rapport till leif

DEMAND BASED TACTICAL PLANNING OF THE ROUNDWOOD SUPPLY CHAIN WITH RESPECT TO STOCHASTIC DISTURBANCES

Daniel Hultqvist, Mid-Sweden University Leif Olsson, Mid-Sweden University

Rapportserie FSCN – ISSN 1650-5387 2003:19Internal FSCN report number: R-03-44

June, 2003

Mid Sweden UniversityFibre Science and Communication Network

SE-851 70 Sundsvall, Sweden

Internet: http://www.mh.se/fscn

FSCN/System Analysis and Mathematical ModelingInternet: Http/www.mh.se/fscn

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INNEHÅLLSFÖRTECKNING

Abstract..............................................................................................................................41. Introduction...................................................................................................................52. Description of the roundwood supply chain problem...................................................6

2.1 Overview of the roundwood supply chain...............................................................62.2 Harvesting, forwarding and small forest owners.....................................................8

2.2.1 Harvesting the own forest.................................................................................82.2.2 Small forest owners..........................................................................................9

2.3 Roundwood transportation and roads....................................................................102.4 Roundwood storage...............................................................................................122.5 Import and domestic roundwood markets and swapping of roundwood..............142.6 The demand distribution problem at the mills.......................................................162.7 A sample problem..................................................................................................172.8 A real large scaled problem...................................................................................18

3. Definitions...................................................................................................................193.1 Sets........................................................................................................................193.2 Parameters.............................................................................................................203.3 Costs......................................................................................................................213.4 Variables................................................................................................................22

3.4.1 Continuous variables......................................................................................223.4.2 Binary variables..............................................................................................22

3.5 Probabilities...........................................................................................................233.6 Uncertainty measures............................................................................................23

4. The mixed integer quadratic model.............................................................................254.1 Mathematical formulation.....................................................................................25

4.1.1 The objective function....................................................................................254.1.2 The constraints................................................................................................26

4.2 Explanation of the model.......................................................................................294.2.1 The objective function....................................................................................294.2.2 The constraints................................................................................................30

4.3 Modelling..............................................................................................................324.3.1 Scenario properties.........................................................................................324.3.2 Non-considered aspects..................................................................................32

5. Result from modelling.................................................................................................325.1 The sample problem..............................................................................................325.2 Solving a large-scaled problem.............................................................................325.3 The value of the uncertainty measures..................................................................32

6. Discussion and concluding remarks............................................................................326.1 Conclusions from the results.................................................................................326.2 Do we need a stochastic model?............................................................................326.3 Future extensions...................................................................................................32

7. Acknowledgements......................................................................................................32Appendix A: Estimation of costs.....................................................................................32A.1 Harvest costs and purchase costs from small forest owners.....................................32

A.1.1 Harvest costs from the own forest.....................................................................32A.1.2 Stumpage prices and delivered roundwood from small forest owners..............32

A 2. Transport and road maintenance costs.....................................................................33


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A 3. Roundwood storage costs........................................................................................33A 3.1 Storage cost........................................................................................................33A 3.2 Storage of purchased roundwood......................................................................33

A 4. Total cost for import and domestic roundwood.......................................................34References.......................................................................................................................35

FSCN/System Analysis and Mathematical ModelingInternet: Http/www.mh.se/fscnR-03-44Sida 4 (39)

DEMAND BASED TACTICAL PLANNING OF THE ROUNDWOOD SUPPLY CHAIN WITH RESPECT TO STOCHASTIC DISTURBANCES

Daniel Hultqvist and Leif OlssonMid-Sweden University, Sundsvall, Sweden

ABSTRACT

Brödtext

Keywords: Mixed-Integer Quadratic Programming, Forest Logistics, Scenario Modelling, Uncertainty, Roundwood Storage, Forest Roads, Roundwood market, Flexibility.


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1. INTRODUCTION

In recent years the forest companies in Sweden have moved their focus from harvest decisions, which have been relatively well investigated, to forest logistics and especially road investments (Olsson 2003a; Olsson 2003b and Arvidsson et al. 2000), and storage of roundwood (Persson et al. 2002; Persson & Elowsson. 2002 and Liukko & Elowsson. 1999). None of these questions have been accurate answered, although case studies indicate that optimal road investments can be calculated in an efficient manner, as in Olsson (2003a and 2003b). Hence, the Forest research institute of Sweden (SkogForsk) develop a decision support system for planning of road investments during 2003.

In difference to road investments, value losses at roundwood storage are another story. Although studies exist, as in Persson et al. (2002), Persson & Elowsson (2002) and Liukko & Elowsson (1999), it is in general very hard to estimate the cost of value losses and how roundwood of low quality affect the pulp and paper process at different mills . The problem will differ with assortment, weather and industry process, as well as storage location etc. Hence, a lot of the parameters are affected by the uncertainty and chaos of the real world (Prigogine & Stengers. 1984). Hence, in general they are very hard to predict. Furthermore, we can’t assume that the decisions in harvesting, transport and forest industry are independent, since decisions have to be flexible, as put forward by Barros & Weintraub (1982) and Weintraub & Navon (1976). Hence, we have to optimise the whole roundwood supply chain to get an optimal solution. There are also studies that indicate substantial gains if we integrate the process at the mills with roundwood supply chain planning (Arlinger et al. 2000 and Berg et al. 1995).

We describe and test an optimisation model that includes so many aspects as possible of the roundwood supply chain from harvesting into the industry processes, in this paper. Further, it is possible to fulfil the demand at the mills from the own forest, purchase from small forest owners, or buy roundwood at the domestic markets or import.

In contrast to other attempts to optimise the roundwood supply chain as in, for instance Karlsson (2002) and Bredström et al. (2001) with purely deterministic models, we focus on how to fulfil the demand of different assortments at the mills with respect to uncertain road bearing capacity. Further, seasonal roundwood price trends estimated from historical data (Skogsvårdsorganisationen) are also incorporated in the model. However, prices are not assumed to be stochastic processes (Hjort. 1996), as in for instance Gong (1998) or Lohmander (1999).

We develop a scenario optimisation model (Birge & Louveaux. 1997 and Kall & Wallace. 1994) that can be represented as a minimum cost network flow model (Rardin. 1998) with dynamic information constraints (Lohmander & Olsson. 2003). The deterministic equivalent is directly implemented as a convex mixed integer quadratic model (Wolsey. 1997). The reason why to use a scenario model is motivated with the Value of the Stochastic Solution (VSS), and the Expected Value of Perfect Information (EVPI), that describes the difference between the solution under uncertainty and certainty. Someone may argue that it is only to use what-if analysis instead of a scenario optimisation model. However, this is in general not true, as well described in Wallace (2000). The uncertainty measures are further described in Birge & Louveaux (1997), and in section 3.5.


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We describe the features of the model with a sample problem and show that it’s possible to solve a full-scaled problem with commercial software (Anon. 2002) and a standard PC without using any decomposition technique in reasonable time, with a reasonable amount of scenarios and time periods. This is an interesting result since most people usually assume that it’s impossible to incorporate randomness in large scaled optimisation problems. However, we show in this paper that this is not always the case anymore. Furthermore, our intention is to decompose the scenario optimisation model and solve with parallel processors in the future. Useful decomposition algorithms already exist for this type of problem as described, in for instance, Linderoth & Wright (2003) and give us the possibility to include more uncertainty and time periods in a future model, and still solve the problem in reasonable time. An outline of the remainder of this paper follows.

We start with the description of the round supply chain problem and assumptions in the next section. In section three, we present parameters, sets, variables, cost functions used in the model, as well as probabilities and uncertainty measures. In section four, we present the mixed integer quadratic model in mathematical terms, and explanation in words. In section five, we present results from the optimisation of a sample example, as well as some results from a full-scaled problem. The last section contains the discussion. Finally, a Appendix with cost estimations can be found in the end.

2. DESCRIPTION OF THE ROUNDWOOD SUPPLY CHAIN PROBLEM

2.1 Overview of the roundwood supply chain

We begin with an overview of our roundwood supply chain problem. We intend to include all essential aspects in the scenario model to solve on annual basis. Motivations are stated in detail in the following subsections, and the problem is depicted in Figure 2, and described below.

First step is to harvest the trees with the harvester and put the trees in log piles. Then, we pick up the log piles with a forwarder and transport to roadside.

However, here we assume no costs are related to roundwood storage in the forest, since good planning will decrease forest storage to a low level. This is also a independent optimization problem, which can be solved without taking the rest of the supply chain into account.

Observe that instead of these two machines we can sometimes use a combined machine called a harwarder (Figure 1), or buy roundwood from small forest owners directly at roadside.

We put the log piles at roadside on a truck or store it a while if we can’t use the road.

If possible, we haul the roundwood on the roads to a terminal storage, a better road or directly to the mills.


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If it’s not possible to use a sufficient amount of the roads to secure the roundwood demand at the mills. We then have to purchase wood at some external market, transport roundwood from the security stock or in advance move the roundwood to an accessible road so that it can be transported to the mills. This is very common during the thawing in the spring and periods with heavy rains, mostly during the fall.

Infrastructure investments are not included as an option in the model since these are long-term decisions as described in Olsson (2003a and 2003 b).

The model can handle railway transportations, even if it’s not included in Figure 2.

Figure 1: A harwarder in a Swedish forest.

The description above is of course simplified to get the overall picture, and more details are given in following subsections. However, one essential part in our model is that the road transportation of the roundwood is affected of the bearing capacity of the roads as described in Hossein (1999) and Löfroth et al. (1992). Further, bearing capacity on roads depends on weather phenomena and has to be modelled as uncertain. In the model presented here we use a scenario optimisation approach to incorporate uncertain bearing capacity in the model. This is further described in section 2.3., and in Lohmander & Olsson (2003).


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Figure 2: An over all view of the roundwood supply chain from forest to mill. Red arrows indicate that hauling is uncertain under some periods and scenarios during a year. This uncertainty is handled with scenarios in the optimisation model.

2.2 Harvesting, forwarding and small forest owners

We divide this subsection into three different harvest types.

1. Harvest the own forest.2. Harvest stumpage purchased from small forest owners.3. Roundwood delivered by small forest owners directly to roadside.

2.2.1 Harvesting the own forest

Studies, as in for instance Arvidsson et al. (2000) and Carlsson et al. (1998) indicate that extraction of logs can be optimized. This is not included in our model. We assume that the extraction is optimized. This and the fact that harvester and forwarder are working in teams makes the harvesting cost independent of this aspect. Further, different costs are assumed if we use external workers, our own workers and if they have to work overtime. This cost function is further described in Appendix A.1.1 , and depicted in Figure 3.


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Figure 3: Estimation of the harvest cost with a quadratic cost function. Observe that this give us a linear marginal harvest cost.

This cost function gives us no longer a linear model. However, the function is convex and quadratic. Hence, it will not add any major difficulties for the solver. Therefore, only has minor affect on the solution time.

We assume that the roundwood will not be stored as log piles in the forest. Of course, that’s not the case. However, storage in the forest should be for a very short time. This storage has only a minor affect on the total solution, and can be skipped in general.

A new type of machine has also been tested that is a combined harvester and forwarder, called harwarder (Figure 1). If this machine is used, there will be no storage in the forest. Hence, to make the modelling easier, we assume that this is always the case.

We also divide harvesting in two different types.

1. Thinning.2. Clear cutting.

The prices and costs for clear cutting and thinning can be found in Appendix A.1.1.

2.2.2 Small forest owners

Small forest owners mostly deliver their roundwood to the roadside or sell it on the stumpage market, as put forward by Rylander (2000). Hence, it is handled as a part of the companies own forest with an additional cost for the stumpage price. If the forest


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owner himself harvest the wood and deliver to mill or roadside, this is seen as an external source. A fixed part of this volume will be billed an average cost for transportation to represent the volume that the forest company has to transport itself. With this approach small forest owners will be an endogenous part of our model. However, the optimization in our model is performed from a mill owner view, and not from the small forest owner’s point of view.

Small forest owners usually have other incomes as well (Bergman, 1998). Hence, it is very important to decide the right price level for companies buying from these owners. Furthermore, most of these owners want to deliver their roundwood during the winter or spring (ref sydved), if at all, since they usually do not need the money. Therefore, we include the prices in the optimisation to decide which prices that are optimal to pay the forest owner.

Another aspect is that many small forest owners, that are spatially rather evenly distributed, sell their roundwood to a few spatially concentrated forest industries. Hence, this is likely to result in an oligopsonized market solution (Johansson & Löfgren, 1985). This indicates that the roundwood price the company have to pay the small forest owner will increase with the volume. For this reason, we have to decide a lower and an upper bound on the purchase cost and estimate a unit cost that depends linearly on the volume. This is done in Appendix A.1.2.

One more thing is that small forest owners are to a higher extent dependent on the weather, as for instance, wind throwing highly affect their forest. However, these aspects haven’t been included in our model. More information how to optimise this subject can be found in, for instance, part three of Lohmander (1987).

The oligopsonized market situation is also an example where using random variables to reflect the behaviour of the market participants can be very dangerous, as shown by Haugen & Wallace (manus), using simple game theoretic models.

2.3 Roundwood transportation and roads

Roundwood is mostly hauled on roads from the forest to the mills. Hence, hauling is highly dependent of the roads. Unfortunately, during the thawing period in the spring and rainy periods, usually in the fall, the bearing capacity is very limited on most of the gravel roads (Hossein 1999 and Olsson & Lohmander 2003). Hence, heavy loads can only be hauled on some roads of high quality during these periods.

Since our model is for a tactical planning period, accurate infrastructure decisions such as upgrading of roads, and construction of new roads (Bergström, 2001) should not be made, since they are better made with a longer planning horizon, as motivated in Olsson (2003a, 2003b) and Olsson & Lohmander (2003). However, road maintenance can and should be made. These costs are included in our transportation costs, and are highly dependent on the weather (Hossein, 1999). For this we use scenarios to cope with uncertain weather as depicted in Figure 4.


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Figure 4: The scenario tree of our optimisation model represent different outcomes of the stochastic variables.

In the scenario tree in Figure 4, we have following definitions.

Low = Low water content in the road allow heavy loads for the period. Med = Normal water content in the road allow normal loads for the period. High = High water content in the road, and the bearing capacity is very limited

on the road for the period.

The bearing limitation on the roads is incorporated as a constraint in the optimization model, and is further described in section 4.2. Further, it is possible to close some roads during critical periods if necessary. We want describe any more about roads here. However, gravel roads are very important for the hauling of roundwood and the interested reader find a lot of information about the subject in Hossein (1999), Löfroth et al. (1992) and Olsson & Lohmander (2003). Surface roads are usually possible to use the whole year, as assumed in our study. Costs for transportation and maintenance of roads are found in Appendix A.2.

We have some other restrictions about the roundwood transportation.

The demand of roundwood is fixed We do not include backhauling in the model

Skriva om – fel! These assumptions are probably not too restrictive in general, since in most cases the destination is already decided, although some savings could be made


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with a non-fixed destination (Bergdahl 2002) and backhauling can only be used in some areas. Furthermore, include backhauling in our model will dramatically decrease the possibility to solve large scaled problem instances (Carlgren et al. 2000 and Carlsson & Rönnqvist 2001), and it can probably be solved separately, without loss of generality, to generate transportation costs to our model.

2.4 Roundwood storage

The most challenging part of the roundwood supply chain is probably roundwood storage, and the following questions.

1. How does storage affect the quality of roundwood?2. How does low or mixed quality of the roundwood affect the processes at the mills?3. What is the cost of quality losses?

Many have tried to answer these questions, for instance Forsberg (2000) and most recently Persson et al. (2002) and Persson & Elowsson (2002). Unfortunately, the answer is substantially different depending on a lot of things (Berg et al. 1995), some of them listed below.

Assortment Weather and Season Industry process Storage time Storage location Sprinkling

Some other difficulties for the modelling are that roundwood log piles can be stored very different. That is, some log piles will be stored one period, some two periods and some even longer, as depicted in Figure 5. This is the case since storage costs are not only dependent on time. As an example, log piles that are frozen during the winter manage storage in spring better then log piles of springwood, due to lower mould probability etc. (Sjöblom, 2003).

The time dependency of the storage constraints depicted in Figure 5 make it impossible to separate these constraints in different blocks as recommended in Birge & Louveaux (1997). These types of constraints are called dynamic information constraints and are described in Lohmander & Olsson (2003).

Scenarios also complicate the situation since roundwood has to be sent from all ancestor scenarios to the present one, as depicted in Figure 6.

The most challenging part is however the storage cost itself. The estimation used in our optimisation is described in Appendix A.3. Nevertheless, further research is necessary to enable better approximations of these costs in the future.

A study by Persson & Elowsson (2001) indicates that spruce pulpwood dry rapidly when stored at roadside, in summer conditions. Hence, in this case, the best policy is to


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store spruce pulpwood under sprinkling at terminal. The situation can however be different for different assortments and locations. In a real world situation, it is important with good knowledge about how quality losses affects each industry process.

Figure 5: Roundwood flow at a typical storage node. Observe that roundwood stored in period t can be used in all descendant periods. This is modelled as roundwood flow in time at the same storage location.


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Figure 6: Roundwood is sent from all ancestor scenarios to the previous one. We call it dynamic information constraints.

2.5 Import and domestic roundwood markets and swapping of roundwood

Swapping roundwood with other companies is considered as virtual links in the network model (Figure 4), between other company mills (k) and internal mills (i). The cost related to this link is the own company’s additional cost for swapping. This cost is assumed to be zero in our study, but can be set to whatever cost that’s available, in a real world situation.

We divide external purchases of roundwood in the domestic roundwood market and the international roundwood market here. Farmers that own small pieces of forestland are another story, and are handled separately in the model, as described in section 2.2. The network representations of the markets are depicted in Figure 7.


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Figure 7: Network representation of roundwood supply from different sources to an internal mill (IM).

Prices on the international roundwood market and the domestic market are estimated as mean values for every time period, from historical data provided by Skogvårdsorganisationerna (SVO), and are presented in Appendix A.4, as well as approximation of the total costs including value losses as described in Appendix A.3.3.

We have no intention to foresee the market price on roundwood, since we do not believe that its possible to predict any roundwood market, at least on our short-term yearly basis, due to many exogenous chaos parameters (Prigogine,1984), for instance: political decisions, weather phenomena and psychological factors , as described in Meyer (2002). To convince the reader, price fluctuations for conifer wood during 1998-2001 are compared in the diagram depicted in Figure 8.


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Figure 8: Import price of conifer wood for every month during a year. Compared for 1998, 1999, 2000 and 2001 with data from SVO. Notice the non-correlated behaviour during the spring. Further, that price seems to be low in the summer and high in the winter.

There has been a lot of research, in forest economy, where the market roundwood prices have been modelled as stochastic processes (Hjorth, 1996), for instance as an first order AR-process in Gong (1998) and Lohmander (1999). Later as a first order ARMA-process in McGough (2002). However, these approximations have been used on long-term basis, and we are rather convinced that this approach works better in the long run, then on annual basis, as in our case.

Due to the unpredictable nature of the roundwood market, we decide to use roundwood market prices from 2001 in our study, since other price and cost estimations also are from this time period.

In our study the most important aspects of the market price trend is that market prices seems to always be lowest in the summer and highest in the winter, independent of the year. If we take a look at historical data from SVO, the only “chaotic” period, with no price trend, seems to be the spring.

Since there are no apparent differences in the market prices depending on weather. The external market variables will be independent of the weather scenarios, explained in Figure 4.

2.6 The demand distribution problem at the mills

An overall view of the demand at a mill can be found in Figure 7, there are however some difficulties. A major problem is that industry processes can handle assortments in a very different manner dependent on quality losses, on the way. If, for instance, we harvest spruce to be stored as spruce pulpwood, it depends on quality losses how it can be used when it arrives to the pulp-mill. This problem can be handled in many different ways. However, we have found that a good approach is to use more assortments in the model, and to downgrade assortments at roundwood storage, as depicted in Figure 9.


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One argument about this approach is that it increases the model size. However, our results in section five indicate that this seems to be no major problem. Further, this is the way it happens in real life.

Figure 9: An example on downgrading of roundwood when storing from period 1 to period 4.Using more types of every assortment is a relevant way to model this.

2.7 A sample problem

For the purpose of development a small sample problem was created.


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Figure 10: A small sample problem with 18 possible harvest areas, one storage, one saw mill and one paper mill.

This is an artificially created problem.

2.8 A real large scaled problem

And a part of Iggesund.


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Figure 11: The district Sveg owned by Holmen Skog in the middle of Sweden.

This is a real problem.

3. DEFINITIONS

3.1 Sets

A = Assortments C = Cross roadsD = Mills

DS= Saw mills, a subset of D.

DP= Pulp and Paper mills, a subset of D.

H =Harvest sites

H s

t t

= Harvest sites that can’t be harvested during period t t , scenario s, due to thawing, a subset of H.L = Terminals

LW= Train terminals

O = Mills own by other companies.


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R = Road

RF= Roads for transporting wood chips

RR= Rail road

S = ScenariosT = Time period.

3.2 Parameters

α jta = A parameter that describes how to much useful fiber there are in assortment a in time period t at mill j.β j = The relative part of the demand at a saw mill j that will be turned into wood chips.

γ jtl

= Calculated part of the volume bought from domestic market that the buyer has to pay the transportation for.

d jtf

= Minimum demand of wood chips at pulp/paper mill j for period t.

d jtal

= Minimum demand of assortment a at mill j for period t.dkta

o ,max= Maximum volume, of assortment a, that can be delivered to mill k for period t.

dktao ,min

= Minimum volume, of assortment a, that has to be delivered to mill k for period t.

d jttot

= Total minimum demand of all assortments at mill j for period t.f jt = The volume of wood chips produced from wood delivered by other contractors to mill j for period t.hits1 . . st

c

= Capacity at location i for scenario (s1,s2,…,st) during period t. Must be greaterthan zero.

htmax

= The maximum number of harvesting hours a team can work in time period t.hia

part= The proportion of the total volume at site i that consist of assortment a.

hitot

= The total volume at site i that can be harvested.

l jainit

= The initial stored volume of roundwood of assortment a at terminal j.

l jtmax

= The maximal volume terminal j can hold at any given time period t.

l jtamin

= The minimal volume of assortment a terminal j must hold at any given time during

period t.M = A large number used to set some binary variables..mts1 .. s t a

d

= Total volume of assortment a that can be bought on the domestic market forscenario (s1,s2,…,st) during period t due to logistic constrictions.

mts1 .. s t ai

= Total volume of assortment a that can be bought on the international market for


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scenario (s1,s2,…,st) during period t.min t

hc ,max thc

= Minimum and maximum harvesting capacity, in hours, available duringperiod t.

min tvc ,maxt

vc= Minimum and maximum transportation capacity, in hours, available

during period t.max t

wc= Maximum number of railway wagons available for period t.

pts1 .. st = Probability of scenario (s1,s2,…,st) in period t.rijts1 .. st

tot

= Maximum volume that can be transported on road (i,j) for scenario (s1,s2,…,st)during period t.

tn = The last time period.t t = The time period that contains the thawing period.

t f= Average transportation time between harvesting sites.

thl = Loading time at harvest site, in average, per volume transported by truck.

t ll = Loading time at storage terminals, in average, per volume transported by truck.

t lu = Unloading time and time it takes to measure the delivered volume at a terminal or

mill for roundwood delivered by truck.t ijts1 . . st

t

= Travel time for road (i,j) for scenario (s1,s2,…,st) during period t.v ijts1 . . st

c

= Loading capacity for trucks on road (i,j) for scenario (s1,s2,…,st) during period t. Must be greater than zero.

w ijc

= Loading capacity for railway wagons on railroad (i,j). Must be greater than zero.x iuts1 .. st a1a2

a

= A parameter that tells if roundwood stored from time period u to period t will change assortment from a1 to a2 in scenario (s1,s2,…,st). Can also represent a part of the volume.

xkjta1 a2

s

= A parameter that tells if roundwood of assortment a1 delivered to mill k will be transferred as assortment a2 to mill j. Can also represent a part of the total volume.

3.3 Costs

c jts1 . . s t aed

= The cost of buying assortment a on the domestic market for mill j in scenario (s1,s2,…,st) during period t.

c jts1 . . s t aed , fixed

= The cost of transporting assortment a, bought on the domestic market, to mill j for scenario (s1,s2,…,st) during period t.c jts1 . . s t a

ei

= The cost of buying assortment a on the international market for mill j for scenario (s1,s2,…,st) during period t.

c ijf

= The cost of transporting wood chips from saw mill i to pulp/paper mill j.


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c ts1 .. st

h1

= The linear cost for harvesting scenario (s1,s2,…,st) during period t. c ts1 .. st

h2

= The quadratic cost for harvesting scenario (s1,s2,…,st) during period t. c i

h , crew= A fixed cost for transportation of the crew to and from the harvest area.

c th , fixed

= The fixed cost for harvesting during period t.c its1 .. s t

hl

= The cost of loading the harvested roundwood at harvest site i for scenario(s1,s2,…,st) during period t.

ch , tran= The cost of transportation of internal equipment between harvesting sites.

c juts1 .. s t al

= The cost of storing at terminal j from time period u to time period t withscenario (s1,s2,…,st) for assortment a without watering.

c jl , fixed

= The cost of having the terminal j itself.

c jll

= The cost of loading at terminal j.

c jlu

= The cost of unloading at terminal or mill j.c its1 .. s t

ph

= The cost of pre-hauling the harvested roundwood to an accessible road for areas that can’t be accessed during the thawing period in the spring.

c ijts1 . . s t

v

= The cost for transport on road (i,j) for scenario (s1,s2,…,st) during period t.

c ijw

= The cost of transportation on railroad form i to j, dependent on volume transported.

c ijw , fixed

= The fixed cost per train set on railroad from i to j.

c jy

= The cost of having an extra assortment a at saw mill j.

3.4 Variables

3.4.1 Continuous variables

d jta = Volume of assortment a used at mill j for period t.dkta

o= Volume of assortment a transported to mill k, own by another company, for

time period t.e jts1 . . st a

d

= Volume of assortment a bought on the domestic market at mill j for scenario(s1,s2,…,st) during period t.

e jts1 . . st ai

= Volume of assortment a bought on the international market at mill j for scenario (s1,s2,…,st) during period t.hits1 . . st = Part of area i harvested in scenario (s1,s2,…,st) during time period t.hts1 . . st

h

= Number of hours used for harvesting in scenario (s1,s2,…,st) during period t.l juts1 . . s t a = Volume of assortment a stored at j for scenario (s1,s2,…,st) from period u to t.l js1 . . st n

afinal

= The volume of assortment a that is in store at the end of time period tn with


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scenario (s1,s2,…,stn).v ijts1 . . st a = Volume of assortment a transported on road (i,j) for scenario (s1,s2,…,st) during period t.v ijts1 . . st

f

= Volume of wood chip transported from saw mill i to pulp/paper mill j in scenario (s1,s2,…,st) during period t.w ijts1 . . s t a= Volume of assortment a transported on railroad (i,j) for scenario (s1,s2,…,st) during period t.

3.4.2 Binary variables

y its1 .. s t = Is 1 if the area is harvested during time period t with scenario (s1,s2,…,st) otherwise 0.

y jas

= Is 1 if the assortment is in use at the saw mill otherwise 0.y is1 .. . s t t−1a

t t

= I 1 if either y

i (t t−1) s1 . . st t−1 or y it t s1 .. s t t is 1 otherwise 0.

zit( t+1)= Is 1 if y its1 .. s t or any of

y i( t+1) s1 . . s t+1 is 1 otherwise 0.

zi1

= Is 1 if any of zit( t+1) is 1 otherwise 0.

zi2

= Is 1 if y its1 .. s t and

y ius1 . . su is 1 and u>t+1 otherwise 0.zt

h= Is 1 if anything is harvested during period t otherwise 0.

z jl

= Is 1 if anything is stored at storage j during any time period and scenario, otherwise 0.

zijw

= Is 1 if anything is transported on railroad from i to j.

3.5 Probabilities

Except for in the first period, representing the winter season where everything is frozen, we have three scenarios in the second period and three scenarios per scenario in period three, as depicted in Figure 4. An interesting question is:

What is the probability that scenario k happens during period t?

The answer is that it depends on how we define wet, normal and dry weather. Hence, we assume that its equal likely that it will be dry, wet or normal weather conditions in our general model. However, applied in reality, the scenario distribution can be estimated from historical weather data, for any geographical area, and generated with for instance moment-matching scenario generation (Höyland et al. 2003). We also have to remember that a uniform distribution is a much better approximation of the reality then a distribution with all the mass in the mean, as in a deterministic model. Further, the important part is to include flexibility and options in our solutions.


R-03-44Sida 24 (39)

We use a discreet uniform distribution (Blom, 1984) in the rest of this paper. Hence in period two all probabilities are 1/3 and in period three 1/9, if we uses scenarios as in Figure 4.

3.6 Uncertainty measures

We use two types of uncertainty measures in the same way as described in Birge & Louveaux (1997). These measures are the Value of the Stochastic Solution (VSS) and the Expected Value of Perfect Information (EVPI). Another way to measure uncertainty is to use simulation or sometimes called scenario analysis. It is not used here, since uncertainty analysis is not the core part of this paper.

The VSS measures the value of including randomness. It can be calculated in the following steps, as well described in Kall & Wallace (1994).

Solve the mean value (EV) problem to get a first stage solution. Fix the first stage solution and solve for all other second stage scenarios, with

the mean value scenarios in all other periods. Fix the variables in scenario 2, and if we have only three scenarios, we solve the

problem for all scenarios with fixed variables in period one and two. If we have more periods its only to repeat! We call the solution from this nested calculation the expectation of the EV

solution or simply EEV. Calculate the scenario model solution or stochastic programming solution called

SP. The VSS is in the minimisation case calculated as:

VSS = EEV-SP.

VSS with the above definition is always greater then equal to zero, since the stochastic solution (SP) always is less then equal the “worse” EEV solution in a minimisation.

The EVPI measures the cost of uncertainty and is calculated in following steps.

Solve the deterministic problem for all of our scenarios. In our case with three periods we have nine scenarios, hence we have to solve nine problems.

Calculate the mean value of this solution and we get the “wait and see solution” WS.

The EVPI is then in the minimisation case calculated as:

EVPI= SP-WS.

EVPI is with the above definition also always greater then equal to zero, since the stochastic solution (SP), in a minimisation, always is greater then equal to the “better” wait and see solution (WS).


R-03-44Sida 25 (39)

Calculating this uncertainty measures give us interesting information. A high value on the VSS means that it is of substantial value to solve the scenario model, and maybe it will be valuable to include more scenarios. Hence, a deterministic model would probably not give us accurate solutions. However, if VSS = 0 there are no value in incorporating randomness in the model. Unfortunately, as pointed out in Birge & Louveaux (1997), and shown in section 5.3 and 6.2, such situations rarely occur in the real world.

A high value on the EVPI means that it would be of great benefit to know what would happen in the future, and indicates that we should be careful to draw to many conclusions from our solutions, since our solutions are highly uncertain. A high EVPI is also good for people working with for instance weather forecasts to motivate people support their future research.

With this uncertainty measures we can quantifiably measure the need of a stochastic model and not only put forward that its better then a deterministic model, and we also measure the uncertainty in our solutions.

4. THE MIXED INTEGER QUADRATIC MODEL

4.1 Mathematical formulation

4.1.1 The objective function

min ∑t∈T

∑s1 .. st∈S

p ts1 . . s tc ts1 .. st

h 2 (hts1 .. st

h )2 + p ts1 .. stc ts1 . .st

h1 hts1 .. st

h

+ (I)∑t∈T

c th , fixed zt

h

+ (II)

∑i∈H

∑t ∈T

∑s1 . . s t∈ S

pts1

.. stci

h ,crew h its1

.. st

+ (III)

∑i∈H

ch ,tran (zi1+ zi

2 ) + (IV)

∑i∈H

∑t ∈T

∑s1 . . s t∈ S

pts1

.. stcits

1.. s

t

hl hits1

.. sthi

tot

+ (V)

∑(i , j )∈R

∑t ∈T

∑s1 .. st∈ S

∑a∈ A

p ts1 . . s tc ijts

1.. s

t

v v ijts1

.. sta

+ (VI)

∑i∈H s

t t

∑s1 . .st∈S

p t t s1 . . s tt

c itts

1.. s

t t

ph hitts

1. .s

t t

hitot

+ (VII)


R-03-44Sida 26 (39)

∑j∈ L

∑u∈T

∑t∈T

∑s1 .. st∈S

∑a∈ A

pts1 .. stc juts

1.. s

ta

l l juts1

.. sta

+ (VIII)

∑j∈ L∪D

∑i :(i , j )∈R

∑t∈T

∑s1 .. st∈S

∑a∈ A

pts1 .. s tc j

lu v ijts1

.. sta

+ (IX)

∑j∈ L

∑i :( i , j )∈ R

∑t∈T

∑s1 .. st∈S

∑a∈ A

pts1 .. stc j

ll v jits1

.. sta

+ (X)

∑j∈ L

c jl , fixed z j

l

+ (XI)

∑j∈D

∑t ∈T

∑s1. .st∈ S

∑a∈ A

p ts1 . . stc jts

1.. s

ta

ed e jts1

.. sta

d

+ (XII)

∑j∈D

∑t ∈T

∑s1. .st∈ S

∑a∈ A

p ts1 . . stc jts

1.. s

ta

ed ,fixed e jts1

.. sta

d

+ (XIII)

∑j∈D

∑t ∈T

∑s1. .st∈ S

∑a∈ A

p ts1 . . stc jts

1.. s

ta

ei e jts1

.. sta

i

+ (XIV)

∑j∈Ds

c jy( ∑

a∈ A

y jas −1)

+ (XV)

∑( i , j )∈ RR

cijw, fixed zij

w

+ (XVI)

∑(i , j )∈R R

∑t∈T

∑s1 .. st∈S

∑a∈ A

c ijw

wijts1 . . s t a

wijc

+ (XVII)

∑(i , j )∈ RF

∑t∈T

∑s1 .. st∈S

p ts1 . . s tc ij

f v ijts1

.. st

f

(XVIII)

4.1.2 The constraints

hits1 . . st¿ y its1 .. st ∀ i∈H−H s

t t , t∈T , s∈S (1a)

y its1 .. s t¿ y is1 .. .st t−1 a

t t∀ i∈H s

t t , s∈S; t=t t−1 , tt (1b)

∑t∈T

hits1 . . st¿1

∀ i∈H , s∈S (2)


R-03-44Sida 27 (39)

∑t∈T

y its1 .. s t¿2 zi

1

∀ i∈H−H s

t t , s∈S (3a)

∑u∈T

y ius1 . . s t+ y is1 .. .st t−1 a

t t +∑v∈T

y ivs1 .. st¿2 zi

1

∀ i∈H s

t t , s∈S (3b)

y its1 .. s t+ y i(t+1 )s

1.. s

t+1¿1+zit(t+1 ) ∀ i∈H−H s

t t , t∈T , s∈S ;t<t n (4a)


1.. s

t+1¿1+zit(t+1 ) ∀ i∈H s

t t ,t ∈T , s∈S; t <t t−2 (4b)

yi (t t−2) s1 . . st

+ y is1 .. .st t−1 a

t t ¿1+zi (t t−2) (tt−1) ∀ i∈H s

t t ,t ∈T , s∈S (4c)


1.. s

t¿1+zit (t+1) ∀ i∈H s

t t ,t ∈T , s∈S; t t ¿ t< tn (4d)

∑i∈H

zit ( t+1)≤h ts1 . . . st

h

htt

max +1∀ t ∈T , s∈S; t <tn (5)

y its1 .. s t+ yius

1.. s

u¿1+zi

2∀ i∈H−H s

t t , t∈T ,u∈T , s∈S ;u>t +1 (6a)

y its1 .. s t+ yius

1.. s

u¿1+zi

2

∀ i∈H s

t t ,t ∈T ,u∈T , s∈S ;u>t+1 ; t , u≠tt−1 , t t (6b)

y is1 .. . s t t−1a

t t + y ius1 .. su¿1+zi

2

∀ i∈H s

t t ,t ∈T ,u∈T , s∈S ;u>tt ;u< tt−2 (6c)

∑i∈H

hits1 . . sthi

tot

hits1 . .st

c +t f ∑i∈H

y its1

.. st=hts

1. .s

t

h

∀ t ∈T , s∈S; t≠t t−1 , tt (7a)

∑i∈H

hi (tt−1) s1 . . s t t−1hi

tot

hi ( tt−1) s1

.. stt−1

c +t f ∑i∈H

y i ( tt−1) s1

.. stt−1

+ ∑i∈H s

tt

hitt s1 . .st t

hitot

hitt s1 .. st t

c =h (t t−1)s1

. .st t−1

h

∀ s∈S (7b)

∑i∈H−H

stt

hit t s1 . . st t

h itot

hit t s1 .. st t

c +t f ∑i∈H

y its1 . .st t

=hts1 .. st t

h

∀ s∈S (7c)


R-03-44Sida 28 (39)

min thc≤hts1 . . st

h ¿maxthc zt

h

∀ t ∈T , s∈S (8)

∑(i , j )∈ R

v ijts1 . . st a− ∑( j ,i)∈ R

v jits1

.. sta =hits

1. .s

thia

part hitot

∀ i∈H , t∈T , s∈S ,a∈ A (9)

∑( j ,i )∈ R

v jits1 . . st a− ∑( i , j)∈R

v ijts1 .. st a =0∀ i∈C ,t ∈T , s∈S , a∈ A (10)

∑a∈ A

v ijts1 .. s t a¿ rijts1

. .st

tot

∀ (i , j )∈R , t∈T , s∈S (11)

min tvc≤∑

i∈H∑a∈ A

v ijts1 .. s t a t hl+ ∑j∈L∪D

∑a∈ A

v ijts1 .. stat lu+∑

j∈ L∑a∈ A

v jits1 .. st at ll+ ∑( i, j )∈ R

∑a∈ A

v ijts1 .. st a

v ijts1 .. st

c t ijts1 .. st

t ¿max tvc

∀ t ∈T , s∈S (12)

l jainit+ ∑

( i , j )∈ R

v ij1 a− ∑( i , j )∈ R

v ji1 a−∑u>1

l j 1us1 . . su a=0∀ j∈ L ,u∈T , s∈S , a∈ A (13)

∑v∈T

∑a1∈ A

l jvts1 . . su a1x jvts1 .. sua1a2

a + ∑(i , j)∈R

v ijts1 .. st a2− ∑

( j ,i)∈ R

v jitas1 .. sta2−∑

u∈T

l jtus1 .. sua2=0

∀ j∈ L ,t∈T , s∈S ,a2∈ A ;u>t , v< t (14)

∑v∈T

∑a1∈ A

l jvtn s1 . . sua1x jvt ns1 .. sua1a2

a + ∑(i , j )∈ R

v ijtn s1 .. stna2− ∑

( j , i)∈R

v jitns1 .. st na2−l ja2 s1 .. st n

final =0

∀ j∈ L , s∈S , a2∈ A ;v<tn (15)

∑a∈ A

∑v∈T

∑u∈T

l jvus1 . . su a¿ l jtmax z j

l

∀ j∈ L ,t∈T , s∈S ;v≤t , u>t (16)

l jtamin≤∑

v∈T∑u∈T

l jvus1 . . sua ∀ j∈ L ,t∈T , s∈S ,a∈ A ; v≤t ,u> t (17a)

l jtn amin ¿ l js

1.. s

t na

final

∀ j∈ L , s∈S , a∈ A (17b)

dktao ,min≤d kta

o ≤d ktao,max ∀ k∈O ,t ∈T ,a∈ A (18)

∑(i , j )∈ R

v ijts1 . . st a2+e jts

1. .s

ta

2

d +e jts1. .s

ta

2

i +∑k∈O

∑a1∈ A

dkta1

o xkja1

a2

s =d jta2


R-03-44Sida 29 (39)

∀ j∈D ,t ∈T , s∈S , a2∈ A (19)

∑a∈ A

α jta d jta=d jttot

∀ j∈D ,t ∈T (20)

∑t∈T

d jta≤My jas

∀ j∈DS , a∈ A (21)

d jta≥d jtal ∀ j∈D ,t ∈T ,a∈ A (22)

v jits1 .. st

f ¿ β jd jttot +f jt ∀ j∈DS ,i∈DP ,t ∈T , s∈S (23)

∑(i , j )∈ Rf

v ijts1 . . st

f ¿d jtf

∀ j∈DP , t∈T , s∈S (24)

∑j∈D

e jts1 . . s t ad ¿mts

1. .s

ta

d

∀ t ∈T , s∈S , a∈ A (25)

∑j∈D

e jts1 . . s t ai ¿mts

1. .s

ta

i

∀ t ∈T , s∈S , a∈ A (26)

∑(i , j )∈ R

vijts1 . . st a= ∑( j ,i )∈ RR

w jits1 .. st a∀ j∈ LW ,t ∈T , s∈S , a∈ A (27)

∑a∈ A

wijts1 . . s t a

wijc

¿maxtwc zij

w

∀ (i , j )∈RR ,t∈T , s∈S (28)

d jta≥0 ∀ j∈D ,t ∈T ,a∈ Ae jts1 . . st a

d ¿0 ∀ j∈D ,t ∈T , s∈S , a∈ Ae jts1 . . st a

i ¿0 ∀ j∈D ,t ∈T , s∈S , a∈ Ahits1 . . st

¿0 ∀ i∈H , t∈T , s∈S (29)l juts1 . . s t a¿0 ∀ j∈ L ,u∈T , t∈T , s∈S ,a∈ Al js1 . . st n

afinal ¿0 ∀ j∈ L , s∈S , a∈ A

v ijts1 . . st a ¿0 ∀ (i , j )∈R , t∈T , s∈S , a∈ A

v ijts1 . . st

f ¿0 ∀ (i , j )∈RF , t∈T , s∈Sw ijts1 . . s t a¿0 ∀ (i , j )∈RR ,t∈T , s∈S , a∈ A


R-03-44Sida 30 (39)

y its1 .. s t∈ {0,1 } ∀ i∈H , t∈T , s∈S

y jas ∈ {0,1 } ∀ j∈DS , a∈ A

y is1 .. . s t t−1a

t t ∈ {0,1 } ∀ i∈DS , s∈S , a∈ Azit( t+1)∈ {0,1 } ∀ i∈H , t∈T ;t<tn

zi1∈ {0,1 } ∀ i∈H (30)

zi2∈ {0,1 } ∀ i∈H

z jl∈ {0,1 } ∀ j∈ L

zth∈ {0,1 } ∀ t ∈T

zijw∈ {0,1 } ∀ (i , j )∈RR

4.2 Explanation of the model

4.2.1 The objective function

The purpose of objective function is to minimize the cost of supplying the industries with the demanded volumes.

I) The cost of harvesting.II) The fixed (initial) cost for harvesting.III) The cost of transporting the crew to and from the harvest site.IV) The initial cost of harvesting at a site. This includes transportation of

equipment and some maintenance of roads.V) The cost of loading the harvested volume onto trucks.VI) The cost of transportation by truck.VII) The cost of pre-hauling to accessible road during thawing period.VIII) The cost of storing.IX) The cost of unloading and measuring incoming volumes at storage terminals

and mills.X) The cost of loading outgoing volumes at the terminals.XI) The fixed cost for having/using storage terminals.XII) The cost of buying wood on the open domestic market.XIII) The cost of collection some of the bought volume on the domestic market.XIV) The cost of buying wood on the international market.XV) The extra cost of having more than one assortment at a saw mill.XVI) The fixed cost for operating a rail road system.XVII) The volume dependent cost of rail road transportation, including loading and

unloading from truck to rail road. XVIII) The transportation and purchase cost of wood chip from saw mills to pulp

and paper mills.


R-03-44Sida 31 (39)

4.2.2 The constraints

These are the models constraints

1) The relative part harvested in each time period and in each scenario must be less than 1. This constraint also sets the binary variable that indicates if anything is harvested during a specific time period and scenario.

2) The sum of the parts harvested in all time periods for each path through the scenario tree can’t be greater than 1.

3) Restricts the number of time periods that an area can be harvested in to 2. It will also set the binary variable that indicates that some volume is harvested at the area.

4) Checks if an area is harvested during two consecutive time periods, if so a binary variable is set to 1.

5) The number of areas that is harvested in two following time periods is restricted to the number of harvesting teams available.

6) This constraint is constructed to indicate if an area is harvest during two non-consecutive time periods, if that is the case a binary variable is set to give the extra cost this implies.

7) All the activities that a harvesting team is involved in are made into hours. This includes not only the time that the team is harvesting but also transfer time. If the next time period is the thawing period, the number of hours needed to harvest areas that can’t be harvested during that time is added to this time period.

8) The number of hours that can be used to harvesting is restricted, both minimum and maximum. A binary variable for each time period is set if any harvesting is done to give the fixed harvesting costs.

9) The difference between outgoing and ingoing flow at a harvest node must equal the volume harvested at the node.

10) At a cross road the ingoing and outgoing volume must be the same for each assortment.

11) The total volume transported on a road can be restricted. This is useful on small roads that are not built to take a lot of traffic.

12) The number of hours that can be used for transporting of roundwood is restricted. This includes loading, unloading and driving time.

13) At the first time period, what is already in storage and what is transported into the storage must equal what is transported out of there and what is stored for later use.

14) The sum of what is stored from earlier time periods to be used in this period, with possible change of assortment classification, and the volume transported into the storage must equal what is stored for later use or transported out of the storage.

15) In the last time period, what is stored for this period or transported to the storage minus the volume transported out of there must equal a fixed volume. This volume is set in advance and represents how much the storage must hold at the end of the last time period.

16) The total volume that can be stored at a storage must be less than, or equal, to what the storage can hold.


R-03-44Sida 32 (39)

17) For each assortment in a storage, a minimum volume (safety level) can be set. This forces the model to always keep to amount of a given assortment at the storage.

18) The volume delivered to mills with other owners can have a maximum and a minimum.

19) The volume transported to a mill, the volume of roundwood bought on the open market and the volume delivered from other mill owners due to swapping is summed into a continuous variable.

20) The sum of all assortments must equal the total demand at the mill for each time period.

21) Sets the binary variable that gives the cost of extra assortments at saw mills.22) For each time period and assortment at a mill the delivered volume must not be

less than the set minimum volume.23) At saw mills a portion of the demand will be returned into the system as wood

chips to be delivered to pulp and paper mills.24) The delivered volume of wood chip at a mill must meet the demand.25) There is a limitation on how much roundwood that can be bought from the

domestic market. This is due to that the domestic market is small but also that it might be impractical to buy from to far away.

26) Even on the open international market there might be a limitation on how much can be bought. This can be due to limitation in transportation capacity and other practical reasons.

27) The volume transported into a rail road terminal is transported out of there by train.

28) There is a limitation on how much roundwood that can be transported by a rail road in each time period. When a rail road system is used a binary variable is set.

29) All continuous variables are equal to or greater than zero.30) All binary variables are 0 or 1.

4.3 Modelling

4.3.1 Scenario properties

4.3.2 Non-considered aspects

5. RESULT FROM MODELLING

5.1 The sample problem

5.2 Solving a large-scaled problem

5.3 The value of the uncertainty measures

6. DISCUSSION AND CONCLUDING REMARKS

6.1 Conclusions from the results


R-03-44Sida 33 (39)

6.2 Do we need a stochastic model?

6.3 Future extensions

7. ACKNOWLEDGEMENTS

FSCN at Mid-Sweden University, Computer Science department at Umeå University, the Kempe foundation and the Gunnar Sundblad research fund have funded this research. We also want to thank numerous people at Holmen Skog for answer forestry related questions, and providing the dataset. Without that dataset we haven’t been able to test our model on a large scaled problem. At last, thanks to Peter Lohmander at the Swedish University of Agriculture Sciences for many useful discussions and advices.

APPENDIX A: ESTIMATION OF COSTS

A.1 HARVEST COSTS AND PURCHASE COSTS FROM SMALL FOREST OWNERS

A.1.1 Harvest costs from the own forest

A.1.2 Stumpage prices and delivered roundwood from small forest owners

The purchase cost is calculated in kr/m3fub as the linear function

C(v)=A+Bv.

Depicted in Figure A.1, where the constants A, B > 0, and

A= Lower bound on the cost.B= The slope calculated such that the highest possible cost is C(vMAX).


R-03-44Sida 34 (39)

Figure A.1: Calculation of the purchase cost C(v)=A+Bv. A and B is estimated from data supplied by SVO, and is of course different for stumpages and delivered roundwood.

The total cost C(v)*v used in the optimization is once more convex and quadratic. Hence, well suited for our model.

A 2. TRANSPORT AND ROAD MAINTENANCE COSTS

A 3. ROUNDWOOD STORAGE COSTS

A 3.1 Storage cost A 3.2 Storage of purchased roundwood


R-03-44Sida 35 (39)

Figure A.2: The estimated value losses when storing roundwood depends, not only on how long time the roundwood is being stored, but also on what type of process it will be used in, storing conditions and so on.

Figure A.2: The estimated value losses when storing roundwood depends on the time, and are rather different depending on many other aspects.

A 4. TOTAL COST FOR IMPORT AND DOMESTIC ROUNDWOOD

Table I and II below summarize the cost we use to buy roundwood at the domestic and the import market. However, an additional cost for quality losses is also added, as described in Appendix A.3.

Table A.1: Import prices on international market 2001.

TT2001ie GT2001ie BM2001ie GM2001ie LM2001ie

1:a kv. 455 425 358 335 3452:a kv 450 421 354 331 3503:e kv 422 394 332 310 3254:e kv. 455 425 358 335 334


R-03-44Sida 36 (39)

Table A.2: Prices on the domestic roundwood market 2001.

TT2001m

GT2001m

BM2001m

GM2001m

LM2001m

1:a kv. 399 376 249 235 2292:a kv 387 361 250 235 2293:e kv 393 365 244 228 2224:e kv. 409 378 231 227 218

REFERENCES

1.Anon. 2002. Lingo - The modelling language and optimizer. Lindo Systems Inc, Chicago, Illinois, USA.

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