A Minimum Concave-Cost Dynamic NetworkFlow Problem with an Application to Lot-SizingStephen C. Graves and James B. OrlinSloan School of Management, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139

.-. .. c .: . ...: . .. . ...,

We consider a minimum-cost dynamic network-flow problem on a very special net-work. This network flow problem models an infinite-horizon, lot-sizing problem withdeterministic demand and periodic data. We permit two different objectives: minimizelong-run average-cost per period and minimize the discounted cost. In both cases wegive polynomial algorithms when certain arc costs are fixed charge functions, andothers are linear.

1. INTRODUCTION

In this article we consider two minimum-cost dynamic network-flow problems basedon the network of Figure 1. We define a dynamic flow x = (xi) to be feasible if itsatisfies the system of constraints

X0o - X12 + X2i = dl (1.1)

Xoj+X_,- 1,; -X _- xj/,/+ + xi+1,i/di for 1=2,3,4,... (1.2)

xf 0.

where d = (d1) is a vector of prespecified nonnegative integers.Associated with each arc(i,i) is a cost function ci(') which we assume to be con-

cave, nonnegative, and nondecreasing. Finally, we assume that the data is periodicwith period n. In particular,

::. :.:.: .:-.:::: ::, ..? dj d- n- for I 2 3, ..i .o .+ . . -. . .. , , for = 1,2 , 3,...

for 1=1,2,3,...

Ci,() =Ci+n,i+n(), for i=- 1 or/+ 1, i,j 1.

In this paper we will consider two different objectives. In Section 2 we considerthe problem of minimizing the long-run average-cost per period of a feasible dynamicnetwork flow. In Section 3 we consider the problem of minimizing the discounted cost.

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60 GRAVES AND ORLIN

dI d2 d3 dk dk+IFIG. 1. The dynamic network.

In both cases we show that the problem is easy if the "period length" of the optimalsolution is a small multiple of n. However, it is unknown whether these problems canbe solved in polynomial time in general. For the special case that the cost functionsare "linear plus a fixed cost," we provide polynomial algorithms for both problems.

An Application to Lot Sizing

This network flow problem may model an infinite-horizon, lot-sizing problemwhere demand is deterministic and is given by dj for demand in time period j. Theflow x0 i denotes production in period j. The flows xi, + 1 and xi + 1,i denote inventoryand backorders, respectively; xi + 1 represents inventory carried from time j to time+ 1, while x+ 1, represents a backordering of demand from time j to time j + I.

Constraints (1.1) and (1.2) are just the inventory balance equations for time period 1and time period j (j > 2), respectively. Finally the cost function coj() is for the costof production, while cj i + (') and c + , i() give the cost of holding inventory and thecost of backordering, respectively.

Because of the possible application to lot-sizing, we will henceforth refer to theproblems in this paper as the average-cost and discounted-cost periodic lot-sizingproblems.

These problems are extensions of the finite-horizon, lot-sizing problem first studiedby Wagner and Whitin [18]. The representation of the infinite-horizon problem as asingle-source network-flow problem directly extends that given by Zangwill [20, 21]for the finite-horizon problem. By assuming that the costs and the demand followa cyclic pattern, we consider a very special version of the infinite-horizon problem.Nevertheless, this problem statement may be a useful representation of settings with astrong seasonal or cyclic demand component. This cyclic property may occur due toa natural product seasonality, or may be induced from the composition of cyclicpurchasing patterns of a set of customers, i.e., customer A buys 100 units once everythree period ... Furthermore, the study of the infinite-horizon problem shouldprovide insight to and supplement the work on planning horizons for the dynamiclot-size problem (Wagner and Whitin [18], Eppen et al. [4], Zabel [19], Lundin andMorton [14], Chand and Morton [21).

Erickson et al. [5] solve the related periodic lot-sizing problem in which the sched-ule x is required to have the same period length as the data. This problem reduces to aconstrained minimum-cost circuit problem, for which they give an O(n3 ) algorithm.

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NETWORK FLOW PROBLEM 61

The periodic lot-sizing problems considered here are special cases of the generalconcave-cost network flow problems for dynamic/periodic graphs. Orlin [16] provedthat this latter problem is P-SPACE hard (and thus is also NP-hard). In other words,any problem that is solvable using a polynomially bounded amount of workspace maybe transformed in polynomial time into a (general) concave-cost dynamic networkflow problem. For further definitions concerning P-SPACE complexity, see Gareyand Johnson [6].

Although the (general) concave-cost dynamic network flow problem is P-SPACEhard, Orlin [17] gave a polynomial time algorithm for the corresponding convex-costdynamic network flow problem.

In Section 2 of this article we consider the undiscounted periodic lot-sizing problemand we reduce it to a minimum cost-to-time ratio circuit problem. We also providea polynomial time algorithm for the case in which the production costs are linear plusa fixed charge and the backorder and inventory costs are linear.

In Section 3, we model the discounted periodic lot-sizing problem as a discretesemi-Markov decision chain. We also provide a polynomial time algorithm for thefixed charge case.

2. REDUCING THE MINIMUM AVERAGE COST PROBLEM TO THEMINIMUM COST-TO-TIME RATIO CIRCUIT PROBLEM

A feasible flow x = (xi) is said to be periodic with period p if

Xo = Xo, +p for all j sufficiently large,

and

Xi 1= Xi +p, +p for all sufficiently large, and i =j + or j - 1.

A flow x = (xii) is said to be a spanning tree flow if there is no circuit C of arcs suchthat xii > O for all (i,)E C.

Lemma 1. There is an optimal dynamic flow for the minimum average cost problemthat is a periodic spanning tree flow.

Proof The fact that there is an optimal periodic flow is a corollary of the optimalityof stationary solutions for finite state Markov Decision Chains. (See Orlin [16] for amore detailed explanation.)

Since all costs are assumed to be concave, within the class of periodic solutions wecan restrict attention to spanning tree flows. (The proof is essentially the same asproving the optimality of spanning tree solutions for finite concave-cost network-flowproblems and does not rely on any infinitary axioms such as the axiom of choice.) I

Because of the application to lot-sizing, we will adopt some of the lot-sizing termi-nology. In particular, we will refer to a period j in which x 0i > 0 as a productionperiod. We refer to period j as a regeneration period if i = and di > 0 or if x -1, =xI, - = 0 and d > 0. In a production context a regeneration occurs at period j if

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62 GRAVES AND ORLIN

there is neither backorders nor inventory at the start of the period. The following isan immediate consequence of the optimality of spanning tree solutions. (Zangwill[20] gives the same result for finite horizon lot-sizing problems.)

Corollary 1. There is an optimal solution to the minimum average cost periodic lot-sizing problem such that between every two successive regeneration periods i and there is exactly one production period k. I

Before proceeding, we will need a bound on the number of periods between succes-sive regeneration points. Our bound is not the tightest possible but is within a constantfactor of the tightest bound.

Let D = d + · + dn; let H(y) be the cost of storing y units of inventory for nconsecutive periods; let B(y) be the cost of back-ordering y units of inventory for nconsecutive periods. Thus, we have

H(y) = 12 (y) = 3(y) + - - +Cn,nl +(Y)

and

B(y) = c 2 (y) + C32 (y) .- + Cn+,n(y).

Finally let c* = co (d ) + -' + Con(d,) be the cost of satisfying all demands in thefirst n periods by production in each period.

Lemma 2. Suppose that (c, d) is an instance of the minimum average cost problemwith period n and that limy,, B(y) = lim_*oo H(y) = o. Then there is an optimumsolution x = (xi) that is a periodic spanning tree solution such that (i) there are aninfinite number of regeneration points and (ii) the number of periods between twosuccessive regeneration points is at most 2(k' + )n, where

k= min [k: k integer, c* < H(kD) and c* < B(kD)] .

Proof. Suppose that i and are successive regeneration points and that p is the pro-duction period between i and j. Suppose further that / - i> 2(k' + 1)n. Hence, wehave/- p >(k' + 1)n orp- i>(k' + 1)n or both.

Consider first the case that - p > (k' + I)n. Suppose we modify schedule x byproducing in periods j - n,...,j - 1 so as to just meet demand, and decreasing pro-duction in period p by D units. This will reduce the inventory in each period betweenp and j. In particular, in the new schedule the inventory level in period k for p < k <j- n is the same as that in period k + n in the original schedule. The inventory levelfor period k for j - n < k < j is zero in the new schedule. Hence, the inventory savingsof the new schedule over the original schedule is just the inventory holding costs in theoriginal schedule for the periods p,p + 1, . p + n - 1. But these savings are at leastH(k'D). Since the increase in production cost is at most c*, it follows, by assumption,that the new schedule is an improvement.

We next consider the case p- i> (k' + l)n. In this case we modify the original

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NETWORK FLOW PROBLEM 63

schedule by producing in periods i, i + 1,.. , i + n - 1 so as just to meet demand, andreducing the production in period p by D units. This will reduce the backorder costsin each period between i and p. Similar to the previous case, the reduction in back-order costs is at least B(k'D), while the increase in production costs is at most c*.Thus the modified schedule gives a strict improvement.

We have thus shown that we can strictly improve the cost of any schedule in whichthere are successive regeneration points i, for which - i > 2(k' + )n, which com-pletes the proof.

For notational convenience, we define k* = 2(k' + 1), where k' is given above.If we consider only solutions satisfying the properties given in Lemma 2, we can

transform the minimum average-cost problem into a minimum cost-to-time ratiocircuit problem, which may be viewed as a minimum average-cost infinite-path prob-lem. In order to define this latter problem more precisely, we first define the follow-ing parameters:

D(i,f) = dkk=i

i-1H(i,j) = E Ck, k+ [D(k + 1,1)]

k=i

=0

-1 k Di

B(i,/)= Ck+l,k [D(i, k)k=i

=0

for 'i <

for i<j

(2.1)

(2.2)

for i=

for i<j (2.3)

for i=/

A (i,j, k) = Ck [D(i,j - 1)] + B(i k) + H(k,j - 1) for i k <,

= +oo

(2.4)

otherwise

Thus A (i,i,k) is the cost of producing in period k so as to satisfy demands betweensuccessive regeneration points i and j. The term cok [D(i,i - 1)] is the cost of produc-tion; the term B(i,k) is the backorder cost; and the term H(k,i - 1) is the inventoryholding cost.

We now construct an infinite graph Go = ( ,E°), where V ® = {1,2,3,. . .}, andE ° consists of all edges (i, j), where i < < i + k*n with k* defined as in Lemma 2,and the cost associated with edge (i,j) is

aii = min A (i, j, k).i<k<j

(2.5)

Thus ai is the minimum cost of satisfying demand in periods i, . . ., - 1 subject tothe constraint that there is one production period in this interval. Alternatively, it isthe minimum cost of satisfying demand between regeneration points i and . An in-finite path in Ed corresponds to an infinite sequence of regeneration points, and the

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64 GRAVES AND ORLIN

average cost per period of the infinite path is the average cost per period of the cor-responding minimum average-cost concave network flow. Note that this is an infinitehorizon variant of Zangwill's [21] interpretation of the n-period dynamic lot-sizeproblem as a shortest path problem.

The parameters of the infinite-path problem are periodic with period n, that is,ai= ai+rn,j+rn for r = 0, 1,2 .... We now show that the average-cost shortest-pathproblem reduces to a minimum cost-to-time ratio circuit problem (as per Dantzig et al.[3]) on the graph G (V,E), where V {1,2, .. . n} and E = {(i,i): 1 < i, n}.

For every edge (i,j + rn) e E for 1 < i, j < n and r a nonnegative integer, we definean associated edge e = (i,) CE with transit time te = r and with an edge cost c, =ai, j +rn. Between any two pair of nodes (i,j) EE there are at most k* distinct edges;if i <j, then the transit time te = r may take the values r = 0, 1,2, . .. , k - 1, whilefor i > j, r = 1,2, ., k*. Any directed circuit C on G defines a periodic spanning treeflow for the average cost problem. The cycle length of the periodic spanning tree flowis the sum of the transit times of the edges of C, while the cost of the periodic span-ning tree flow over one cycle is just the sum of costs of the edges of C. Hence, theaverage cost of the periodic spanning tree flow given by C is just the cost of C dividedby the cycle length of C, which is called the cost-to-time ratio of the directed circuitC. Consequently, to solve the minimum average cost problem we need solve the mini-mum average cost-to-time ratio problem, which determines a directed circuit C on Gwith the minimum cost-to-time ratio.

We note that the above instance of the minimum cost-to-time ratio circuit problemis efficiently solvable in the case that k* is small. For example, Lawler [12], Megiddo[15], and Karp and Orlin [9] all provide efficient algorithms in this case. (The latterpaper provides an O [(nk*)3 i algorithm.)

While these algorithms are all quite efficient for many instances of the average costproblem, they are exponential in the worst case because k* may grow exponentiallylarge in the size of the data. Indeed, the difficultly with solving the above instancesof the minimum cost-to-time ratio circuit problem is that there is no known way ofpruning the exponentially large set of edges. Thus there is no known polynomial-time algorithm.

Below we provide a polynomial-time algorithm for the special instance of the lot-size problem in which the production costs are "linear plus a fixed charge" and wherethe inventory and backorder costs are linear functions.

We say that a function f(.) is a fixed-charge function is there are integers a and bsuch that a > 0, b > 0, and

a+ bx if x>OX) = 0 if x = 0.

It is easy to see that fixed-charge functions are concave.A large part of the lot-sizing literature deals exclusively with problems in which the

production costs are fixed charge. In this context the fixed charge a denotes a setupcost incurred whenever production is initiated, regardless of the production quantity;in addition to this fixed cost, there may be a variable cost that is directly proportionalto the production quantity at the rate b. In what follows we assume that the produc-

tion cost is a fixed-charge function, while both the inventory and backorder costs arelinear-cost functions.

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NETWORK FLOW PROBLEM

A standard technique for determining a minimum cost-to-time ratio circuit is aniterative procedure based on the following observation.

Remark. Let G be a directed graph, and for each edge e let Ce and te denote its costand transit time. Let X be a real number and let e = Ce - Hte be the reduced cost ofedge e. Then any circuit C in C has a cost-to-time ratio of at least X if and only if thereduced cost of C is nonnegative (e.g., Lawler [ 11 ] ).

The technique based on this remark is to use binary search to find the minimumvalue of X for which there is no circuit with negative reduced cost. At each iterationfor each pair (i,/) we select that edge e (i,) with minimum reduced cost Ce andignore all other edges from i to j. Once these edges are selected, the remaining numberof steps in each iteration is 0(n3 ) using a standard algorithm such as Bellman [1] forcomputing the existence of negative-cost circuits. Moreover, the number of iterationsis polynomially bounded because we start with a search interval that is at most [max(ICe I: e E)], and we may stop when our search interval is smaller than (nk*)- 2 .(See Zemel [22] for more details on binary search.)

We now claim that for fixed X we can determine in polynomial time those edges ofG with minimum reduced cost. We accomplish this for a specified pair i,f-.of verticesas follows.

min ce - .te e is an edge of E from i to j}

= in aii tn - Nt: t = 0, 1,2, . . k - 1 for i <j,t

and t 1,2,..., k* for i>j} (2.6)

Now after substituting the definition for aij given by (2.5) into (2.6), we obtain

min {A [i, + (r + s)n, k +rn] - (r + s): i < k + rn <j + (r + s)n,r,s,k

1 < k < n and r + s k*}

If the production cost in period k is

Cok (X) 0

I+ vkx if x >0

if x = 0,(2.7)

then using (2.4) we can break up the above minimization into two parts as follows:

min {fk + min[D(i, k + rn - )vk + B(i, k + rn) + D(k + rn,j + (r + s)n - 1)vk1 •k<n r,s

+ H(k + rn,j + (r + s)n - 1) - (r + s)]

= min {fk + min [D(i , k + rn - 1)vk + B(i, k + n) - Xr]1 kn r

+ min[D(k,/+ sn - i)vk + H(k,j + sn - 1) - Xs]}. (2.8)

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66 GRAVES AND ORLIN

In the last simplification we use the fact that both the demand and cost functions areperiodic with period n.

Now suppose the holding cost and backorder cost in period k, < k < n, are givenby

Ck,k+ (X) = hkx, (2.9)

and

Ck,k- (X) = bk (2.10)

respectively with hk,bk > 0; then for fixed i, j, k in (2.8), the functions B and H areconvex quadratic functions of r and s, respectively. This is shown in the followinglemma.

Lemma 3. For fixed values i and j, let f1 (t) = H(ij + tn) and let f2(t) = B(i,j + tn).Then both fi and f2 are convex quadratic functions of t.

Proof It is easy to verify thatH(i,j + tn) - H(i,j + (t - 1)n) is a nondecreasing linearfunction of t. Similarly, B(i,j + tn) - B(i,j + (t - )n) is a nondecreasing linear func-tion of t. Thus both f1 and f2 are convex quadratic functions. I

We note that for each i,i we need determine just three points along the quadraticfunction, e.g., H(i,), H(i,j + n), and H(i,j + 2n), to specify the coefficients of thefunction. Thus, to determine the coefficients for the quadratic functions H(i, j + tn)and B(i,j + tn) for all i,i, it suffices to determine H(i,k) and B(i,k) for all 1 < i nand i < k < i + 3 n. But this clearly takes just O(n2 ) steps since the evaluation of eachinstance of H(i,k) and B(i,k) requires 0(1) steps.

As a consequence of Lemma 3, the inner minimizations in (2.8) are over convexquadratic functions and can be done in a constant number of steps for given values ofi, j, and k. (The minimization of (ax2 + bx + c: x > 0, integer) for a > 0 occurs at oneof the points [-b/2a], -b/2a] or 0.) Since k may take on values 1, 2, ... , n, thecomplexity of (2:8) is O(n); since the minimization (2.8) need be performed for eachpair (i,j) of vertices, the overall complexity for selecting the edges with minimumreduced cost is 0(n3 ).

3. THE DISCOUNTED PERIODIC LOT-SIZING PROBLEM

In this section, we consider the periodic lot-sizing problem in which the objective isto determine the minimum discounted cost for satisfying demands over an infinitehorizon.

We let p (p > 0) denote the discount rate per n periods. We assume that p is alreadyincorporated into the cost functions, i.e.,

coj(') = p- o,+n (), for > 1,

ci(') = p-lCi+n,j +n(), for i=j- I or j+l and i, j> l.

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NETWORK FLOW PROBLEM 67

For the discounted case we define a regeneration period as before. Moreover, thecounterparts of Lemma 1 and Corollary 1 extend to the discounted case. However,there no longer may be an infinite number of regeneration periods, as is illustrated bythe following example.

Example. Let n = 1, p = 1/2, Co(x) =10x, C12(x) = lOx, c 21(x) = x, and d = 1. Thenthe optimal solution is to backorder in each period, and the resulting discounted costis 4.

Nevertheless, we can guarantee that the number of periods of backordering is boundedby making appropriate assumptions on the cost of backordering as in Lemma 4.

.' . . .:: . .. .' : . ..

Lemma 4. Suppose that (c,d) is an instance of the minimum discounted cost problemwith period n. Then there is a periodic spanning tree solution. If in addition

lim c,+(y)=oo, for j=,...,n

and if

lim inf c+ 1, (y)- co,(y)>O for j = , n,y -- oo

then there is an infinite number of regeneration points.

Proof Let = ,+- Cok (dk). Then an upper bound on the cost of a schedule sat-isfying demands from period i on is (1 - p)-'. It follows that the maximum numberof consecutive periods in which inventory is stored is at most tn, where

= min(k: k integer and c j+ l (kD) > (1 - p)- for j = 1, n).

Any schedule x with more than kn consecutive periods of inventory starting in period jcan be improved upon by producing to meet demand in periods j, j + 1, j + 2, . . . Thisfollows since there must be at least lD units in inventory in period j, and ci,+ 1 (kD) >c;(1 - p)-' by definition.

Let

k = min{k: k integer and ci 1t, (k'D) - co, (k'D) > O

'. ¢.:'!i:!:' ::?:.:111: i;i:::."::'7.: -Ifor k' k andan l,..., n}.

We claim that the maximum number of consecutive periods of backordering is atmost kn. To see this, suppose that x is a schedule in which there are greater than knperiods of backordering starting in period i. Let x' be obtained from x by producingin period i + kn to meet all demand in periods i, i + 1,..., i + kn, and adjusting pro-duction and backorders in other periods accordingly. Then the increase in productioncost in period i + kn is less than the savings in backorder cost in period i + kn, and thusx has a lower cost than x. U

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68 GRAVES AND ORLIN

We now define D (, ), H(, -), B(-, ), and A (, -,) in the same way as in (2.1)-(2.4)except that we use the discounted costs. We also define aj as in (2.5).

Let zi denote the minimum discounted cost of satisfying all demands over theinfinite horizon starting in period i with zero inventory. Then the following recursionuniquely determines the values for z.

(3.1)

Similar to the undiscounted case, we can efficiently solve for the z's if there is a boundk on t in (3.1) such that k is a small integer. The more difficult case is the one in whichk is large, or possibly infinite.

Below we will solve the fixed charge version of this problem; that is, the productioncost is a fixed charge function while the holding and backorder cost functions arelinear. The proof is based on the following lemma.

Lemma 5. For each integer i,j with 1 < i, j < n and each integer t with j + tn > i,there are coefficients h , h h b, b and bj, such that

H(i,/ + tn) = h + h.t + hpP

and

B(i, + tn) = b + b i +b tpt + b. pt. (3.3)

Proof. Consider first the costs of storing inventory. We consider the first orderdifference

H(i, + (t + 1)n) - pH(i,j + t) = H(i,j + (t + )n) - H(i + n,j + (t + )n)

i+n- 1

= Ck,k+l(D(k + ,j+(t + l)n).k=i

Let f(t) = H(i,j + tn). By the above, the difference f(t + 1) - pf(t) is linear in t, be-cause ck,k + (D(k + 1, + (t + )n)) is linear in t for fixed j,k. By the theory of lineardifference equations (see for example Luenberger [13]), we may express f(t) asf(t) = h b + hit + h pt, for some choice of parameters h for r= 0,1,2.

We now consider the costs due to backordering. Let g(t) = B(i,j + t). Then

j +tn +n-1

g(t + 1)- g(t) = Sk=j+tn

Ck+l, k (D(i, k))

= Pt Ck+ ,, k (D(i,k + tn)).k=j

Since ck +l,k (D(i,k + tn)) is linear in t for fixed i,k, it follows that g(t + 1) - g(t) =(tl + 2 t)Pt. By the theory of linear difference equations, we may express g(t) asg(t) = b + bltpt + bp t for some choice of parameters b for r = 0,1,2. I

(3.2)

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NETWORK FLOW PROBLEM 69

We observe that we can calculate b, h for 1 i, j <n and r = 0,1,2 in O(n2 )steps by successively calculating B(i,j) and H(iJ) for all i,i with 1 < i < j < i + 3n.The coefficients are easily computed from these function values.

We show below that we can calculate z via policy improvement (see Howard [8]),where each iteration in policy improvement takes O(n2 ) steps.

Let us assume that we wish to calculate z i via recurrence relation (3.1) for the caseof a fixed charge problem.

zi = inf {ai /+tn + p tzi: 1 <j <n, t>O}

= inf{A(i, i+ (r + s)n,k + rn)+pr+z: l <,k < n, r,s> 0,

and i < k + rn <j + (r + s)n} (3.4)

We now substitute the production cost (2.7) into the cost A(,,-) obtaining the fol-lowing relation.

z i = inf [B(i,k + rn) + prf k + pk ik + rn) + prVk D(k + rn + 1, j + (r + s)n - 1)

+ prH(kj + sn) + pf+szi]

=inf[B(i,k+rn)+prvk D ( i,k + rn )+ p k : I< k < n, k + rn > i]

(3.5)

(3.6)

where

k = inf[ fk + ok D(k + l, j + sn - 1) + H(k, j + sn) + pzi:

l < jn, j+sn>k] (3.7)

By substituting (3.2) into (3.7) and observing that vkD(k + , j + sn - 1) is linear ins for fixed j, the minimization (3.7) may be rewritten as

Zk =inf a7 + as apS for I n, s(k - j)/n].

For each pair of fixed values j,k, the above minimization problem is solvable in 0(1)steps by setting the derivative to 0. If s' is the value for which the derivative is 0, thenwe can show that the minimum occurs at [s'j or [s'] or [(k - j)/n]. Thus we cancompute zk for k = ,..., n in O(n2) steps.

By substituting (3.3) and the solution of (3.7) into (3.6) we find that

z i = inf[a + ap r + a rp : 1 < j n, r (j- i)/n].

Let ri be the value of r that minimizes the above for fixed j. By taking the derivativewe can easily show that the above function is either unimnodal up or unimodal down,depending upon the sign of a . Consequently, if r' is the value for which the derivativeis zero, then the infinum must occur at [r' j or [r'] or [(/ - i)/nl or oo.

Thus we find the minimum in 0(1) steps for each J, and thus can perform one itera-tion of policy improvement in O(n2 ) steps.

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70 GRAVES AND ORLIN

Implementation of Linear Programming

We may solve the discounted problem in polynomial time via the Khachian ellip- .

soidal algorithm [10] as described by Groetschel et al. [7]. To see this, we rewrite(3.1) as its equivalent linear program:

min zl + + Zn (3.8)

subject to

Zi >ai, tn + Pt z ]

for i,= 1,2 ... ,n, k = 0, 1,2,..., and t >(i-i)ln. (3.9)

Given any vector {zi}, we can discover in 0(n 2 ) steps, as shown above, whether it isfeasible (and hence optimal) to (3.8)-(3.9), and if not, we find a violated constraint.Therefore, the ellipsoidal algorithm runs in polynomial time despite the infinitenumber of constraints implied by the linear program (3.8)-(3.9).

References

[ 1] R. E. Bellman, On a routing problem. Q. App. Math. 15 (1958) 87-90.[2] S. Chand and T. E. Morton, Perfect Planning Horizon Procedures for the Dy-

namic Lot Size Inventory Model. Working paper, August 1979.[31 G. B. Dantzig, W. Blattner, and M. R. Rao, Finding a cycle in a graph with

minimum cost to times ratio with application to a ship routing problem. InTheory of Graphs (P. Rosenthal, Ed.). Dunod, Paris, Gordon and Breach, NewYork ( 1 96 7) 77-84.

[4] G. D. Eppen, F. J. Gould, and B. P. Pashigian, Extensions of the planning hori-zon theorem in the dynamic lot size model. Management Sci. 15 (1969) 268-277.

[51 R. E. Erickson, C. L. Monma, and A. F. Veinott, Jr., Minimum concave-costnetwork flows. Math. Operations Res., in press.

[6] M. R. Garey, and D. S. Johnson, Computers and Intractibility:'A Guide to theTheory of NP-Completeness. Freeman, San Francisco (1979).

[7] M. Groetschel, L Lovasz, and A. Schrijver, The ellipsoid method and its conse-quences in combinatorial optimization. Combinatorica 1 (1981) 169-197.

[8] R. A. Howard, Dynamic Programming and Markov Processes. Wiley, New York(1960).

[9] R. M. Karp and J. B. Orlin, Parametric shortest path algorithms with an appli-cation to cyclic staffing. Discrete App. Math. 3 (1981) 37-45.

[10] L. G. Khachian, A polynomial algorithm for linear programming. Dokl. Akad.Nauk. SSSR 244 (1979) 1093-196.

[ 11 ] E. L. Lawler, Combinatorial Optimization: Networks and Matroids. Holt, Rine-hart and Winston, New York (1976).

[12] E. L. Lawler, Optimal cycles in doubly weighted linear graphs. In Theory ofGraphs (P. Rosenthahl, Ed.). Dunod, Paris, Gordon and Breach, New York(1967), pp. 209-214.

[ 13] D. G. Luenberger, Introduction to Dynamic Systems: Theory Models and Appli-cations. Wiley, New York (979).

[14] R. Lundin and T. Morton, Planning horizons for the dynamic lot size model:Zabel vs. protective procedures and computational results. Operations Res. 23(1975) 711-734.

C�

''··;

.. .. ��t t·:·;··,·.·�..�·��· �·..;. ·.,;

NETWORK FLOW PROBLEM

[15] N. Megiddo, Combinatorial optimization with rational objective functions. InProceedings of the 10th A CM Symposium on the Theory of Computing (1978),pp. 1-12.

[16] J. B. Orlin, The complexity of dynamic languages and dynamic optimizationproblems. In Proceedings of the 13th ACM Symposium on the Theory ofComputing (1981), pp. 218-227.

[171 J. B. Orlin, Minimum convex cost dynamic network flows. Math. OperationsRes. 9 (1984) 190-207.

[18] H. M. Wagner and T. Whitin, Dynamic version of the economic lot size model.Management Sci. 5 (1958) 89-96.

[19] E: Zabel, Some generalizations of an inventory planning horizon theorem.Management Sci. 10 (1964) 495-471.

[20] W. I. Zangwill, A backlogging model and a multi-echelon model of a dynamiceconomic lot size production system-A network approach. Management Sci.15 (1969) 506-527.

[21] W. 1. Zangwill, Minimum concave cost flows in certain networks. ManagementSci. 14 (1968) 429-450.

[221 E. Zemel, On search over rationals. Operations Res. Lett. 1 (1981) 34-38.

Received November 4, 1983Accepted April 17, 1984

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71

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