., Dauz`ere-P´er`ez, S., Prins, C., Louly, M. A. O. (2006). Integrated optimization of production and distribution for several products International Conference on Service Systems and Service Management, IEEE, Vol. 1, pages 272-277, 2006 Chandra, P., Fisher, M. L. (1994). Coordination of Production and Distribution Planning European Journal of Operational Research, Vol. 72, pages 503-517, 1994 Ereng¨ uc¸, S¸. S., Simpson, S. C., Vakharia, A. J. (1999). Integrated production/distribution planning in supply chains: An invited review European Journal of Operational Research, Vol. 115, pages 219-236, 1999 Fumero, F., Vercellis, C. (1999). Synchronized Development of Production, Inventory, and Distribution Schedules Transportation Science, Vol. 33, No. 3, pages 330340, 1999 Gosier, R., Simchi-Levi, D., Wright, J., Bentz, B. A. (2008). Past the tipping point Accenture/ILOG White Paper, 2008 Hofmann, C. (1995). Interdependente Losgr¨oßenplanung in Logistiksystemen Verlag f¨ ur Wissenschaft und Forschung, Stuttgart, 1995 Kuhn, H., Liske, T. (2008). Simultane Anlieferungs- und Produktionsplanung in der Automobilzulieferindustrie in: D. C. Mattfeld, H.-O. G¨ unther, L. Suhl, S. Voß(eds.), Informations- und Kommunikationssysteme in Supply Chain Management, Logistik und Transport, pages 3956, Books on Demand, Norderstedt, 2008 Reeves, C. R. (1995). Modern Heuristic Techniques for Combinatorial Problems McGraw-Hill, London, 1995 Sarmiento, A. M., Nagi, R. (1999). A review of integrated analysis of production-distribution systems IIE Transactions, Vol. 31, pages 1061-1071, 1999 Tempelmeier, H. (2006). Material-Logistik SpringerVerlag, Berlin, 2006

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

Toth, P., Vigo, D. (2002). The Vehicle Routing Problem Siam, Philadelphia, 2002 Voß, S., Woodruff, D. L. (2003) Introduction to Computational Optimization Models for Production Planning in a Supply Chain Springer-Verlag, Berlin, 2003 V¨olkl, S. (2006). Zwischenbetriebliche Koordination von Losgr¨oßen in Supply Chains Josef Eul Verlag, Lohmar, 2006 Yung, K.-L., Tang, J., Ip, A. W. H., Wang, D. (2006). Heuristics for Joint Decisions in Production, Transportation, and Order Quantity Transportation Science, Vol. 40, No. 1, pages 99-116, 2006

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