Dynamic - Cash Flow Based - Inventory Management

INFORMS Applied Probability Society Conference 2013 -Costa Rica Meeting Dynamic - Cash Flow Based - Inventory Management Michael N. Katehakis Rutgers University July 15, 2013 Talk based on joint work with J. Shi, Robinson School of Business, B. Melamed, Rutgers University

Outline The talk is organized as follows. We present a new model in which inventory decisions for a single item under random demand are made taking into account cash flow issues related to sale generated profits as well as borrowing costs to finance purchases. We present the optimal solution for the single period problem. We present the structure of optimal solution for the N period problem.

A Simple Example Suppose you have 10, 000 USD $ to invest in a retailer business.

A Simple Example Suppose you have 10, 000 USD $ to invest in a retailer business. Each period (day, month etc.) you may order some units to satisfy random demand.

A Simple Example Suppose you have 10, 000 USD $ to invest in a retailer business. Each period (day, month etc.) you may order some units to satisfy random demand. If you order less and there is cash left, you may deposit the excess cash to a bank account at an interest i =1%. If you order more and there is no su cient cash, you may borrow from alenderatahigherloanrate:` =7%. Leftover units are carried over to next period subject to holding cost. Excess cash may be deposited to a bank account for interest. Objective: to maximize the expected total net wealth at the end of the horizon (e.g., year).

Notation p: sellingpriceperunit c: ordering cost per unit s: salvage price per unit (disposing cost if negative) h: holding cost per unit

Notation p: sellingpriceperunit c: ordering cost per unit s: salvage price per unit (disposing cost if negative) h: holding cost per unit i : the interest rate for deposits `: the interest rate for a loan, where i<`apple p c 1 D: the single period demand, with pdf f( ) and cdf F ( )

States and Actions at the beginning of a period States:

States and Actions at the beginning of a period States: (x, y): asset-capital state at the beginning of the period x denotes the on-hand inventory level y = Y/c: the capital level in units of the product Net worth: = x + y A negative x represents a backorder quantity A negative y represents a non-negative amount of loan: Y = yc. At the beginning of the period it is possible to purchase y units of the product if the available capital is Y, (where y = Y/c > 0).

Cash Flow Dynamics - Single Period: When q 0 is placed while the state is (x, y), and if the demand during the period is D, then at the end of the period we have: Cash flow from sales: R(D, q, x) = p min(q + x, D)+s [q + x D] + = p(q + x) (p s) [q + x D] + where [z] + is the positive part of z, andtheequalityholdsby min{z, t} = z [t z] +

System Dynamics - Multi Period: For n =1, 2,...,N 1, x n+1 = [x n + q n D n ] + y n+1 = [R n (D n,q n,x n )+K n (D n,q n,y n )]/c n+1 where R n (D n,q n,x n ) = p n (x n + q n ) (p n + h n )[x n + q n D n ] + K n (D n,q n,y n ) = c n (y n q n ) (1 + i n )1 {qnappley n} +(1+`n)1 {qn>y n}

System Dynamics - Multi Period: For n =1, 2,...,N 1, where x n+1 = [x n + q n D n ] + y n+1 = [R n (D n,q n,x n )+K n (D n,q n,y n )]/c n+1 R n (D n,q n,x n ) = p n (x n + q n ) (p n + h n )[x n + q n D n ] + K n (D n,q n,y n ) = c n (y n q n ) (1 + i n )1 {qnappley n} +(1+`n)1 {qn>y n} and at the beginning of period n : (x n,y n), system state q n(x n,y n) the order quantity D n, the demand

System Dynamics - Continued. At the end of period N, the revenue consists of:

System Dynamics - Continued. At the end of period N, the revenue consists of: Inventory sales: R N (D N,q N,x N ) = p N min{x N + q N,D N } h N [x N + q N D N ] + = p N [q N + x N ] (p N s)[q N + x N D N ] + where h N = s,

System Dynamics - Continued. At the end of period N, the revenue consists of: Inventory sales: R N (D N,q N,x N ) = p N min{x N + q N,D N } h N [x N + q N D N ] + where h N = s, Capital form the bank: = p N [q N + x N ] (p N s)[q N + x N D N ] + K N (D N,q N,y N )=c N (y N q N ) (1 + i N )1 {qn appley N } +(1+`N )1 {qn >y N }

Dynamic programming formulation Objective: For a risk-neutral newsvendor, the objective is to maximize the expected value of the total asset at the end of period N: E [R N (D N,q N,x N )+K N (D N,q N,y N )] D.P. Equations: V n (x n,y n )=supe[v n+1 (x n+1,y n+1 ) x n,y n ], n =1, 2,,N 1 q n where the expectation is taken with respect to D n. For the final period N: V N (x N,y N )=sup q N E [R N (D N,q N,x N )+K N (D N,q N,y N )].

Single Period - Main Theorem: Theorem 1 For any given initial state (x, y), theoptimalorderquantityis 8 < ( x) +, x+ y 2 [,1), q (x, y) = y, x + y 2 [, ], : x, x + y 2 ( 1, ], where the critical values of and are: = F 1 p c[1 + `] p s = F 1 p c[1 + i] p s,, F 1 ( ) is the inverse function of F ( ).

Single Period - Main Theorem (Cont ): Theorem 1 For any initial state (x, y), i) V (x, y) =G(q (x, y),x,y) is given by: 8 px (p s)l(x)+cy(1 + i), x > ; >< p (p s)l( )+c(x + y )(1 + i), x apple, apple x + y; V (x, y) = p (x + y) (p s)l(x + y), apple x + y< ; >: p (p s)l( )+c (x + y )(1 + l), x+ y<, where L(x) = R x (x t)f(t)dt; 0

Optimal order quantity when x =0. * 0 q (, y) 0 y Note: (Loan utilization) When x + y< then, y<q (x, y) = x = y +( x y). (Capital Full-utilization) When apple x + y<, q (x, y) =y = Y/c. (Capital Under-utilization) When x + y,itisoptimaltoorderq =( x) +.

Remarks 0 apple i apple ` implies that apple, (since F 1 (z) is increasing in z).

Remarks 0 apple i apple ` implies that apple, (since F 1 (z) is increasing in z). When i = ` =0: p c = = F 1. p s This is the classical Newsvendor model

Remarks 0 apple i apple ` implies that apple, (since F 1 (z) is increasing in z). When i = ` =0: p c = = F 1. p s This is the classical Newsvendor model The critical value p c[1 + i] = F 1 p s is the critical value for the classical Newsvendor problem in which no loan is involved, but the unit price c(1 + i) reflects the opportunity cost of cash not invested in the bank at interest i; the case Y = 1 of our model. The critical value p c[1 + `] = F 1 p s is the critical value for the classical Newsvendor problem when all units are purchased by a loan at an interest `; the case Y =0of our model.

Outline of proof. Use concavity and monotonicity properties of the single period expected value of total assets G(q, x, y) = E[ R(D, q, x)] +K(q, y) Z q+x = p(q + x) (p s) (q + x z)f(z)dz 0 +c (y q) (1 + i)1 {qappley} +(1+`)1 {q>y} Gq,x,y q ( ) Gq,x,y q ( ) Gq,x,y q ( ) 0 x y x+ q 0 x y x+ q 0 x y x+ q (a) x y (b) x y (c) x y the derivative of G(q, x, y) with respect to three cases for the values of x + y

Multi Period: Theorem 2 The ( n, n) orderingpolicy. For given system state (x n,y n) at the beginning of period n =1, 2,,N,there exist constants n = n(x n,y n) 0 and n = n(x n,y n) 0 with n apple n: 8 < q (x n,y n)= : ( n x n) +, x n + y n n; y n, n apple x n + y n < n; n x n, x n + y n < n

Theorem 2 - Continued Further, n is the unique value of q n 0 for which the equation below holds: apple @Vn+1 E (p 0 n + h 0 @x n) @V n+1 1 {xn+q n+1 @y n>d n} = apple c 0 n (1 + in) @Vn+1 p0 n E, n+1 @y n+1 and n is the unique value of q n 0 for which the equation below holds: apple @Vn+1 E (p 0 n @x + h0 n ) @V n+1 1 {xn+q n+1 @y n>d n} = apple c 0 n (1 + `n) @Vn+1 p0 n E n+1 @y n+1 where the expectations are taken with respect to D n conditionally on (x n,y n), p 0 n = pn/c n+1, h 0 n = hn/c n+1 and c 0 n = cn/c n+1.

Approximation With a Myopic Optimal Policy Myopic Policy (I) ˆ n, ˆn : We construct the following modified salvage value cost structure: hn, n < N, ŝ n = s, n = N, and the corresponding critical values given by ˆ n = Fn 1 pn ˆn = F 1 n c n [1 + `n] ; p n ŝ n pn c n [1 + i n ]. One can interpret the new salvage values ŝ n as representing a fictitious additional holding cost for leftover inventory in period n that is carried over to period n +1. p n ŝ n

Approximation With a Myopic Optimal Policy - (Cont ) Myopic Policy (II) : n, n : We construct the following modified salvage value cost structure: s n = cn+1 h n, n < N; s, n = N, and the corresponding critical values given by n = Fn 1 pn n = F 1 n c n [1 + `n] ; p n s n pn c n [1 + i n ]. One can interpret the new salvage values s n as representing a fictitious income from inventory liquidation (or pre-salvage at full current cost), at the beginning of the next period n +1. p n s n

Myopic Upper & Lower Bounds Theorem 3 The following are true:

Myopic Upper & Lower Bounds Theorem 3 The following are true: i) For the last period N, N =ˆ N = N and N = ˆN = N.

Myopic Upper & Lower Bounds Theorem 3 The following are true: i) For the last period N, N =ˆ N = N and N = ˆN = N. ii) For any period n =1, 2,...N 1, ˆ n apple n apple n, ˆn apple n apple n.

Remarks on Numerical Studies for the Single Period Model Amajorreasonforthetwothresholdvalues and rates, i and `. is the two distinct financial It is of interest to see how sensitive of the variation between the two threshold values with respect to the di erence between i and `. We did experiment the single period model with Uniform demand distribution of D U(0, 100) and Exponential demand distribution of D Exp(50). We set the selling price as p =50;costc =20;salvagecostperunits =10. We fix the interest rate as i =2%and change the loan rate ` from 2% to 50%. It shows that the value of does not change with respect to `. For any `, =74.00 for Uniform demand, while =67.35 for Exponential demand.

Numerical Study: Single Period Model Consider D U(0, 100) and D Exp(50). p =50; c =20and s =10. =74.00 for the Uniform, while =67.35 for the Exponential. Fix i =2%and vary ` from 2% to 50%. of Single Period Newsvendor Problem (1) For both demand distributions, is decreasing in `; (2) and of Uniform demand are larger than those of Exponential demand.

Single Period Model (Cont ) / of Single Period Problem

Single Period Model (Cont ) / of Single Period Problem / is not significantly sensitive to `/i. While `/i =25, / =1.48 for the Uniform demand, and / =1.94 for the Exponential demand.

Optimal Threshold Values in Ending Period N =2 70 68 β =66 66 64 Order Up to Level 62 60 58 α =57 56 54 52 50 0 20 40 60 80 100 120 Initial Asset ξ Optimal Threshold Values in Period 2 versus Initial Asset We assume i.i.d. Uninform demand distributions, D U(0, 200), for each period and p =50, c =35, s =10and holding cost h =5. *We fix the interest rate as i =5%and the loan rate ` =10%.

Optimal Threshold Values in the First Period 140 Upper Bound 120 Upper Bound Order Up To Levels 100 80 α (ξ) β (ξ) 60 40 Lower Bound Lower Bound 20 0 20 40 60 80 100 120 140 160 180 Total Asset ξ Optimal Threshold Values in Period 1 versus Initial Initial Asset For ( ), ˆ =41.61, =114.43; For ( ), ˆ =47.94, =131.84. If the net worth 2 (42, 145), thenthefirmwouldorder with all available capital. *NOTE The zigzag shape!

Sensitivity of Loan Rate Table : Sensitivity w.r.t. the Loan Interest Rate ` q1 (0, 0) q 2 (0, 0) V 1(0, 0) V 2 (0, 0) V 1 /V 2 5% 111 66 1097.13 432.30 2.54 6% 114 64 1062.63 409.58 2.59 7% 106 62 1034.16 387.48 2.67 8% 104 61 996.45 366.00 2.72 9% 107 59 965.47 345.15 2.80 10% 110 57 935.08 324.88 2.88 11% 110 55 902.47 305.23 2.96 12% 107 54 872.85 286.20 3.05 13% 111 52 837.24 267.80 3.13 14% 104 50 808.67 249.98 3.23 15% 108 49 775.81 232.77 3.33 16% 108 47 744.96 216.20 3.45 17% 105 45 712.47 200.25 3.56 18% 109 43 682.37 184.87 3.69 19% 109 42 652.67 170.12 3.84 20% 105 40 624.45 156.00 4.00 (1) q in the ending period is relatively sensitive to the loan rate `, anditisdecreasingin`; (2) the ( n, n)-policy yields a positive value,and V 1 and V 2 are both decreasing in `. (3) The value of time (V 1 /V 2 )atasmallloanrateissmallerthanthatatalargeloanrate. The initial states x n =0and y n =0in each period n =1, 2.

Conclusions We studied a new model of cash-flow based inventory/financial single-item inventory system.

Conclusions We studied a new model of cash-flow based inventory/financial single-item inventory system. The optimal order policy for each period is characterized by two threshold variables, so-called ( n, n)-policy. We showed that the (, ) policy yields a positive expected value even with zero values for both initial inventory and capital.

Related Work Related studies of problems in which inventory and financial decisions are made simultaneously were done by Li, Shubik, and Sobel (2013) Infinite horizon, maximize the expected present value of dividends (prior to dissolution). The same objective is considered in the rest of the references. Buzacott,and Zhang (2004) Their models allowed di erent interest rates on cash balance and outstanding loans. This paper also demonstrated the importance of joint consideration of production and financing decisions in a start-up setting in which the ability to grow the firm is mainly constrained by its limited capital and dependence on bank financing. Dada and Hu (2008) assume that the interest rate to be charged by the bank is endogenous and the newsvendor s problem may be modeled as a multi-period problem that explicitly examines the cost of reorganization when bankruptcy risks are significant. Accordingly, such single-period model could be used as a building block for considering such models when liquidity or working capital is an issue. This paper treats the system as a game between bank and inventory manager, within which a comparative statics of the equilibrium are presented and a non-linear loan schedule is proposed. Chao et al (2008) The firm can not borrow. Chao et al (2013) Related work.

Current Work Piecewise Type of Loan and Deposit Functions L(x) can have a more complex form in practice. In this section we investigate the often occurring case in which L(x) is a piecewise linear function, i.e., it has the form: L(x) =(1+`(m) ) x, x 2 (x (m 1),x (m) ], where x (m 1) <x (m), x (0) =0and `(m) <`(m+1) for m =1, 2, 3,... Similarly, the deposit interest function to be a piecewise linear function of the form: M(y) =(1+i (k) ) y, y 2 (y (k 1),y (k) ], where y (k 1) <y (k), y (0) =0and i (k) apple i (k+1) for k =1, 2, 3,... We assume that the loan interest rates are always greater than the deposit interest rates, that is ī<`(1) where ī =sup k {i (k) }.

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