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1 Stochastic Dynamic Optimization of Forest Industry Company Management INFORMS International Meeting 2007 Puerto Rico Peter Lohmander Professor SLU Umea,

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Presentation on theme: "1 Stochastic Dynamic Optimization of Forest Industry Company Management INFORMS International Meeting 2007 Puerto Rico Peter Lohmander Professor SLU Umea,"— Presentation transcript:

1 1 Stochastic Dynamic Optimization of Forest Industry Company Management INFORMS International Meeting 2007 Puerto Rico Peter Lohmander Professor SLU Umea, SE , Sweden, Version

2 2 Abstract Forest industry production, capacity and harvest levels are optimized. Adaptive full system optimization is necessary for consistent results. The stochastic dynamic programming problem of a complete forest industry company is solved. The raw material stock level and the main product prices are state variables. In each state and at each stage, a linear programming profit maximization problem of the forest company is solved. Parameters from the Swedish forest industry are used as illustration.

3 3 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

4 4 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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42 42 Optimization of the variable profit during a year: Linear programming code Background:

43 43 ! 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;

44 44 !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;

45 45 !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

46 46 Optimization of the variable profit during a year: Optimal results

47 47 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

48 48 TPKOSTTI ETIMMER TPKOSTMA EMAV CSV PRODSV CLINER PRODLIN PSV PLINER PSTI STIMMER PSMAV SMAV PSFLIS SFLIS AVVKAP TIMANDEL SVKAP

49 49 TTIMMER PRODFL PRODSP EFLIS LINERKAP TRETP TFIBER

50 50 Row Slack or Surplus Dual Price LRETP KAPAVV KAPSV FFIBERK E KAPLINER

51 51 ”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.

52 52 ”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?

53 53 Optimal variable profit (KSEK/Year) as a function of * stock level * harvest level * price of external pulpwood (PWP) * price of imported waste paper (IWPP):

54 54 s=1s= m PWP (SEK/ton) hm IWPP (SEK/ton)

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

56 56 s= m PWP (SEK/ ton) hm IWPP (SEK/ ton)

57 57 Optimization of the expected present value of the company during over an infinite horizon: Linear programming in a Markov chain

58 58 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).

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61 61 Min Z = s.t.

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64 64 Min Z = s.t.

65 65 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

66 66 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) + )));

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

68 68 data: g =

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70 ; p1 = ; p2 = ; enddata end

71 71 Global optimal solution found at step: 94 Objective value: E+09

72 72 Variable Value Reduced Cost B Z E

73 73 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

74 74 F( 1, 5) E

75 75 F( 1, 5) E F( 2, 5) E F( 3, 5) E

76 76 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)

77 77 DEC( 1, 5, 1) DEC( 1, 5, 2) DEC( 1, 5, 3)

78 78 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)

79 79 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)

80 80 Optimal harvest levels in the forest district when r = 2%, 5% or 10%

81 81 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%

82 82 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

83 83 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

84 84 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.

85 85 Let’s study the optimal decisions with changing pulp wood prices if the rate of interest is 10%!

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87 87 Optimal harvest levels in the forest district when r = 15% (or higher)

88 88 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

89 89 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

90 90 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

91 91 Optimal harvest levels in the forest district when r = 15% (or higher): Harvest as much as possible for all possible pulp wood prices!

92 92 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..

93 93 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.

94 94 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.

95 95 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.

96 96 Abstract Forest industry production, capacity and harvest levels are optimized. Adaptive full system optimization is necessary for consistent results. The stochastic dynamic programming problem of a complete forest industry company is solved. The raw material stock level and the main product prices are state variables. In each state and at each stage, a linear programming profit maximization problem of the forest company is solved. Parameters from the Swedish forest industry are used as illustration.

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98 98 Contact: Peter Lohmander Professor SLU, Dept. of Forest Economics SE Umea, Sweden Personal home page:


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