Furniture supply chain tactical planning optimization using a time decomposition approach

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European Journal of Operational Research 189 (2008) 952–970 www.elsevier.com/locate/ejor

Furniture supply chain tactical planning optimization using a time decomposition approach M. Ouhimmou

b

a,b

, S. D’Amours a,b,*, R. Beauregard S. Singh Chauhan d

b,c

, D. Ait-Kadi

a,b

,

a De´partement de Ge´nie Me´canique, Universite´ Laval, Que´bec, Canada G1K7P4 Centre Interuniversitaire de Recherche sur les Re´seaux d’Entreprise, la Logistique et le Transport (CIRRELT), Universite´ Laval, Que´bec, Canada G1K7P4 c De´partement des Sciences du Bois et de la Foreˆt, Universite´ Laval, Que´bec, Canada G1K7P4 d John Molson School of Business, Concordia University, Montreal, Canada, QC H3G 1M8

Received 1 October 2005; accepted 1 January 2007 Available online 15 July 2007

Abstract We study the supply chain tactical planning problem of an integrated furniture company located in the Province of Que´bec, Canada. The paper presents a mathematical model for tactical planning of a subset of the supply chain. The decisions concern procurement, inventory, outsourcing and demand allocation policies. The goal is to define manufacturing and logistics policies that will allow the furniture company to have a competitive level of service at minimum cost. We consider planning horizon of 1 year and the time periods are based on weeks. We assume that customer’s demand is known and dynamic over the planning horizon. Supply chain planning is formulated as a large mixed integer programming model. We developed a heuristic using a time decomposition approach in order to obtain good solutions within reasonable time limit for large size problems. Computational results of the heuristic are reported. We also present the quantitative and qualitative results of the application of the mathematical model to a real industrial case. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Integrated planning; Furniture industry; Tactical decisions; Large scale optimization

1. Introduction A supply chain is a network of facilities that procure raw materials, transform them first into intermediate goods and second into final products, to finally deliver the products to end customers through a distribution system [16]. Material flows downstream from suppliers to customers, while information flows in the opposite direction. The furniture supply chain studied in this paper consists of hardwood timber suppliers, sawmills * Corresponding author. Address: Centre Interuniversitaire de Recherche sur les Re´seaux d’Entreprise, la Logistique et le Transport (CIRRELT), Universite´ Laval, Que´bec, Canada G1K7P4. E-mail address: [email protected] (S. D’Amours).

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.01.064

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transforming logs into green wood boards, kilns drying green wood boards, furniture mills manufacturing furniture, warehouses, retailers, subcontractors and finally customers. The main raw material of the supply chain is hardwood logs of various species and qualities. These logs are transformed in sawmills into boards using sawing policies. Depending on the inventory level and the customer known demand, the sawmill planner selects a sawing policy to be applied to the raw material. These primary products are gathered into drying groups based on their optimal drying time. Then, final products, the dried boards, are shipped to their customers, the furniture maker. In the framework of the model devised here, the furniture maker is considered to be the end-customer. This paper aims at modeling the wood supply chain section of the whole furniture supply chain (Fig. 1). The sawing process is divergent while furniture making itself is an assembly process. This means that from each hardwood log, several boards (semi-final products) are produced depending on the sawing policy (sawing or breakdown pattern) used. Drying process takes a long time (from 2 to 5 weeks) to transform green board into dried board. Also the log supplier’s capacity is seasonal (depends on available harvesting periods). Finally, the customer’s demand is known, variable and seasonal. The combination of all these factors makes the planning of the furniture supply chain complex and difficult. The furniture company needs a decision support system (DSS) to support the tactical planning process of the supply chain. The supply chain planning matrix [23] allows to make planning decisions according to two dimensions, ‘‘planning horizon’’ and ‘‘process’’. The process involves the procurement, production, distribution and sales functions while the planning horizon is divided as follows: 1. Long-term (strategic level) is concerned with long-term issues such as the number and location of manufacturing plants and warehouses, new product introductions, etc. 2. Mid-term (tactical level) is concerned with shorter term decisions such as synchronized planning policies for procurement, production, distribution and sales. 3. Short-term (operational level) is concerned with day-to-day decisions such as scheduling and truck loading.

Fig. 1. The furniture company’s supply chain.

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Fleischmann et al. [10] also give a different description to the supply chain planning matrix and general descriptions of different issues in supply chain management. Two other authors [4,11] have adapted this matrix to the forest industry supply chain in Sweden and Canada. Frayret et al. [11] reported that tactical decisions concern issues surrounding the definition of the more-or-less generic rules for guiding daily operations. These rules tend to satisfy the strategic objectives while respecting the capacities of the supply chain. Cohen and Lee [8] presented a framework for the integration of the procurement, production and distribution processes. They used three different models: inventory management, production control and management of the distribution to develop a comprehensive model which establishes an optimal policy for inventory management. Chandra and Marshall [5] developed a mathematical model for one factory with multi-products and multi-periods. This model aims at studying the effect of the coordination of the production and distribution planning process. Gutierrez and Graves [14] proposed a linear program to optimize the operations of production, distribution and inventory of a manufacturing company. The model counts 80,000 variables and 30,000 constraints. In the forest industry supply chain, Maness and Adams [18] proposed a model to integrate the processes of bucking and sawing. The model is formulated as a mixed integer program. This model simultaneously determines the optimal bucking and sawing policies based on known demand and final product price (integration of stem bucking and log sawing). The system developed takes into account only one sawmill, with raw material distribution, over one period of time and with a deterministic and known final product demand. In 2001, Maness and Norton [19] developed an extension of the model to take into account several planning periods. Reinders [22] developed a decision support system for the strategic, tactical and operational planning of a sawmill (the operations of bucking and sawing are done in the same unit). The model considers only one sawmill and does not take into account other processes like planing and drying. Lide´n and Ro¨nnqvist [17] introduced an integrated optimization system, CustOpt, allowing a wood supply chain to satisfy the customer’s demand at minimum cost. The model considers the bucking, sawing, drying, planing, and grading processes. The integrated system is used as a decision support tool at the tactical level (3 months planning horizon). The system was tested using two to five harvesting districts, two sawmills and two planing mills. Farrell and Maness [9] developed a relational database approach to create an integrated linear programming-based decision support system that can be used to analyze short-term production planning issues in a wide variety of secondary wood product manufacturers. The DSS is generic and able to analyze production strategies in the highly dynamic environment of secondary manufacturers. Finally, Bredstro¨m et al. [1] studied the supply chain problem of a large international pulp company with five pulp mills located in Scandinavia. They developed two mixed integer models that determine daily supply chain decisions over a planning horizon of 3 months. The algorithm used is based on column generation and the shortest path network. They also used an aggregation approach to solve the model within practical time limits. As for the supply chain management issues directly linked to the furniture industry, many contributions [2,3,13,15] have dealt with the optimization of the cutting list at the mill level in order to meet demand and minimize wood lost. The cutting lists define how to group the dimension parts together so the associate cutting processes can be performed using as little wood boards as possible. Moodley [20] have studied the impact of using information technology as Internet on the performance of the furniture supply chain in South Africa. Grushecky et al. [12] studied the decline of the North American wood furniture and its impact on the entire hardwood supply chain. However, to our knowledge, no one has used optimization model and operations research techniques to tackle the planning problem of the whole wood supply chain of the furniture industry. This raises the need for a multi-suppliers, multi-facilities, multi-products, multi-periods and multi-customers model. This paper aims to contribute to the definition of such model, propose an efficient solution method and apply it to real industrial case. The mathematical formulation, an efficient method to solve this model and an application of this model to real industrial case are the major contributions of this paper. In this paper, we develop an efficient optimization model and rules of decision to support supply chain planning at the furniture company. The focus is on supporting mid-term planning (tactical decisions over a one year planning horizon). Decisions to be made concern procurement contract, cyclic inventory level, demand allocation policies and outsourcing. The model includes procurement, sawing and drying processes.

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The case study used in this paper considers a subset of the supply chain of a Canadian wood furniture company. We consider forest products (logs) suppliers, sawmills, kilns, subcontractors and final lumber customers (furniture mills). We assume that the demand is known, dynamic and deterministic over the planning horizon and must be satisfied. Sawing policies list for each raw material as well as production process data are known upfront. In fact, we formulated the wood supply chain planning problem as mixed integer programming model then we developed a heuristic to obtain good solutions within reasonable amount of time. We applied the mathematical model to our industrial partner and showed potential saving cost using integrated planning and global optimization approach of the supply chain. The outline of this paper is as follow. The next section gives a detailed problem description including the supply chain business units and planning process while Section 3 presents the mathematical formulation of the problem. Section 4 presents the proposed heuristics. Comparisons of computational results of the heuristics versus CPLEX [7] are reported in Section 5. The application of the mathematical model to a real industrial case (the supply chain of our project partner) and the difficulty we faced are described in section five. Finally, some future research and conclusion are presented in Section 6. 2. Problem description The furniture supply chain, from harvesting trees in the forest to delivering furniture to final customers, involves many business units and processes. The main raw material in the supply chain is hardwood logs and the optimization of transformation, inventory and transportation of this raw material can significantly reduce the operation and procurement costs of the whole supply chain. We begin by a detailed description of the furniture supply chain, especially the wood supply chain, and then we describe the processes involved in the raw material transformation. Finally, we describe the actual operations planning in the wood supply chain and its related challenges. Our industrial partner, is a leader in the production and distribution of high-quality residential furniture. The company’s supply chain include many logs suppliers, two sawmills, more than 16 kilns dryers, a plant specializing in the fabrication of components, and a veneer plant, 10 furniture manufacturing facilities and two warehouses. 2.1. Business units of the furniture supply chain Hardwood logs come from both public land, on the basis of the supply and forest management agreements (SFMA) with the Government of Que´bec, or from private land mainly from the USA or Que´bec private woodlot owners. The SFMA agreement authorizes the owner of a sawmill to harvest, in one territorial unit, a volume of timber of predetermined species and quality, every year. Procurement from private land is based on an annual contract between the furniture company and logs suppliers owning these forests. Usually, the furniture company gives priority to supplying from the pubic forest over the private forest in order to meet their commitments with the government. The logs are delivered and stored at different sawmill yards. Logs are transported by trucks owned by subcontractors who have agreements with the furniture company. These companies also assure the transportation of green hardwood boards between sawmills and kilns and transportation of dried hardwood boards between kilns and furniture mills. The transformation of logs into boards takes place in different sawmills according to sawing policies based on raw material quality, inventory levels of green and dry wood boards and customer’s needs. The drying process transforms green wood boards into dry wood boards by reducing the moisture content of wood to 6–8%. Many other processes take place in the furniture mills before the final product is achieved. The dry wood board is cut into dimension parts and panels. These items are glued and machined, assembled, sanded, painted, packaged and shipped to warehouses. Occasionally, furniture is directly shipped to the retailer. In this paper, we consider only a limited part of the whole supply chain called the wood supply chain. This subset of the supply chain is responsible for supplying all the furniture mills in dried hardwood boards. Thus, we consider logs suppliers, sawmills and kilns in the furniture supply chain as shown in Fig. 1. This network must be planned to respond effectively to the furniture mills demand. Based on the furniture sales forecasts and the furniture bill of material, the furniture mills calculate their weekly needs of dry wood boards over a year. These forecasts are sent to the wood supply chain planner

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and are updated every month. This information on customer’s needs combined with business units capacities and process characteristics are used to realize the procurement, sawing, drying planning as well as to compute the targeted end of period inventory levels for each product and each location of the wood supply chain. Moreover, as customer’s needs change over time, the wood supply chain must be capable to adapt the plans in order to satisfy demand. 2.2. Tactical planning problem The furniture company has identified a need for a decision-making system that can propose solutions, or generic rules to answer certain questions such as: 1. 2. 3. 4.

What What What What

is the annual volume of raw material to be procured from each log supplier? is the inventory level of each product in each business unit? are the demand allocation policies? is the proportion of sawing and drying to be outsourced?

This decision-making system can support the wood supply chain planner to plan the wood supply chain operations, anticipate the operational planning level and also can be used to face emergency situations when customer’s needs change, log supplier fails to respect agreements, etc. This model describes a mid-term planning for a furniture company and will assist tactical planning. The reason why we consider a 1 year planning horizon is to reflect the seasonality aspect of procurement, production and demand. These processes are all seasonal and their cycles typically span over a year. The fact the furniture company is used to signing wood procurement contracts with logs suppliers for a 1 year period is another major reason to motivate the 1 year plan. The mathematical model will not be used at the operational level but will be used to help guide daily operations by for example, informing the operational plan that the seasonal inventory level for each product is set to a certain value, etc. Indeed, our model will be used at the tactical level and is an anticipating model that summarizes the impact of tactical decisions. The tactical model could be rerun each month or even each week but it is important to run it for the whole next year (entire seasonal cycle) to catch seasonality as opposed to the operational plan that cannot catch it because of its short horizon planning (1 and 3 months). In the next sections, we will describe, in detail, the processes involved in the different business units of the studied supply chain. The characteristics of these processes had a great impact on the way we formulated the model and consequently on its complexity. The main processes used in the limited supply chain are sawing, drying in kiln, air drying and transportation. 2.3. Sawing process The furniture manufactured by the company is made from hardwood of different species. The sawmill saws more than 10 different species (maple, oak, cherry, etc.). The sawing consists of transforming, using a sawing policy, the log into green wood boards and by-products such as chips and sawdust. The green wood board is produced with specific thickness and grade. In the North American furniture industry the board thickness is expressed in quarter (1/4) in. (for example 1 in. = 4 * 1/4 = 4/4, 1.25 in. = 5 * 1/4 = 5/4, etc.). The grading process is based on the National Hardwood Lumber Association (NHLA) [21] rules. A set of possible sawing policies can be applied to each log (Fig. 2). Each sawing policy will give a particular outcome expressed as a volume of green board types and an amount of byproducts (Table 1). On average the planner uses five sawing policies for each log (raw material). These sawing policies may differ between sawmills. 2.3.1. Drying process The drying process transforms green wood boards into dry wood boards by reducing the moisture content of wood to 6–8%. The main drying method used in the furniture company is kiln drying where green wood boards are kept in a kiln dryer for a required time. This time is what is required to dry the green wood boards

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Fig. 2. The list of sawing policies and their outcomes for each hardwood log.

Table 1 Example of a real sawing policy and its expected outcomes for a maple log Species of log

Sawing policy

Thickness

Grade

Outcome (%)

Maple

Policy_1

4/4

Select 1 Comm 2 Comm 3 Comm Select 1 Comm 2 Comm 3 Comm Select 1 Comm 2 Comm 3 Comm Sawdust Chips

2.25 12.45 8.50 1.80 4.75 10.50 9.50 5.25 2.20 8.75 10.25 1.80

5/4

6/4

Byproducts

25%

30%

23%

22%

and it depends on many factors such as, species, thickness of board, drying season, etc. The thicker the board, the longer it takes to dry (Table 2). We have also gathered green boards into drying groups, based on their drying time. One major technical constraint in drying is that different boards can only by dried together under specific conditions. Therefore, drying groups defining the boards that can be dried together are created to tackle this constraint. Some furniture mills require air dried wood boards instead. These boards are stored in the sawmill yard for a certain time until the moisture content is between 25% and 30%. The air drying time also depends on season, thickness, etc. This time is longer than with kiln drying but it costs less. Some kiln units are next to sawmills and others are next to furniture mills (final customers). This geographical distribution of the business is taken into account in planning by including transportation costs between supply chain business units. Table 2 Drying groups and their corresponding drying time Specie of green wood board

Thickness

Grade

Drying time (weeks)

Drying group

All All All All

4/4 5/4 6/4 8/4

All All All All

1 2 3 4

Group Group Group Group

species species species species

grades grades grades grades

1 2 3 4

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Table 3 Flows between supply chain business units Type of product

Origin (from)

Destination (to)

Logs Green wood boards Air dried wood boards Air dried wood boards Dry wood boards

Logs supplier (forest) Sawmill Sawmill Sawmill Kiln

Sawmill Kiln Final customer (furniture mill) Kiln Final customer (furniture mill)

2.3.2. Transportation The company negotiates contracts for transportation activities with third party logistics (3PL) companies. Some companies transport raw material (logs) from supplier to sawmill and others transport semi-final and final products between sawmill, kiln and furniture mills (green, air dried and kiln dried wood) (Table 3). Different types of trucks are used depending on the product to transport. Transportation capacities of trucks vary depending on the product to be transported and also depending on season (weather constraints). The possible flows of different products within the supply chain are given in Table 3. 3. Mathematical formulation Sets and index The sets used to formulate the mathematical model are listed in the table below: s2S a2A k2K c2C t2T r2R j2J f2F p2P v2V g2G

set set set set set set set set set set set

of of of of of of of of of of of

forest products (wood logs) suppliers, sawmills, kilns, customers, time periods, raw materials, intermediate products, final products, all products (P = R [ J [ F), sawing policies, drying groups.

Parameters csrt purchasing cost for raw material r from supplier s in time period t, cpt procurement cost for final product p in time period t, cpod transportation cost per volume unit of product p from origin o to destination d, hapt inventory cost per volume unit of product p in sawmill a for each time period t, ca sawing cost per unit in sawmill a, cca setup cost in sawmill a, ck drying cost per unit in kiln k, cck setup cost in kiln k, qsrt maximum capacity of supplier s of raw material r in each time period t, qat maximum available operating time of sawmill a over time period t, ea production rate of sawmill a, qk maximum capacity of kiln k, dtg drying time of intermediate products group g, wvrj proportion of intermediate product j produced when we apply sawing policy v to raw material r, dfct demand of customer c of final product f in time period t.

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Decisions variables Xodpt volume of product p that is transported from origin o to destination d in time period t, volume of raw material r consumed by sawmill a over time period t, Yrat volume of intermediate product j consumed by kiln k over time period t, Yjkt volume of final product p procured from the market to satisfy the customer c in time period t, Zcpt inventory level of product p by in location l the end of time period t, Iplt r = 1 if we process raw material r in sawmill a in time period t, 0 otherwise, bat = 1 if we start drying process of group g in kiln k in time period t, 0 otherwise, agkt = 1 if the kiln k is busy at the beginning of period t, 0 otherwise. /kt Objective function The objective for the furniture supply chain is to minimize the total cost including procurement cost, transportation cost, inventory cost and production cost. The procurement cost (raw material and final products) can be expressed as: XXXX XXX csrt X sart þ cpt Z cpt : ð1Þ s

a

r

t

c

p

t

The transportation cost between forest products suppliers–sawmills–kilns–customers is expressed as: XXXX XXXX XXXX crsa X sart þ cjak X akjt þ cfkc X kcft : s

a

r

t

a

k

j

t

k

c

f

We express the total inventory cost of raw material, intermediate and final products as: XXX XXX XXX hart I rat þ hajt I jat þ hkft I fkt : a

r

t

a

j

t

k

r

t

j

k

Constraints X X sapt 6 csrt 8r; 8s; 8t; a X X r ea Y rat þ bat Y rat 6 qat r

t

k

g

ð3Þ

t

f

The production and setup cost (sawing and drying) is expressed as: XXX XXX XXX XXX ca Y rat þ ck Y jkt þ cck agkt þ cca brat : a

ð2Þ

t

t

a

r

ð4Þ

t

ð5Þ 8a; 8t;

ð6Þ

r

r YX 8r; 8a; 8t; rat 6 cat bat g akt þ /kt 6 1 8k; 8t;

ð7Þ ð8Þ

g g aX 8k; 8t; 8g; 8t þ 1 6 x 6 t þ dtg  1; kt 6 /kx Y jkt 6 qk 8j; 8k; 8t;

ð9Þ ð10Þ

j g YX 8j; 8k; 8t; jkt 6 qk akt X sart þ I raðt1Þ ¼ I rat þ Y rat 8r; 8a; 8t; s X X wvri Y rat þ I jaðt1Þ ¼ I jat þ X akjt 8j; 8a; 8t; v2V r k X Y fkt þ I fkðt1Þ ¼ I fkt þ X kfpt ; 8f ; 8k; 8t;

ð11Þ ð12Þ ð13Þ ð14Þ

c

YX 8j; 8f ; 8k; 8t; fkt  Y jkðtdtg Þ ¼ 0 X kcft þ Z cft P d cft 8f ; 8c; 8t;

ð15Þ ð16Þ

k

brat ; agkt ; /kt 2 f0; 1g; X odpt ; Y rat ; Y jkt ; I plt ; Z cft P 0:

ð17Þ ð18Þ

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All constraints are classical like flow balancing, demand satisfaction, capacity constraint or setup constraint, except the constraints (8) and (9) which formulates the drying process in the kiln. The constraints achieve the following: (5) express a raw material supplier capacity constraint in volume units, for each raw material, each supplier and each time period, (6) express a sawmill capacity constraint define in available operating time for each period, (7) is a logical constraint which forces a setup when a new raw material is processed at the sawmill, (8) ensure that one cannot dry more than one drying group in each kiln at the same time (we cannot mix products that have not the same drying time in the kiln), (9) assure the kiln is locked during the drying process and will be available when the drying process is finished, (10) express the kiln capacity constraint define in volume units per time period, (11) is a logical constraint which forces a setup each time a drying process is launched, (12) express a flow balancing constraint for raw material at each sawmill location, for each time period, (13) assure a flow conservation constraint of semi-final product at each sawmill location, for each time period, (14) guarantee a flow conservation constraint of final product at each kiln location, for each time period, (15) indicate that each semi-final product becomes a final product after the drying process, (16) express that demand satisfaction of customer’s requirements, demand needs to be met. If the capacity of in-house business units cannot meet demand, the dry wood can be bought on the market, (17) state the binary variables, and finally, (18) state that all variables cannot be negative. We did not include the air drying process and the seasonal variation of the drying time in the mathematical formulation in order to keep the model simple and easy to follow. All these specific characteristic of the furniture industry are included in the programming implementation phase of the mathematical model for the industrial case study. The mathematical formulation of the variation of the drying time is presented in detail in the Appendix. The mathematical formulation of the air drying process is similar to the drying process in kiln except it uses natural drying process (store products for a certain number of periods in the yard) and takes much more time. 4. Solution methods 4.1. Problem is NP-hard The complexity arises partially due to the setup cost at sawmills and principally due to the kiln operation. In this section we analyze the problem complexity with respect to a single kiln. Let us assume that semi-final products (green wood boards) are available at the kiln whenever required. Since dry wood boards are shipped to customer at a particular date, we assume the date of delivery is fixed for all type of products and is known. At this point we need to decide the sequence of groups in which they enter the kiln. Let us assume that we have G groups with delivery dates t1, t2, . . ., tG. Each group required a specific drying time denoted by dg, g 2 G and has an inventory cost hg, g 2 G. In our problem late delivery is not permitted, i.e. seasoning completion time for a particular group Cg 6 tg. Furthermore, there is inventory holding cost associated with each group, proportional to early production. The objective is to schedule as many groups as possible so as to minimize inventory cost and additional costs incurred due to procurement from external forest products suppliers. The single kiln problem can be formulated as follows: Min

X g2G

hg ðtg  C g Þ;

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subject to C g 6 tg g 2 G; C g ¼ eg þ d g ; epðiÞ < epðiþ1Þ

i ¼ 1; 2; . . . ; G  1;

where eg is the period in which group g enters the kiln and p(i) gives the index of group entering in the ith sequence. The above problem in literature is referred to as weighted earliness problem. Lemma 1. Weighted earliness problem is NP-hard. Proof. See Chand and Schneeberger [6].  We have developed two algorithms, one for planning one kiln with one drying group (Section 4.2) and one for planning the entire network from wood supply to furniture mills (Section 4.3). 4.2. Algorithm for the one kiln and one product type problem We suppose that we have a single kiln with limited capacity and we have only one drying group. We suppose also that one objective of this problem is to utilize the maximum kiln capacity if it is economical because in reality the demand exceed capacity in certain periods. Let e be the incremental cost per unit if one unit is purchased from an external supplier and let h be the inventory holding cost per unit per period. Now the problem at hand is to find the best planning for the kiln and the load for each selected planning. We suppose that the incremental cost e and the drying time are constant over the planning horizon. The following algorithm finds the best  planning and the load. Let D ¼ he , b c gives the greatest integer lower than he. D represents the maximum stocking duration by which product is cost effective over outsourcing. b = drying time, C = kiln capacity, A = setup cost, di = demand in period i and P = set of periods where kiln has to be scheduled. We define a function cost as follows:  Xj¼tþb1    Xj¼t1 Xj¼tþD Xj¼k1   F ði; tÞ ¼ |{z} A þ ed ðk  tÞhMin Max 0; C  d d kþ j ; d k þ eMax 0; k C : k¼i k¼tþ1 j¼t k¼t |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1 2

3

4

Function F(i, t) calculates the incremental cost in the case when a production process is started in the kiln in period t  b instead of period i  b where i 6 t. The cost function means how much we will save (costs) if we start production in period t  b instead of period i  b. Note that b is the number of treatment periods required for the product in the kiln. Which means that if we start production at t  b then the production will finish at period t and the kiln will be available at period t (Fig. 3). So the main idea behind the cost function is to calculate the difference cost between starting production in two given periods t1 and t2 and choose the best of them (the lowest cost decision). This idea will be used for the heuristic to propose a good planning for the kiln. The first term of the cost function corresponds to the setup cost. The second term expresses the cost for fulfilling demand between periods i and t  1 by buying products (all demand) from

1

2

3

i-b

t-b

i

t

t+1

………………..

T

b b F(i,t)

Fig. 3. Calculation of the cost function F(i, t) as the marginal increase if we start production in the kiln in period i  b instead of period t  b.

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external forest suppliers. The third term expresses the inventory cost. This cost corresponds to the inventory cost of products stored in order to satisfy demand of next periods. Finally, the fourth term expresses the cost of buying products (simply the difference between cumulated demand and kiln capacity) to satisfy demand from period t to t + b  1,from external forest products suppliers. Now we can construct the algorithm as follows: Algorithm (H) 1. Set P = /, i = b 2. Set Mincost = F(i, i), Schedule_in_period = i  b, 3. For j = i + 1, . . ., T a. if F(i, j) < Mincost then i. Set Mincost = F(i, j), ii. Set Schedule_in_period = j  b, b. If F(i, j) > F(i, j  1) then i. P = P + {Schedule_in_period}, PiþD  ii. Load in period j  b the quantity L ¼ Min k¼i d k ; C iii. For k = i to k = i + D 1. set temp = Min(dk, C) 2. set dk = dk  temp 3. set L = L  temp iv. i = j + 1, go to 2. 4. If i > T stop. The algorithm (H) tries to find the best planning for the kiln in order to satisfy customer’s demand while reducing the setup, the inventory and the outsourcing cost. The function Mincost represents the total cost to satisfy customer’s demand over the planning horizon. Schedule_in_period represents the period to start a new drying process. First two step initializes P, which is the set of periods where we start a new drying process in the kiln, to empty set and i = b since products cannot proceed before period b, the Mincost to initial cost F(i, i) and Schedule_in_ period to period i  b. The step 3 checks periods (j = i + 1, . . ., T) which are more economical than period i and stored in Schedule_in_period. We follow step 3 for a given period i, untill the cost in next period is less economical than the previous period (if it is not economical in period j then it will not be in period j + 1). Note that when we select a new period as Schedule_in_period, we update the demand accordingly (we subtract the demand). Numerical experiments We generated 16 instances of problems to compare the algorithm and CPLEX (Table 4). A mixed integer program associated to the problem described above was formulated. We developed a C++ program for implementing both the mixed integer program and the algorithm. For each instance, we randomly simulated 10 experiments. The instance is defined by the planning horizon and the drying time. The kiln capacity, setup cost, outsourcing cost, inventory cost and demand are randomly generated in the set {40, . . ., 100}, {20, . . ., 80}, {2, . . ., 5}, {0.5, . . ., 1.5} and {kiln capacity * 0.75, . . ., kiln capacity * 1.60}, respectively. The computational results are reported in Table 4. For all instances, the algorithm is faster then CPLEX and both the mean and variance of the solution quality never exceed 2%. The above algorithm is useful to compute a near optimal schedule in a short time. This schedule can be used as an initial solution for the mixed integer program if the optimal schedule is of prime importance. 4.3. Algorithm for the general case We tested two methods to solve the supply chain tactical planning problem. First, we used the optimization software CPLEX 9.1.2, and second we used a heuristic based on time decomposition over the planning horizon. The main steps of the heuristic can be described as follows. Let Pi be the supply chain tactical problem for a horizon planning with length and its starting period is period 1 (Fig. 4):

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963

Table 4 Computational results of different configurations of the algorithm and CPLEX Instance

Planning horizon (T)

Drying time (b)

Mean gap (%)

Variance gap (%)

Mean processing time, CPLEX (ms)

Mean processing time, algorithm (ms)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

50 50 50 50 100 100 100 100 200 200 200 200 500 500 500 500

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

1.93 0.65 0.36 2.62 1.17 0.01 0.87 1.05 0.47 0.09 0.42 0.45 0.10 0.01 0.32 0.24

1.36 1.47 1.53 0.35 0.94 0.30 0.30 0.27 0.48 0.33 0.15 0.10 0.22 0.15 0.06 0.05

343 171 171 218 250 328 406 409 609 421 656 921 2863 3060 3440 3859

5 8 7 6 12 13 14 11 20 23 21 26 40 46 42 48

1 1

…

2 k P1

i

…

2k

…

…

n-1

i+1 ik

… (i+1)k

…

… (n-1)k

n …

nk T

P2 Pi

Pi+1 PT

Fig. 4. Decomposition of planning horizon into small and equal horizon.

1. 2. 3. 4. 5. 6. 7.

Divide the planning horizon into n equal intervals with k period’s length each (Fig. 4). Solve the problem Pi and get the solution (only values of binary variables that are equal to 1). Set these binary variables equal to 1 and add them as new constraints in the problem Pi+1. Solve the problem Pi+1 and get the solution (values of all binary variables (=1)). Increment i (i = i + 1). If i (>n) then stop. Go to step 3.

All binary and continuous variables of the model depend on time. Based on this observation, if we reduce the length of the planning horizon, the size and consequently the resolution time of the mixed integer problem will also be reduced. Thus, we split the horizon planning into small intervals that would be considered as the new planning horizon. The main idea of the heuristic is to successively solve smaller problems. For each iteration, we recover the solution of the current problem Pi and use it as new constraints for the next problem Pi+1 in order to quickly solve it. This is possible because the new added constraints reduce the set of feasible solutions of the problem Pi+1. Note that we fix only binary variables that are equal to 1 (=1), which means that other null binary variables (=0) can take other values in the solution of the new problem. The continuous variables are not fixed either, so their value can also change in the solution of the new problem. We developed a C++ program to implement the heuristic and the input data of the model is stored in a database. The architecture of our tool is similar to the DSS developed by Farrell and Maness [9]. We used CPLEX 9.1.2 version to solve the mixed integer program. 4.4. Numerical experiments We generated a set of 24 instances according to four dimensions: size, demand pattern, flow magnitude and capacity structure (Table 6). The last three factors were added because they cannot be expressed by the size

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dimension and they have a great impact on the complexity of the model and therefore in resolution time. The size of each instance is given by the number of logs suppliers, sawmills, kilns, final customers, raw material, green wood, air dry wood, dry wood, planning horizon length, drying group, dry wood suppliers and sawing policies (Table 5). The capacity structure is determined by the number of forest products suppliers that can provide each raw material. For low capacity instances denoted by ‘‘c’’, this value is low and for high capacity instances denoted by ‘‘C’’, the value is high. The flow magnitude is determined by the number of semi-final products that result from each raw material and by the number of customers requiring each final product. For low flow magnitudes instances denoted by ‘‘f’’, the value is small and for high flow magnitudes instances denoted by ‘‘F’’, the value is high. Finally, for constant demand, the demand pattern is denoted by ‘‘d’ and for random demand, it is denoted by ‘‘D’’.

Table 5 Characteristics and size of tested problems Logs Sawmills Kilns Final Raw Green Air suppliers customers material wood dry wood Small size Medium size Large size

Dry Horizon wood planning length

Drying Dry Sawing group wood policies suppliers

P1 2 P2 6

1 2

4 10

2 6

1 3

16 48

16 48

16 48

52 52

4 4

2 6

16 48

P3 6

3

20

6

4

64

16

64

52

4

6

64

Table 6 Computational results of different configurations of the heuristic and CPLEX Problem

Heuristic (n = 10)

Heuristic (n = 17)

Heuristic (n = 25)

CPLEX Time (seconds)

Gap (%)

No. of binary var.

No. of constraints

Time (seconds)

Gap (%)

Time (seconds)

Gap (%)

Time (seconds)

184 163 197 48 16 561 139 67

2.96 3.53 3.12 1.37 0.00 2.51 1.95 1.69

147 134 76 74 29 437 69 66

3.49 3.48 3.39 1.09 0.00 2.40 1.13 1.52

149 237 75 1 43 590 101 93

3.53 2.70 3.69 1.41 0.00 2.48 1.20 1.61

184.00 163.00 197.00 48.00 16.00 561.00 139.00 67.00

14.13 7.15 54.34 Infinite 0.00 20.90 5.51 Infinite

44,330 43,914 44,354 43,938 43,290 43,290 43,938 43,938

884 884 884 884 884 884 884 884

24,073 24,071 25,261 25,259 24,072 24,072 25,260 25,228

Medium size P2_cfd 3049 P2_cfD 1565 P2_cFd 1687 P2_cFD 1538 P2_Cfd 351 P2_CfD 14,826 P2_CFd 16,097 P2_CFD 14,342

1.23 0.74 1.95 0.00 1.46 3.08 1.58 1.72

2226 1395 1873 1929 542 16,103 16,103 16,013

2.47 2.78 1.11 0.00 1.73 3.19 3.19 3.19

2031 2434 2340 2438 815 7906 7906 7906

2.18 2.74 0.69 0.00 1,42 3.01 3.01 3.01

3049.00 1565.00 1687.00 1538.00 351.00 14,826.00 16,097.00 14,342.00

1.28 41.81 23.31 9.66 Infinite 47.12 3.15 7.38

644,459 540,747 545,885 648,445 540,747 540,747 544,637 544,637

1872 1872 1872 1872 1872 1872 1872 1872

20,946 196,170 199,718 212,851 196,183 196,183 199,728 199,656

Large size P3_cfd 9095 P3_cfD 10,662 P3_cFd 2198 P3_cFD 2016 P3_Cfd 1489 P3_CfD 27,312 P3_CFd 30,404 P3_CFD 7404

2.85 1.86 0.00 0.00 2.53 2.25 0.64 0.61

7645 8039 3352 2711 1397 22,439 36,309 7444

3.31 2.90 0.00 0.00 2.35 2.99 0.65 0.60

7958 5895 3903 3751 1914 29,046 34,309 7434

3.04 3.61 0.00 0.00 2.03 3.04 0.64 0.65

9095.00 10,662.00 2198.00 2016.00 1489.00 27,312.00 30,404.00 7404.00

3.10 1.96 0.00 0.00 1.82 2.33 Infinite Infinite

846,734 846,734 855,305 855,305 846,734 846,734 855,305 855,305

4784 4784 4784 4784 4784 4784 4784 4784

369,132 369,149 386,818 386,818 369,019 369,200 387,287 386,835

Small size P1_cfd P1_cfD P1_cFd P1_cFD P1_Cfd P1_CfD P1_CFd P1_CFD

Gap (%)

No. of continuous var.

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For the dimension size, we have three major categories: small, medium and large size. For each category, we instantiate one problem size who have eight possible combinations because of the three other dimensions (capacity structure, flow magnitude and demand pattern). For each problem Pi (i = 1, . . ., 6) in 5, we instantiate eight test problems (Pi_cfd, Pi_cfD, Pi_cFd, Pi_cFD, Pi_Cfd, Pi_CfD, Pi_CFD) (see Table 6). For each of the 24 instances, we consider four different configurations of the heuristic based on the number n. This parameter is included in order to study the impact of the heuristic configuration (n = 10, 17, 25) on the resolution time. The total number of continuous and binary variables and constraints for all instances are reported in Table 6. All tests we performed using a computer with an AMD Opteron Processor 244 (1.79 GHz) with 4 GB of RAM. The computer uses Microsoft Windows Server 2003 Enterprise x64 Edition operating system, a 64bit operating system. We also used the 64-bit CPLEX 9.1.2 version with default parameters. We used the default setting of CPLEX except that we set the tolerance from the optimal integer solution to 1%. Since the data used in the model (cost, demand, capacity and process data estimations) is from a real world case, it often contains a margin of error larger than 1%, it is useless to solve the problem to optimality in practice. We also gave priority to the variables based on time and process (earliest time and first process in the supply chain process have higher priority) [13]. Table 6 reports the results obtained by the heuristic and CPLEX methods for all instances. Columns time and gap provide the CPU time (seconds) and the solution quality, respectively. Obviously, the solution of heuristic may have a gap (1%). In order to compare the performance of our heuristic with CPLEX, we solve the same problem using CPLEX till the gap is equal to the one obtained by the heuristic. The results show that in most cases, our heuristic is performing better than CPLEX. The solution quality obtained by our heuristic is

600 Heuristic (n=10) 500

Heuristic (n=17) Heuristic (n=25)

Time (s)

400 300 200 100 0 P1_cf d

P1_cf D

P1_cFd

P1_cFD

P1_Cf d

P1_Cf D

P1_CFd

P1_CFD

Problem Instance

Fig. 5. Comparison of different heuristic’s configurations and different instances of problem 1.

Time (s)

18,000 16,000

Heuristic (n=10)

14,000

Heuristic (n=17)

12,000

Heuristic (n=25)

10,000 8,000 6,000 4,000 2,000 0 P2_cf d

P2_cf D

P2_cFd

P2_cFD

P2_Cf d

P2_Cf D

P2_CFd

P2_CFD

Problem Instance

Fig. 6. Comparison of different heuristic’s configurations and different instances of problem 2.

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Time (s)

40,000 35,000

Heuristic (n=10)

30,000

Heuristic (n=17)

25,000

Heuristic (n=25)

20,000 15,000 10,000 5,000 0 P3_cfd

P3_cfD

P3_cFd

P3_cFD

P3_Cfd

P3_CfD

P3_CFd

P3_CFD

Problem Instance

Fig. 7. Comparison of different heuristic’s configurations and different instances of problem 3.

better than the one obtained by CPLEX for the same resolution time. The average total gap is 1.63% for the heuristic (Table 6). CPLEX was not able to find feasible integer solution in 50% of the instances tested and for the other 50% the average gap is 16.75%. For small size problems (P1), we observe that the combination ‘‘CfD’’ takes more time to solve than the others. For this size of problems, results show that the more you split the planning horizon (n is high), the longer takes the heuristic (Fig. 5). For medium size problems (P2), the combination ‘‘CfD’’, ‘‘CFd’’ and ‘‘CFD’’ are longer to solve. For the problem 4, the heuristic configuration with n = 25 is shorter than the other configurations for the hard problem to solve (Fig. 6). For large problems (P3), the combination ‘‘CfD’’, and ‘‘CFd’’ are also longer to solve (Fig. 7). The heuristic configuration with n = 10 seems to be the best for the combination ‘‘CFd’’, and with n = 17 to be the best for the combination ‘‘CfD’’.

5. Validation of the model on real industrial case The supply chain studied in this industrial case includes more than 40 logs suppliers, 2 sawmills, 16 kilns, 10 customers (furniture mills), 11 raw material, 135 final and semi-final products. The mixed-integer program obtained for the wood supply chain tactical planning over 52 weeks of planning horizon is very large. The model has more than 2,000,000 continuous variables, 8000 binary variables and 3,000,000 constraints. Note that during the implementation of the mathematical model we include the air drying process and the seasonal variation of the drying process. The mathematical formulation of these constraints is presented in the Appendix. We solved the mathematical problem on a 64-bit operating system with AMD64 processor and 4 GB RAM to overcome the over memory problem due to the size of the problem. The complete problem took approximately 2 hours to provide a near optimal solution (2% gap) using the heuristic. CPLEX was not able to find even a feasible solution within 2 days of run time for the same problem. Data collection required by the mathematical model was one of the biggest hurdles we came across. The supply chain’s business units used different technology to store their data (Microsoft Excel, text file and different database). The challenge was to import all these data into a unique database, created specifically to host data required by the mathematical model. We develop Visual Basic programs to export all these data to a Microsoft Access database (DB). We also had to ensure that all these data were homogeneous among all business units using integrity and consistency mechanisms. For example, products codes and names are not uniform across the supply chain so we standardized them before to import the data related to products to our database. This was a challenging task. Some data corresponding to parameters of the mathematical model did not even exist in any support (paper or electronic), so either experiment were performed to get these data or contact employees, logs suppliers or subcontractors to obtain the corresponding data. The optimization tool’s architecture is described in Fig. 8. System used CPLEX 9.1.2 as a backend solver. First, the Microsoft Access database is used to store both all input data (parameters of the model) and results (decisions variables of the model). We developed Visual basic modules (Forms) to export input data from different data sources to the database (DB). The heuristic is developed using a C++ program. This program

Displaying results

Forms (Parameters)

Data input

M. Ouhimmou et al. / European Journal of Operational Research 189 (2008) 952–970

Storing Data (Microsoft Access Database)

Read Parmaters data from DB

967

Reports (Decisions variables)

Write decisions variables values in DB

C++ program Write the MIP

Solve the MIP and get decisions variables values

CPLEX optimization software

Fig. 8. The global architecture of the optimization tool.

establishes the read/write connection with the database. The interface with CPLEX is assured by the C++ program. We compared and presented the results of the optimization tool versus the solution obtained by their planning process for the year 2004–2005. The major result is that our solution was able to reduce the total operations cost by more than 22%. The results are presented in percentage format for confidentiality reasons. For both solutions, we present the percentage of each component of the total cost. In Table 7, the last column provides the variation (increase/decrease) between both solutions for each component of total cost. All the different components of cost, except wood market cost, have been reduced. The wood market cost has been increased because our model suggests buying more volume, for some specific products, from the market and decreasing the amount produced internally. The main reason for the increasing of wood market cost is that customer’s demand configuration did not fit with the sawing policies used and the hardwood logs procured by logs suppliers. Consequently, it costs much less to buy some specific products from the wood market than to manufacture them by the company. One of the key findings was that the sawing and drying capacities were sufficient and that the company had no need to outsource sawing and drying process. In addition, the model yielded much lower inventory levels in the supply chain while capturing the seasonal phenomena of customer known demand and supplier capacity. Also, the material flows between business units were optimized and consequently reduce transportation cost. Finally, one major result is that the optimization tool allowed for the evaluation of raw material contracts negotiated with forest products suppliers, as the model gives the percentage used of each contract with each supplier. This result can be used to re-evaluate the raw material contracts and ranks the logs suppliers. The planning managers were used to making tactical planning decisions based on their own experiences and some home-made tools like spreadsheets. They were aware that the tactical planning problem is complex and challenging. However, there was no tool to support them to conduct simulations to evaluate the impact of regular or emergency situations such as when customer’s needs change or a log supplier fails to respect agreements. With our solution they can now answer all these challenging questions in a few minutes using the Table 7 Comparison of the actual practice planning solution (second column) with the optimization solution obtained by the heuristic (third column) for different costs (first column) Manual solution (%) Raw material cost Sawing cost Drying cost Inventory cost Transportation cost Outsourcing cost Wood market Total

48.1 30.79 3.88 3.29 6.36 2.01 5.54 100

Heuristic solution (%) 30.92 23.04 3.33 1.42 4.79 0.00 36.51 100

Variation (manual vs. heuristic) & & & & & & %

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optimization tool. Moreover, they can perform several simulations since the run time is relatively small. Beyond these specific results and according to top managers, the most useful aspect was the resulting evolution in wood sourcing tactical planning and thinking of the company. Their interaction with us was helpful in their evolution towards a leaner, more agile operation of the wood supply chain and now they have a more global vision when they take decisions opposite to local vision as they used to have before this project. 6. Conclusion and future work Supply chain planning in the wood furniture industry is a complex task. The combination of raw materials, with process and customer known demand produces complex interrelationships that require simultaneous analysis in order to find the optimal tactical decision. The purpose of this paper is to provide a decision support system for furniture supply chain tactical planning. This DSS help the planner to establish optimal seasonal inventory level, amounts of raw material to procure from each supplier (raw material contracts), proportion of flow exchanged between business units and allocations of customer’s demand. Much effort was spent to assure that the mathematical model represents the real supply chain of our industrial partner. This was accomplished through close interaction with the supply chain planners. Since the resulting MIP is large and cannot be solved within a reasonable time by CPLEX, we developed a simple and efficient heuristic to solve the problem within reasonable time for an industrial case problem. We also presented the quantitative and qualitative results of the application of the mathematical model to a real industrial case. Acknowledgements This research is funded by the Research Consortium in E-Business for The Forest Products Industry (FOR@C). The authors would like to thank also the industrial partner of the project for its collaboration and support. Appendix Constraint (15) indicates that each product j must spend exactly the corresponding drying time dtg in the kiln: Y fkt  Y jkðtdtg Þ ¼ 0

8j; 8f ; 8k; 8t:

ð15Þ

Constraint (15) is valid when dtg is constant over time. Since drying time depends on time, other constraints are required. We have two cases: drying time dtg (t) depends on tome and either increases or decreases over time. We define w as a binary variable to express that constraints (20) and (21) are mutually exclusive: If dtg ðtÞ  dtg ðt þ 1Þ ¼ d P 1 so Y fkðtþ1þxÞ ¼ Y jkðtþ1dtg ðtÞþxÞ þ Y jkðtþ1dtg ðtþ1ÞþxÞ 8j; 8f ; 8k; 8t; 8x ¼ 1; . . . ; d;

ð19Þ

Y fkðtþ1dtg ðtÞþxÞ 6 cak  w 8f ; 8g; 8k; 8t; 8x ¼ 1; . . . ; d;

ð20Þ

Y fkðtþ1dtg ðtþ1ÞþxÞ 6 cak  ð1  wÞ 8f ; 8g; 8k; 8t; 8x ¼ 1; . . . ; d:

ð21Þ

The three constraints (19)–(21) assure that, when drying time decreases, between seasons (e.g. from 4 weeks in winter to 3 weeks in summer) alternatives drying processes may overlap. Because of our formulation, which formulates the drying process based on the finishing period, we need to add these constraints so only one of two possibilities (drying processes) is selected. The example below (Table 8) illustrates the case where the drying time up to (t = 10) is 5 and for (t P 11) is 1. If we start a drying process for example at (t = 8), we expect that the drying process will finish at the period (8 + 5 = 13). We repeat the same procedure at (t = 12), we see

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969

Table 8 The impact of the drying time decreasing over the time on the kiln planning dtg(t) = 5 Entry period to kiln Exit period from kiln

dtg(t + 1) = 1

t4

t3

t2

t1

t

t+1

t+2

t+3

t+4

t+5

6 11

7 12

8 13

9 14

10 15

11 12

12 13

13 14

14 15

15 16

Table 9 The impact of the drying time increasing over the time on the kiln planning dtg(t) = 1 Entry period to kiln Exit period from kiln

dtg(t + 1) = 5

t4

t3

t2

t1

t

t+1

t+2

t+3

t+4

t+5

6 7

7 8

8 9

9 10

10 11

11 16

12 17

13 18

14 19

15 20

12–15 |fflffl{zfflffl}

that the expected finish period is (12 + 1 = 13). Thus, the two processes have different starting period (8 and 12) and have the same finish period (13) and we have to choose only one process and the constraints (19)–(21) allow to do that: If dtg ðt þ 1Þ  dtg ðtÞ ¼ g P 1 so Y fkðtþ1dtg ðtþ1ÞxÞ ¼ 0 8f ; 8g; 8k; 8t; 8x ¼ 1; . . . ; g;

ð22Þ

Y fkðtþ1dtg ðtÞþxÞ 6 cak  w 8f ; 8g; 8k; 8t; 8x ¼ 1; . . . ; g:

ð23Þ

Constraints (22) and (23) assure that, when drying time increases, between seasons (e.g. from 3 weeks in summer to 4 weeks in autumn) there is no products produced by the kilns for a certain number of periods. The example below illustrates the case where the drying time up to (t = 10) is 1 and for (t P 11) is 5. If we look at the last row of Table 9, we see that we skip from period 11 to period 16 which means that for periods 12–15 there is no products produced by the kiln. References [1] D. Bredstro¨m, J.T. Lundgren, M. Ro¨nnqvist, D. Carlsson, A. Mason, Supply chain optimization in the pulp mill industry – IP models, column generation and novel constraint branches, European Journal of Operational Research 156 (2004) 2–22. [2] U. Buehlmann, D.E. Kline, J.K. Wiedenbeck, Understanding the relationship of lumber yield and cutting bill requirements: A statistical approach to the rough mill yield estimation problem, in: Annual Meeting of the Forest Products Society, Technical Session Presentation, Merida, June 23, 1998. [3] C. Carnieri, G.A. Mendoza, W.G. Luppold, Optimal cutting of dimension parts from lumber with a defect: A heuristic solution procedure, Forest Products Journal 43 (9) (1993) 66–72. [4] D. Carlsson, M. Ro¨nnqvist, Supply chain management in forestry – case studies at So¨dra Cell AB, European Journal of Operational Research 163 (2005) 589–616. [5] P. Chandra, L.F. Marshall, Coordination of production and distribution planning, European Journal of Operational Research 72 (1994) 503–517. [6] S. Chand, H. Schneeberger, Single machine scheduling to minimize weighted earliness subject to no tardy jobs, European Journal of Operational Research 34 (1988) 221–230. [7] http://www.cplex.com. [8] M.A. Cohen, H.L. Lee, Strategic analysis of integrated production–distribution systems: Models and methods, Operations Research 36 (1988) 216–228. [9] R.R. Farrell, T.C. Maness, A relational database approach to a linear programming-based decision support system for production planning in secondary wood product manufacturing, 183-19, Decision Support Systems 40 (2005) 143–405.

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