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

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1 Stochastic Dynamic Optimization of Forest Industry Company Management INFORMS International Meeting 2007 Puerto Rico Peter Lohmander Professor SLU Umea, SE-901 83, Sweden, http://www.Lohmander.com Version 2007-06-21

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 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 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 Optimization of the variable profit during a year: Linear programming code Background: http://www.lohmander.com/SkogIndEk1/SI1.html http://www.lohmander.com/SkogIndEk1/SI1.html

43 ! SMB2; ! Peter Lohmander 2003-10-15; 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 !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 !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 Optimization of the variable profit during a year: Optimal results

47 Local optimal solution found at step: 10 Objective value: 1373354. Variable Value Reduced Cost TPROF 1373354. 0.0000000 INKK 236846.2 0.0000000 INTKOSTN 563850.0 0.0000000 FORSI 2174050. 0.0000000 PKTI 380.0000 0.0000000 KTIMMER 160.0000 0.0000000 PKMAV 200.0000 0.0000000 KMAV 471.5128 0.0000000 PKFLIS 250.0000 0.0000000 KFLIS 0.0000000 50.00000 PRETURPL 50.00000 0.0000000 KRETURPL 100.0000 0.0000000 PRETURPI 730.0000 0.0000000 KRETURPI 105.1282 0.0000000 AVVK 70.00000 0.0000000 AVV 570.0000 0.0000000

48 TPKOSTTI 60.00000 0.0000000 ETIMMER 0.0000000 10.00000 TPKOSTMA 70.00000 0.0000000 EMAV 285.0000 0.0000000 CSV 300.0000 0.0000000 PRODSV 80.00000 0.0000000 CLINER 1200.000 0.0000000 PRODLIN 400.0000 0.0000000 PSV 1500.000 0.0000000 PLINER 4900.000 0.0000000 PSTI 330.0000 0.0000000 STIMMER 285.0000 0.0000000 PSMAV 120.0000 0.0000000 SMAV 0.0000000 10.00000 PSFLIS 150.0000 0.0000000 SFLIS 0.0000000 50.00000 AVVKAP 570.0000 0.0000000 TIMANDEL 0.5000000 0.0000000 SVKAP 80.00000 0.0000000

49 TTIMMER 160.0000 0.0000000 PRODFL 64.00000 0.0000000 PRODSP 16.00000 0.0000000 EFLIS 64.00000 0.0000000 LINERKAP 400.0000 0.0000000 TRETP 205.1282 0.0000000 TFIBER 820.5128 0.0000000

50 Row Slack or Surplus Dual Price LRETP 0.0000000 680.0000 KAPAVV 0.0000000 160.0000 KAPSV 0.0000000 600.0000 FFIBERK 0.6043397E-09 -788.9547 KAPLINER 0.0000000 2915.385

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 ”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 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 s=1s=1 123456789m 160200240160200240160200240 PWP (SEK/ton) hm3630 730 830 IWPP (SEK/ton) 1470 139809013678671347006139809013573541336493139809013542001325981 2520 140584013758671356006140584013653541345493140584013622001334981 3570 141359013838671365006141359013733541354493141359013702001343981 4620 5670

55 s= 2 123456789m 160200240160200240160200240 PWP (SEK/ ton) hm3630 730 830 IWPP (SEK/ ton) 1471 2521 140599513760271356186140599513655141345673140599513623601335161 3571 141374513840271365186141374513735141354673141374513703601344161 4621 142149513920271374186142149513815141363673142149513783601363673 5671

56 s=3 123456789m 160200240160200240160200240 PWP (SEK/ ton) hm3630 730 830 IWPP (SEK/ ton) 1472 2522 3572 141390013841871365366141390013736741354853141390013705201344341 4622 142165013921871374366142165013816741363853142165013785201353341 5672 142940014001871383366142940013896741372853142940013865201362341

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

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 Min Z = s.t.

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64 Min Z = s.t.

65 Optimal dynamic analysis r = 5% ! Puerto_20070516_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 b = 1/1.05; Z = @sum( sm(i,j):f(i,j)); min = Z; @for(stock(s): @for(market(m): @for(harv(h)| (s+3-h) #GE#1 #AND# (s+3-h) #LE# 3 : [Dec] f(s,m) >= g(s,h,m) + b*@sum(market(n):TR(m,n)*f(s+3-h,n)) )));

67 @for(mark2(m,n)| n #NE# 5 : TR(m,n) =.05); @for(mark2(m,n)| n #EQ# 5 : TR(m,n) =.6);

68 data: g = 1398090 1367867 1347006 1398090 1357354 1336493 1398090 1354200 1325981 1405840 1375867 1356006 1405840 1365354 1345493 1405840 1362200 1334981 1413590 1383867 1365006 1413590 1373354 1354493 1413590 1370200 1343981 0 0 0 0 0 0 0 0 0

69 0 0 0 0 0 0 0 0 0 1405995 1376027 1356186 1405995 1365514 1345673 1405995 1362360 1335161 1413745 1384027 1365186 1413745 1373514 1354673 1413745 1370360 1344161 1421495 1392027 1374186 1421495 1381514 1363673 1421495 1378360 1363673 0 0 0 0 0 0 0 0 0

70 0 0 0 0 0 0 0 0 0 1413900 1384187 1365366 1413900 1373674 1354853 1413900 1370520 1344341 1421650 1392187 1374366 1421650 1381674 1363853 1421650 1378520 1353341 1429400 1400187 1383366 1429400 1389674 1372853 1429400 1386520 1362341 ; p1 = 1 2 3 1 2 3 1 2 3; p2 = 1 1 1 2 2 2 3 3 3; enddata end

71 Global optimal solution found at step: 94 Objective value: 0.7812784E+09

72 Variable Value Reduced Cost B 0.9523810 0.0000000 Z 0.7812784E+09 0.0000000

73 F( 1, 1) 0.2896002E+08 0.0000000 F( 1, 2) 0.2893005E+08 0.0000000 F( 1, 3) 0.2891080E+08 0.0000000 F( 1, 4) 0.2896002E+08 0.0000000 F( 1, 5) 0.2891953E+08 0.0000000 F( 1, 6) 0.2890029E+08 0.0000000 F( 1, 7) 0.2896002E+08 0.0000000 F( 1, 8) 0.2891638E+08 0.0000000 F( 1, 9) 0.2888978E+08 0.0000000 F( 2, 1) 0.2896792E+08 0.0000000 F( 2, 2) 0.2893821E+08 0.0000000 F( 2, 3) 0.2891998E+08 0.0000000 F( 2, 4) 0.2896792E+08 0.0000000 F( 2, 5) 0.2892769E+08 0.0000000 F( 2, 6) 0.2890947E+08 0.0000000 F( 2, 7) 0.2896792E+08 0.0000000 F( 2, 8) 0.2892454E+08 0.0000000 F( 2, 9) 0.2890947E+08 0.0000000 F( 3, 1) 0.2897583E+08 0.0000000 F( 3, 2) 0.2894637E+08 0.0000000 F( 3, 3) 0.2892916E+08 0.0000000 F( 3, 4) 0.2897583E+08 0.0000000 F( 3, 5) 0.2893585E+08 0.0000000 F( 3, 6) 0.2891865E+08 0.0000000 F( 3, 7) 0.2897583E+08 0.0000000 F( 3, 8) 0.2893270E+08 0.0000000 F( 3, 9) 0.2890814E+08 0.0000000

74 F( 1, 5) 0.2891953E+08 0.0000000

75 F( 1, 5) 0.2891953E+08 0.0000000 F( 2, 5) 0.2892769E+08 0.0000000 F( 3, 5) 0.2893585E+08 0.0000000

76 DEC( 1, 1, 1) 369.8571 0.0000000 DEC( 1, 1, 2) 0.0000000 -5.285714 DEC( 1, 1, 3) 631.2857 0.0000000 DEC( 1, 2, 1) 619.8571 0.0000000 DEC( 1, 2, 2) 0.0000000 -5.285714 DEC( 1, 2, 3) 381.2857 0.0000000 DEC( 1, 3, 1) 2238.571 0.0000000 DEC( 1, 3, 2) 618.7143 0.0000000 DEC( 1, 3, 3) 0.0000000 -5.285714 DEC( 1, 4, 1) 369.8571 0.0000000 DEC( 1, 4, 2) 0.0000000 -5.285714 DEC( 1, 4, 3) 631.2857 0.0000000 DEC( 1, 5, 1) 619.8571 0.0000000 DEC( 1, 5, 2) 0.0000000 -52.42857 DEC( 1, 5, 3) 381.2857 0.0000000 DEC( 1, 6, 1) 2238.571 0.0000000 DEC( 1, 6, 2) 618.7143 0.0000000 DEC( 1, 6, 3) 0.0000000 -5.285714 DEC( 1, 7, 1) 369.8571 0.0000000 DEC( 1, 7, 2) 0.0000000 -5.285714 DEC( 1, 7, 3) 631.2857 0.0000000 DEC( 1, 8, 1) 619.8571 0.0000000 DEC( 1, 8, 2) 0.0000000 -5.285714 DEC( 1, 8, 3) 381.2857 0.0000000 DEC( 1, 9, 1) 2238.571 0.0000000 DEC( 1, 9, 2) 618.7143 0.0000000 DEC( 1, 9, 3) 0.0000000 -5.285714

77 DEC( 1, 5, 1) 619.8571 0.0000000 DEC( 1, 5, 2) 0.0000000 -52.42857 DEC( 1, 5, 3) 381.2857 0.0000000

78 DEC( 2, 1, 2) 369.8571 0.0000000 DEC( 2, 1, 3) 0.0000000 -23.71428 DEC( 2, 1, 4) 631.2857 0.0000000 DEC( 2, 2, 2) 619.8571 0.0000000 DEC( 2, 2, 3) 0.0000000 -23.71428 DEC( 2, 2, 4) 381.2857 0.0000000 DEC( 2, 3, 2) 2238.571 0.0000000 DEC( 2, 3, 3) 618.7143 0.0000000 DEC( 2, 3, 4) 0.0000000 -23.71428 DEC( 2, 4, 2) 369.8571 0.0000000 DEC( 2, 4, 3) 0.0000000 -23.71428 DEC( 2, 4, 4) 631.2857 0.0000000 DEC( 2, 5, 2) 619.8571 0.0000000 DEC( 2, 5, 3) 0.0000000 -273.5714 DEC( 2, 5, 4) 381.2857 0.0000000 DEC( 2, 6, 2) 2238.571 0.0000000 DEC( 2, 6, 3) 618.7143 0.0000000 DEC( 2, 6, 4) 0.0000000 -23.71428 DEC( 2, 7, 2) 369.8571 0.0000000 DEC( 2, 7, 3) 0.0000000 -23.71428 DEC( 2, 7, 4) 631.2857 0.0000000 DEC( 2, 8, 2) 619.8571 0.0000000 DEC( 2, 8, 3) 0.0000000 -23.71428 DEC( 2, 8, 4) 381.2857 0.0000000 DEC( 2, 9, 2) 12750.57 0.0000000 DEC( 2, 9, 3) 11130.71 0.0000000 DEC( 2, 9, 4) 0.0000000 -23.71428

79 DEC( 3, 1, 3) 369.8571 0.0000000 DEC( 3, 1, 4) 0.0000000 -1.000000 DEC( 3, 1, 5) 631.2857 0.0000000 DEC( 3, 2, 3) 619.8571 0.0000000 DEC( 3, 2, 4) 0.0000000 -1.000000 DEC( 3, 2, 5) 381.2857 0.0000000 DEC( 3, 3, 3) 2238.571 0.0000000 DEC( 3, 3, 4) 618.7143 0.0000000 DEC( 3, 3, 5) 0.0000000 -1.000000 DEC( 3, 4, 3) 369.8571 0.0000000 DEC( 3, 4, 4) 0.0000000 -1.000000 DEC( 3, 4, 5) 631.2857 0.0000000 DEC( 3, 5, 3) 619.8571 0.0000000 DEC( 3, 5, 4) 0.0000000 -1.000000 DEC( 3, 5, 5) 381.2857 0.0000000 DEC( 3, 6, 3) 2238.571 0.0000000 DEC( 3, 6, 4) 618.7143 0.0000000 DEC( 3, 6, 5) 0.0000000 -1.000000 DEC( 3, 7, 3) 369.8571 0.0000000 DEC( 3, 7, 4) 0.0000000 -1.000000 DEC( 3, 7, 5) 631.2857 0.0000000 DEC( 3, 8, 3) 619.8571 0.0000000 DEC( 3, 8, 4) 0.0000000 -1.000000 DEC( 3, 8, 5) 381.2857 0.0000000 DEC( 3, 9, 3) 2238.571 0.0000000 DEC( 3, 9, 4) 618.7143 0.0000000 DEC( 3, 9, 5) 0.0000000 -1.000000

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

81 S = Entering stock level S=1 (570 000 m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = 830520 000 m3 570 000 m3 IWPP = 730520 000 m3 570 000 m3 IWPP = 630520 000 m3 570 000 m3 Optimal harvest levels in the forest district when r = 2%, 5% or 10%

82 Optimal harvest levels in the forest district when r = 2%, 5% or 10% S=2 (621 000 m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = 830571 000 m3 621 000 m3 IWPP = 730571 000 m3 621 000 m3 IWPP = 630571 000 m3 621 000 m3

83 Optimal harvest levels in the forest district when r = 2%, 5% or 10% S=3 (672 000 m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = 830622 000 m3 672 000 m3 IWPP = 730622 000 m3 672 000 m3 IWPP = 630622 000 m3 672 000 m3

84 Optimal harvest levels in the forest district when r = 2%, 5% or 10%: Harvest approx. 50 000 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 Let’s study the optimal decisions with changing pulp wood prices if the rate of interest is 10%!

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

88 S = Entering stock level S=1 (570 000 m3 or less may be harvested this year) Optimal harvest levels in the forest district when r = 15% PWP = 160PWP = 200PWP = 240 IWPP = 830570 000 m3 IWPP = 730570 000 m3 IWPP = 630570 000 m3

89 Optimal harvest levels in the forest district when r = 15% S=2 (621 000 m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = 830621 000 m3 IWPP = 730621 000 m3 IWPP = 630621 000 m3

90 Optimal harvest levels in the forest district when r = 15% S=3 (672 000 m3 or less may be harvested this year) PWP = 160PWP = 200PWP = 240 IWPP = 830672 000 m3 IWPP = 730672 000 m3 IWPP = 630672 000 m3

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 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 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 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 100 000 variables. Such a problem can easily be solved.

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 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 Contact: Peter Lohmander Professor SLU, Dept. of Forest Economics SE-901 83 Umea, Sweden e-mail: Peter.Lohmander@sekon.SLU.se Personal home page: http://www.Lohmander.com

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