1 Optimal activities over time in a typical forest industry company in the presence of stochastic markets - A flexible approach and possible extensions! Peter Lohmander

2 Question How should these activities in a typical forest industry company be optimized and coordinated in the presence of stochastic markets? *Pulp, paper and liner production and sales, *Sawn wood production and sales, *Raw material procurement and sales, *Harvest operations *Transport

3 Approach in three stages A typical forest industry company is defined using real mills and forest conditions in the North of Sweden. For each year (or other period) and possible price and stock state, the variable company profit is maximized using linear programming. (Quadratic programming etc. are other options.) The expected present value of the company over an infinite horizon is maximized via stochastic dynamic programming in Markov chains. In this stage, a standard LP solver is used.

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41 Optimization of the variable profit during a year: Linear programming code Background:

42 ! SMB2; ! Peter Lohmander ; Max = TProf; TProf = - InkK - IntKostn + ForsI; InkK = PKTi*KTimmer + PKMav*KMav + PKFlis*KFlis + PReturpL*KReturpl + PReturpI*KReturpI; IntKostn = AvvK*Avv + TPKostTI*ETimmer + TPKostMA*EMav + CSV*ProdSV + CLiner*ProdLin; ForsI = PSV*ProdSV + PLiner*ProdLin + PSTi*STimmer + PSMav*SMav + PSFlis*SFlis; !Market prices of raw material and raw material constraints; PKTi = 380; PSTi = 330; PKMav = 200; PSMav = 120; PKFlis = 250; PSFlis = 150; PReturpL = 50; PReturpI = 730; [LRetP] KReturpL <= 100;

43 !SMBs forest and harvesting; AvvK = 70; AvvKap = 570; TimAndel =.5; [KapAvv] Avv <= AvvKap; !Roundwood transport costs; TPKostTI = 60; TPKostMa = 70; !SMBs saw mill; PSV = 1500; CSV = 300; SVKap = 80; TTimmer = ETimmer + KTimmer; ProdSV =.5*TTimmer; ProdFl =.8*ProdSV; ProdSp =.2*ProdSV; [KapSV] ProdSV <= SVKap;

44 !SMBs raw material balance; EMav = (1-TimAndel)* Avv - SMav; ETimmer = Timandel*Avv - STimmer; EFlis = ProdFl - SFlis; !SMBs liner mill; PLiner = 4900; CLiner = 1200; LinerKap = 400; TRetP = KReturpL + KReturpI; TFiber = EMav + EFlis + KMav + KFlis; ProdLin =.25*TFiber +.95*TRetP; [FFiberK] TFiber/TRetP >= 4; [KapLiner] ProdLin <= LinerKap; end

45 Optimization of the variable profit during a year: Optimal results

46 Local optimal solution found at step: 10 Objective value: Variable Value Reduced Cost TPROF INKK INTKOSTN FORSI PKTI KTIMMER PKMAV KMAV PKFLIS KFLIS PRETURPL KRETURPL PRETURPI KRETURPI AVVK AVV

47 TPKOSTTI ETIMMER TPKOSTMA EMAV CSV PRODSV CLINER PRODLIN PSV PLINER PSTI STIMMER PSMAV SMAV PSFLIS SFLIS AVVKAP TIMANDEL SVKAP

48 TTIMMER PRODFL PRODSP EFLIS LINERKAP TRETP TFIBER

49 Row Slack or Surplus Dual Price LRETP KAPAVV KAPSV FFIBERK E KAPLINER

50 ”Single period results” In the ”single period optimization model”, you should always harvest all available stands and produce at full capacity utilization in the saw mill and the liner mill.

51 ”Stochastic multi period questions”: Maybe you should, under some market conditions, save some harvest areas for future periods? Maybe also the other decisions in the company are different when we consider many periods and stochastic markets?

52 Optimal variable profit (KSEK/Year) as a function of * stock level * harvest level * price of external pulpwood (PWP) * price of imported waste paper (IWPP):

53 s=1s= m PWP (SEK/ton) hm IWPP (SEK/ton)

54 s= m PWP (SEK/ ton) hm IWPP (SEK/ ton)

55 s= m PWP (SEK/ ton) hm IWPP (SEK/ ton)

56 Optimization of the expected present value of the company during over an infinite horizon: Linear programming in a Markov chain

57 The optimization problem at a general level We want to maximize the expected present value of the profit, all revenues minus costs, over an infinite horizon. This is solved via stochastic dynamic programming. Compare Howard (1960), Wagner (1975), Ross (1983) and Winston (2004).

58 Min Z = s.t.

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61 Min Z = s.t.

62 Optimal dynamic analysis r = 5% ! Puerto_ _2010 HRS; ! Peter Lohmander; Model: sets: stock/1..3/:; market/1..9/:p1,p2; sm(stock,market):f; mark2(market,market):TR; harv/1..5/:; shm(stock,harv,market):g; endsets

63 b = 1/1.05; Z sm(i,j):f(i,j)); min @for(harv(h)| (s+3-h) #GE#1 #AND# (s+3-h) #LE# 3 : [Dec] f(s,m) >= g(s,h,m) + )));

n #NE# 5 : TR(m,n) n #EQ# 5 : TR(m,n) =.6);

65 data: g =

; p1 = ; p2 = ; enddata end

68 Global optimal solution found at step: 94 Objective value: E+09

69 Variable Value Reduced Cost B Z E

70 F( 1, 1) E F( 1, 2) E F( 1, 3) E F( 1, 4) E F( 1, 5) E F( 1, 6) E F( 1, 7) E F( 1, 8) E F( 1, 9) E F( 2, 1) E F( 2, 2) E F( 2, 3) E F( 2, 4) E F( 2, 5) E F( 2, 6) E F( 2, 7) E F( 2, 8) E F( 2, 9) E F( 3, 1) E F( 3, 2) E F( 3, 3) E F( 3, 4) E F( 3, 5) E F( 3, 6) E F( 3, 7) E F( 3, 8) E F( 3, 9) E

71 F( 1, 5) E

72 F( 1, 5) E F( 2, 5) E F( 3, 5) E

73 DEC( 1, 1, 1) DEC( 1, 1, 2) DEC( 1, 1, 3) DEC( 1, 2, 1) DEC( 1, 2, 2) DEC( 1, 2, 3) DEC( 1, 3, 1) DEC( 1, 3, 2) DEC( 1, 3, 3) DEC( 1, 4, 1) DEC( 1, 4, 2) DEC( 1, 4, 3) DEC( 1, 5, 1) DEC( 1, 5, 2) DEC( 1, 5, 3) DEC( 1, 6, 1) DEC( 1, 6, 2) DEC( 1, 6, 3) DEC( 1, 7, 1) DEC( 1, 7, 2) DEC( 1, 7, 3) DEC( 1, 8, 1) DEC( 1, 8, 2) DEC( 1, 8, 3) DEC( 1, 9, 1) DEC( 1, 9, 2) DEC( 1, 9, 3)

74 DEC( 1, 5, 1) DEC( 1, 5, 2) DEC( 1, 5, 3)

75 DEC( 2, 1, 2) DEC( 2, 1, 3) DEC( 2, 1, 4) DEC( 2, 2, 2) DEC( 2, 2, 3) DEC( 2, 2, 4) DEC( 2, 3, 2) DEC( 2, 3, 3) DEC( 2, 3, 4) DEC( 2, 4, 2) DEC( 2, 4, 3) DEC( 2, 4, 4) DEC( 2, 5, 2) DEC( 2, 5, 3) DEC( 2, 5, 4) DEC( 2, 6, 2) DEC( 2, 6, 3) DEC( 2, 6, 4) DEC( 2, 7, 2) DEC( 2, 7, 3) DEC( 2, 7, 4) DEC( 2, 8, 2) DEC( 2, 8, 3) DEC( 2, 8, 4) DEC( 2, 9, 2) DEC( 2, 9, 3) DEC( 2, 9, 4)

76 DEC( 3, 1, 3) DEC( 3, 1, 4) DEC( 3, 1, 5) DEC( 3, 2, 3) DEC( 3, 2, 4) DEC( 3, 2, 5) DEC( 3, 3, 3) DEC( 3, 3, 4) DEC( 3, 3, 5) DEC( 3, 4, 3) DEC( 3, 4, 4) DEC( 3, 4, 5) DEC( 3, 5, 3) DEC( 3, 5, 4) DEC( 3, 5, 5) DEC( 3, 6, 3) DEC( 3, 6, 4) DEC( 3, 6, 5) DEC( 3, 7, 3) DEC( 3, 7, 4) DEC( 3, 7, 5) DEC( 3, 8, 3) DEC( 3, 8, 4) DEC( 3, 8, 5) DEC( 3, 9, 3) DEC( 3, 9, 4) DEC( 3, 9, 5)

77 Optimal harvest levels in the forest district when r = 2%, 5% or 10%

78 S = Entering stock level S=1 ( m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = m m3 IWPP = m m3 IWPP = m m3 Optimal harvest levels in the forest district when r = 2%, 5% or 10%

79 Optimal harvest levels in the forest district when r = 2%, 5% or 10% S=2 ( m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = m m3 IWPP = m m3 IWPP = m m3

80 Optimal harvest levels in the forest district when r = 2%, 5% or 10% S=3 ( m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = m m3 IWPP = m m3 IWPP = m m3

81 Optimal harvest levels in the forest district when r = 2%, 5% or 10%: Harvest approx cubic metres less than the maximum possible in case the pulp wood price is not at the highest level. Harvest as much as possible if the pulp wood price is at the highest level.

82 Let’s study the optimal decisions with changing pulp wood prices if the rate of interest is 10%!

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84 Optimal harvest levels in the forest district when r = 15% (or higher)

85 S = Entering stock level S=1 ( m3 or less may be harvested this year) Optimal harvest levels in the forest district when r = 15% PWP = 160PWP = 200PWP = 240 IWPP = m3 IWPP = m3 IWPP = m3

86 Optimal harvest levels in the forest district when r = 15% S=2 ( m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = m3 IWPP = m3 IWPP = m3

87 Optimal harvest levels in the forest district when r = 15% S=3 ( m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = m3 IWPP = m3 IWPP = m3

88 Optimal harvest levels in the forest district when r = 15% (or higher): Harvest as much as possible for all possible pulp wood prices!

89 General results With this approach, all relevant decisions in the company can be consistently optimized. For instance, we find how the optimal harvest level is affected by the present price in the pulpwood market, the transition probability matrix of prices, the rate of interest, the volume of harvestable stands and all other company relevant conditions such as capacities in the sawmill, the liner mill, transport costs for different assortements on different roads etc..

90 Observation #1 All the sub problems (the optimization problems within each time period) may be solved with continuous or discrete variables via linear programming, quadratic programming or some other optimization method, taking all relevant constraints into consideration.

91 Observation #2 In the ”master problem” (the Markov chain problem over an infinite horizon), the state space is discrete. With standard software, this still makes it possible to use high resolution in the interesting dimensions. For instance, with ten possible stock levels, we may use four exogenous market prices (with ten possible levels in each dimension) and still have no more than variables. Such a problem can easily be solved.

92 Observation #3 This forest sector model can easily be modified and we can instantly calculate how the expected economic value of the company and the optimal decisions change. For instance, we may introduce ”possible” bioenergy power plants and new types of pulp and paper mills and instantly derive the optimal results.

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94 Contact: Peter Lohmander Professor SLU, Dept. of Forest Economics SE Umea, Sweden Personal home page: