MPC MPC R p N p Z p p N (m, σ 2 ) m σ 2 floor( ), rem(v 1 v 2 ) v 1 v 2 r p e u[k] x[k] Σ x[k] Σ 2 L 0 Σ x[k + 1] = x[k] + u[k floor(l/h)] d[k]. Σ k x
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1 MPC Inventory Manegement via Model Predictive Control 1 1 1,2,3 Yoshinobu Matsui 1 Yuhei Umeda 1 Hirokazu Anai 1,2,3 1 1 FUJITSULABORATORIES LTD. 2 2 Kyushu University IMI 3 3 National Institute of Informatics Abstract: At a retail store or wholesales, orderers must order their items adequately For example, there are some situations such that they must order their items so as to reduce their order cost or to equalize the order quantity. In order to handle such problems unitedly, the Model Predictive Control (MPC) framework is effective, because the MPC is an optimal operation method of dynamical systems and the process of stocks can be seen as a dynamical system In this paper, we propose a MPC framework for the inventory management and show its effectiveness by numerical examples. 1 [6, 10, 11] [4] m.yoshinobu@jp.fujitsu.com Model Predictive Control MPC [14] [13] MPC EMS(Energy Management System) [5]. MPC MPC
2 MPC MPC R p N p Z p p N (m, σ 2 ) m σ 2 floor( ), rem(v 1 v 2 ) v 1 v 2 r p e u[k] x[k] Σ x[k] Σ 2 L 0 Σ x[k + 1] = x[k] + u[k floor(l/h)] d[k]. Σ k x r [k] k d p [k] 1 x p [k + 1] x p [k + 1] = x r [k] + u[k] d p [k] k d p [k] x p [k + 1] = x r [k] + u[k] d p [k] 0 u[k] = x r [k] + d p [k] 2: Σ 1: 2 1 t R t h R h k Z L R + x[k] Z + t = kh u[k] Z + t = kh d[k] Z + kh t < (k + 1)h L = 0 Σ Σ x[k + 1] = x[k] + u[k] d[k]. 3 Σ MPC Σ MPC STEPI Σ x r [k]. STEPII x r [k] Σ H u[k],..., u[k + H 1]( 3) STEPIII STEPII STEPIV STEPIII u[k] Σ STEPV k k + 1 STEPI
3 4, 3: H Σ dp : { ξ[k + 1] = F ξ[k] + Gv[k], d p [k] = Hξ[k] + w[k]. Σ H Σ d p [ ] Σ x[ ] d r [ ] d p [ ] d r [ ] x[ ] MPC H = 1 MPC 1 u[k] = x r [k] + d p [k] 1 u[k] x p [k + 1] = x r [k] + u[k] d p [k] 0 Σ dp ξ[k] R p d p [k] Σ dp v[k] R q w[k] R F, G, H d p [ ] Σ dp ξ[1] N (ξ[1], P [1]), k = 1 k N, v[k] N (0, Q), w[k] N (0, R) k N ξ[1] v[k], w[k] P [ ] ξ AR MA ARMA [9] [2, 3, 15] 1 1 H H H 1 STEPi Q, R ξ[1], P [1] [15] k = 1 STEPiii STEPii AIC(Akaike Information Criteria)[1] STEPi STEPi STEPiii d r [k 1] ξ e [k 1] P e [k 1]
4 STEPiv STEPiii 1 STEP [7, 8] STEP1 ξ p [k 1] = F ξ e [k 2] ξ e [1] = ξ[1] STEP2 P p [k 1] = F P e [k 2]F T + GQG T P e [1] = P [1] STEP3 K[k 1] = P p [k 1]H T (HP p [k 1]H T + R) 1 STEP4 ξ e [k 1] = K[k 1](d r[k 1] Hξ p [k 1]) STEP5 P e[k 1] = P p[k 1] K[k 1]HP p[k 1] 1 1 i = 0,..., H 1 d p [k + i] N (µ[k + i], V [k + i]) µ[k + i] = Hξ p [k + i] (1) V [k + i] = HP p [k + i]h T + R (2) ξ p [k + i] = F i+1 ξ e [k 1] (3) P p [k + i] = F (i+1) P e [k 1]F (i+1)t i + F j 1 GQG T F (j 1)T (4) j=1 (1) (2) d p [k + i] ξ p [k + i] w[k +i] (3) (4) ξ p [k + i] = F ξ p [k +i 1] P p [k +i] = F P p [k +i 1]F T +GQG T [7] 5.1 J k f k (u[k]) J = k+h 1 i=k f i (u[i]) k k R( ) R(u[k]) := floor(u[k]/lot) (lot nr) pr +rem(u[k], lot) pr. lot nr pr 1 R( ) 1 pr lot nr J c J c = k+h 1 i=k R(u[i]) J e J e = k+h 1 i=k u[i] u[i 1] 1 l 2 l a[k] P (a[k]) P (a[k]) := 1 a[k]σ 2πσ exp ( ) (y m)2 2σ 2 dy. a[k] = 2 P (a[k]) = , 1
5 4: P (a[k]) H 2 2 P (a[k]),..., P (a[k+ H 1]) k k + H 1 x r [k] + u[k] (µ[k] + a[k] V [k]) 0, x p [k+1]+u[k+1] (µ[k+1]+a[k+1] V [k + 1]) 0,. x p [k + H 1] + u[k + H 1] (µ[k + H 1] + a[k + H 1] V [k + H 1]) 0. xc(a[k],..., a[k+h 1]) J c ( ) J c {P (a[i])} i R(u[i]) (5) J cm = k+h 1 i=k R(u[i]) {P (a[i])} i. (5) J cm J c J cm J c M J cm J c jc J c J cm + M. d p [k],..., d p [k + H 1] x p [k + 1],..., x p [k + H] u[k],..., u[k + H 1] 5 6: 5.3 5: x p [ ] { k P (a[k]) ( ) : k + 1 P (a[k + 1]) J cm J e xc(a[k],..., a[k + H 1]) jc
6 MPC AR 7 1 AR AR 2 J cm J c [12] MILP(Mixed Intejar Linear Programming) 3 z J cm, J e 3 z [0, 1] J (1 z) cm max{j cm } + z J e max{j e } xc(a[k],..., a[k + H 1]), uc jc a[k] = = a[k + H 1] = u[k] x r [k] + u[k] (µ[k] + 2 V [k]) : 6.2 R( ) lot= 500 nr= 50 pr= 460 uc k, 0 u[k] max{j c } max{j cm } max{j e } M = max{j c J cm } z = 0 H = 1 8 MPC 283, 125, , 540, 320 MPC z = 0 H = 1 414, 920 MPC 500 MPC z = 0 H = 4 9 1, 236, Intel (R) Core (TM) i5-2520m CPU 2.5 GHz and 4.0 GByte memory MATLAB MILP intlinprog 10.1
7 6.4 発注平滑化とのトレードオフ MPC が発注コストに対して効果があることが示さ れたが 図 9 を見ると発注量は各期において大きく上 下変動している これにより 商品納品時の作業負担 や生産もとへの生産量安定の観点からは適切な在庫管 理ができているとは言えない そこでパラメータ z を 変えることで できるだけ発注量の上下変動を抑える 平滑化する ことを試みる まず 平滑化のみを考え たとき すなわち z = 1 H = 4 としたときの各発注 量と在庫の様子を図 10 に示す 発注量を完全に平滑 化しているが在庫が大きく増えている さらに差額は +10, 399, 680 円であった そこで z = 0.29 H = 4 図 8: z = 0 H = 1 のときの発注量と在庫量の推移 図 10: z = 1 H = 4 のときの発注量と在庫量の推移 図 9: z = 0 H = 4 のときの発注量と在庫量の推移 とした このときの各発注量と在庫の様子を図 11 に示 す 発注量はそれなりに平滑化されており また 差 額も 1, 123, 780 円であったので 発注コスト 平滑 化の両観点からよい在庫管理ができている また 在 庫量の推移を見てみると MPC では在庫量が一時的に 増え また安全在庫量の水準まで在庫量を減らすとい う在庫管理をしていることが分かる これは 逆にい うと MPC は一時的に在庫を増やすことにより発注量 の平滑化を行い また 全体的に在庫量を減らすこと により発注コストの低減を行っているといえる この ような在庫管理も MPC の長期予測なくして不可能で ある
8 [1] Akaike, H, Information theory and an extension of the maximum likelihood principle, Proceedings of the 2nd International Symposium on Information pp [2] I, [3] II, [4], [5],,,, PC,, vol. 49, no. 2, [6], : z = 0.29 H = 4 7 MPC MPC MPC [7], [8], [9], [10],, [11],,, [12] vol. 57 no. 4 pp [13] S. Bose, J.F. Pekny, A model predictive framework for planning and scheduling problems: a case study of consumer goods supply chain, Computers and Chemical Engineering, vol. 24, pp , [14] J. M. Maciejowski [15],,,,, vol. 49, no. 5, pp , 2004.
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