computational complexity of uncapacitated multi-echelon production planning problems

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Page 1: Computational complexity of uncapacitated multi-echelon production planning problems

Operations Research Letters 8 (1989) 61-66 April 1989 North-Holland

C O M P U T A T I O N A L C O M P L E X I T Y O F U N C A P A C I T A T E D M U L T I - E C H E L O N P R O D U C T I O N P L A N N I N G P R O B L E M S

Esther A R K I N *, Dev J O N E J A * * and Robin R O U N D Y * *

School of Operations Research and Industrial Engineering, Cornell Universi(v, Ithaca, N Y 14853, USA

Received August 1988 Revised October 1988

Recently there has been a flurry of research in the area of production planning for multi-echelon production-distribution systems with deterministic non-stationary demands and no capacity constraints. A variety of algorithms have been proposed to optimally solve these problems, with varying success. This paper investigates the issue of computational complexity of the problem for all commonly studied product structures, i.e. the single item, the serial system, the assembly system, the one-warehouse-N-retailer system, the distribution system, the joint replenishment system, and the general production-distri- bution system. Polynomial time algorithms are available for the single-item, serial and assembly systems. We prove that the remaining problems are NP-complete.

production planning * multi-echelon systems * computational complexity

1. I n t r o d u c t i o n

Consider a multi-echelon production-distribu- tion system which is represented by its bill-of- materials network. In this network, each produc- tion operation or distribution stage is represented by a node. Arcs in the network show how the items flow from one stage to the next in the production-distribution system. Assuming that no item is consumed in its own production, the gen- eral production-distribution system is thus repre- sented by an acyclic directed network. An assem- bly node is one at which several sub-assemblies are combined in the production of the item. This node will have several arcs coming into it, each corre- sponding to one sub-assembly. A Distribution node represents an item which is consumed in the pro- duction of several other items, or which is shipped to several locations in a distribution system. Such a node will have several arcs going out of it. Note

* Research supported in part by Presidential Young Investi- gator grant ECSE-8857642.

** Research supported in part by NSF Grant DMC-8451984, and by AT&T Information Systems and DuPont Corpora- tion.

that in general a node can be both an assembly node and a distribution node.

Several specific classes of networks have been discussed in the literature. The simplest case is the network consisting of a single node. A system in which the raw-material is processed by a fixed sequence of operations, yielding the final product, is called the serial system. The corresponding net- work has a series of nodes with an arc from each node to the next one in the series. External de- mand occurs only at the last node in this network. An assembly system has several sub-assemblies combined together at each stage. Thus it has only assembly nodes and no distribution nodes, and is represented by a tree. External demand in this network occurs at the final assembled stage. The reverse of this is the distribution svstem~ in which each node is a distribution node and there are no assembly nodes. External demand occurs at the terminal nodes in this tree, which have no succes- sors. The two stage distribution system in which arcs exist only from one special node (the warehouse) to each of the other nodes (the re- tailers) is known as the one-warehouse-N-retailer system. The general system can have any combina- tion of nodes and arcs, and is represented by an acyclic directed network.

0167-6377/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland) 61

Page 2: Computational complexity of uncapacitated multi-echelon production planning problems

Volume 8, Number 2 OPERATIONS RESEARCH LETTERS April 1989

We consider a production-distr ibution system represented by a directed acyclic network. Produc- tion planning in such systems is often done using large computer based tools known as Materials Requirement Planning (MRP) systems. The pro- duction-distr ibution system operates in discrete time to satisfy external demand of several items. The demand for each item in each time period over a certain finite time interval in the future (the time horizon) is assumed to be known. We do not require demand to be constant over time, but allow it to vary from one time period to another. The external demand for each item in each time period must be satisfied, and no backorders are permitted.

We assume that production at each stage is instantaneous, and that there are no capacity con- straints on production. The lead times from one stage to the next are assumed to be zero. The costs we are considering are fixed ordering costs or setup costs for the various items which are in- curred whenever the item is ordered or produced, and linear inventory holding costs. We are focus- ing on decisions about when to order or produce each item, and in what lot sizes, in order to minimize the total ordering and inventory holding cost over the planning horizon.

This paper addresses the question of computa- tional complexity of each of the systems discussed above, as well as the joint replenishment system which is described in the next section. We prove that the production planning problem for the one-warehouse-N-retailer system, the distribution system, the general product ion-dis tr ibut ion sys- tem, and the joint replenishment system are NP- complete. The remaining systems (single item, serial, and assembly) can be solved in polynomial time (see Wagner and Whitin [9], Love [5], and Rajagopalan and Cornuejols [6] respectively). The reader is also referred to the papers by Florian, Lenstra and Rinnooy Kan [2] for a discussion of the complexity of single-item problems under a variety of cost functions both with and without capacity constraints, and by Bitran and Matsuo [1] for the single-item problem with capacity con- straints.

2. The joint replenishment problem

The joint replenishment system is composed of several items, for each of which there is external

demand. This external demand must be satisfied in each time period, and backorders are not per- mitted. Each item incurs a fixed setup cost when it is produced, and a linear inventory holding cost. In addition, a joint setup cost is incurred in each time period in which one or more of the items is produced. Thus economies can be effected by ordering several items in the same time period. We use the following notation:

n, i = subscripts for nodes or items, s, t = subscripts for time period, N = number of items in the system, T - 1 = the number of time periods in the time

horizon, dnt = demand for i tem n in time period t, H~ = inventory holding cost of item n per unit

per time period, K n = ordering (setup) cost of item n, K 0 = joint ordering (setup) cost.

Time period T is a dummy time period at the end of the time horizon which will be used in the discussion below.

In the joint replenishment problem, it is easy to show that there exists an optimal policy in which an item is ordered in time period t only if its inventory at the end of period t - 1 is zero. Since we can restrict attention to policies which follow this rule, consider an item n which places two successive orders in time periods s and t, where 1 ~< s < t ~< T. Then the production in time period s must exactly cover the demands in periods s through t - 1. This fact allows us to define:

C,s t = cost for item n incurred in periods s to t - 1 if it is ordered in periods s and t, and nowhere in between

= K . + H . E ~ - ] + l ( i - s )d .~ .

For each item n, consider a network where there is a node for each time period. In this network, from any node s there is an arc to each node t > s, with the associated cost (length) C.s,, as shown in Figure 1. Consider any path from node 1 to node T on this network. If an arc s --* t lies on this path we interpret it as item n being ordered in time periods s and t, and nowhere in between, at a cost of C.s t. The path then repre- sents a feasible ordering policy for item n, and the length of the path gives the total cost of following this policy. Then the shortest path from node 1 to node T on this network gives the optimal ordering

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Cn 13

Fig. 1. The network for item n. There is one such network for each item in the joint replenishment problem.

policy for item n alone. Besides, if the pa th for any item passes through a node s, then the joint cost K 0 is incurred in time period s. The joint replenishment problem is then to find a path on the network for each node in order to minimize the total pa th costs and the joint costs.

2.1. Computational complexity

In this section we prove that the Joint Re- plenishment Problem (JRP) is NP-complete . For a discussion of NP-completeness, the reader is re- ferred to Garey and Johnson [3]. We then use this result to show that the one-warehouse-N-re ta i le r problem is NP-complete . Finally, this leads to the conclusion that the lot-sizing problem in the multi-echelon distribution system and in the gen- eral multi-echelon produc t ion-d i s t r ibu t ion system is NP-complete .

For the purpose of this section, we use the network interpretat ion of the joint replenishment problem, where there is a network of the form shown in Figure 1 for each node in the system. The corresponding decision problem is described below.

Instance. A set of N products, n = 1 . . . . . N, and T time periods, t = 1 . . . . . T, inventory holding costs H . , n = 1 . . . . . N, ordering costs K n, n = 1 , . . . , N, demands dnt, n = 1 . . . . . N, t = 1 . . . . . T - 1, a joint ordering cost K 0, and an integer B. Create the following graph: for each produc t we have nodes (n, t) for all t. Arcs f rom (n, s) to (n, t) exist for all s < t with cost

t - - S

C, s t=K, + H , ~ ( t - s - i ) d n . t _ , . i = 1

The joint setup cost K 0 is incurred for each t where a node (n, t) is used in the solution paths for any n.

Question. Does there exist a set of paths f rom nodes (n, 1) to (n, T ) for all n for which the sum of the jo in t setup costs incurred plus the path costs is not greater than B?

Theorem 1. The above problem (JRP) is NP-com- plete.

Proof. It is easy to see that the problem is in NP. In fact, algori thms are known which run in poly- nomial time for fixed T (Veinott [8]) or for fixed N (Zangwill [10] and K a o [4]). To show complete- ness, we give reduct ion f rom the NP-comple te problem 3-satisfiability (3-SAT). The proof is in two parts. First we provide a reduct ion of an instance of 3-SAT into an instance of the joint replenishment problem which has certain special characteristics. Then we show that these character- istics can actually be at tained in an instance of J R P using suitable data.

The problem 3-SAT is defined as follows (for details, see [3]). Let X = ( x 1 . . . . . x , } be a set of Boolean variables. I f x is in X, then x and its negat ion Y are literals over X. A clause over X is a set of literals over X, such as (x3, Ys, X6)" A truth assignment gives to each variable in X the value of ' t rue ' or 'false' . A truth assignment satis- fies a clause if at least one literal in the clause is true under the assignment. A set of clauses is satisfiable if there exists a truth assignment for X which satisfies every clause. An instance of 3- satisfiability (3-SAT) is defined by the set X with n variables and a set of clauses (C1 . . . . . C,, } with exactly 3 literals in each clause. Question: Is there a satisfying truth assignment for this set of clauses?

Given an instance of 3-SAT, we define an instance of J R P with T = M + 2n + 2 periods and N = n + m products. For each product , consider a network of the form in Figure 1. The time periods correspond to: A special period 1 at the begin- ning, M d u m m y periods after it, two periods corresponding to each variable x i, one for x i and one for the negat ion of x i, ~i, and a special period T at the end. The ' n o d e x , ' will refer to the node for the time period corresponding to xi in this network. The first n products correspond to the variables xi, and the remaining m products corre- spond to clauses Cj. We will show later that M is a polynomial in m, n.

On the network for the produc t corresponding to x,, the cost of a pa th f rom node 1 to x~ to N is

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2, the cost of the path f rom node 1 to ffi to N is 2, and the cost of the path f rom node 1 to x i to .2 i to N is 2 + K 0. A n y other path is of length greater than 2 + K 0. On the network for the product corresponding to Cj, the cost of a path f rom node 1 to Ljk to N is 2, and the cost of any other path is greater than 2. (Ljk are the literals that appear in clause Cj.) We show later how to create a set of demands and costs to fit this pattern. We set

B = 2(m + n) + nK o.

This is clearly a polynomial time construction. We now show that the above-defined instance

of (JRP) has the desired set of paths if and only if the corresponding 3-SAT instance is satisfiable. First, note that if the formula is satisfiable, then the following set of paths has cost B: For each product x i, go f rom node I to either x i or Yi (according to which is true in the satisfying assign- ment), and from there to node T. The cost in- curred will be 2n for the paths and nK o for setups. For each produc t Cj, go f rom node 1 to any literal that is true in clause Ca, and f rom there to T. The cost will be 2m with no addit ional setup cost. Clearly, the total cost then is 2m + 2n + n K o

: B . N o w assume that there is a set of paths f rom 1

to T for each product , such that the sum of path lengths and setup costs is no greater than B. First consider the products corresponding to the x i. Let p of these products use the path f rom 1 to either x i or Yi to T, at a total cost of 2p + p K o. Let q of these products use the path f rom I to x i to if, to T, at a total cost of 2q + 3qK o. Let the remaining n - p - q of these products choose any other path, at a total pa th cost larger than (n - p - q)(2 + K0). Finally, the products for clauses Cj will cost at least 2m. Hence the total cost is greater than or equal to

2p + p K o + 2 q + 3qK o + ( n - p - q) (2 + K0)

+ 2m = 2m + 2n + 2 q K o + n K o = B + 2 q K o,

and is equal to B + 2qK o only if n - p - q = 0. Since the total cost is at most B, this implies q = 0 and n = p . Further, equality holds only if the produc t for each clause C a chooses a path of length exactly 2, which means that the path goes f rom 1 to Ljk to T for which we have already incurred a setup cost. To summarise, the solution satisfies: each produc t corresponding to a variable x i chooses a path f rom 1 to x i or ~i to T, and each

produc t cor responding to a clause Cj chooses a path f rom 1 to Ljk to T for which the setup cost has already been incurred. This is clearly a satisfy- ing truth assignment.

Finally, it remains to be seen that we can indeed generate demands and costs, so that the paths have the lengths described above. For the produc t x i, let dx, denote the demand in time period x i.

K0 dx '= M + 2 ( i - 1 ) + l ' d ~ ' = K ° "

K i = 2 K o, H i = 1.

Demands in other time periods for p roduc t x i are zero. The cost of the path f rom node 1 to x i to T is

2 K i 4- d,, = 5K 0.

The cost of the path f rom node I to ffi to T is

2 K i+ d : , i ( M + 2 ( i - 1) + 1) = 5K 0.

The cost of the pa th f rom node 1 to T is

K i+ ( M + 2 ( i - 1) + 1)dx, + ( M + 2i)d~i

= 3K 0 + ( M + 2 i ) K 0 > 6K 0

for M > 1. The cost of the path f rom node 1 to t to T for t < x i is greater than or equal to

2 K i + dx, + dx, > 6K0,

and for t > ~i the cost is greater than the cost of the path f rom 1 to T. This takes care of all possibilities where p roduc t x i is ordered once or twice. If it is ordered at least three times, the cost of the path will be strictly greater than 3K i = 6K 0, except for the path f rom node 1 to x i to Yi to T, which costs exactly 3K i = 6K 0. Setting K 0 = 0.4 yields the required demand structure. Non-integer demands are easily removed, if necessary, by using suitable holding costs H i such as

d x , = 5 K o = 2 , d~ = 2 ( M + 2 ( i - 1 ) + 1 ) ,

1

/4,.= 5 ( M + 2 ( i - 1) + 1) '

For clause p roduc t Cj, let p be the number of time periods f rom 1 to L21, q be the number of time periods from Ljl to Lj2 , and r be number of time periods f rom Lj2 to Lj3. (p , q, and r depend on the clause, but we omit the subscript.) Let djk denote the demand for p roduc t ~ in the time period corresponding to literal Ljk. The demands

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for product ~ in all other time periods are zero. Let

d/l = ( p + q + r ) q , d j2=r p,

d / 3 = p ( p + q ) , K,+j = 2K0 = 0.8,

0.4 On+ j = p(pq + q2 + pr + 2qr)

The cost of a path f rom node 1 to L/1 to T is

2(0.8) + qH,+/ds 2 + (q + r)H,+jdj3

=1 .6+ 0.4qr + O.4( q + r )( p + q ) = 2 . pq + qZ + pr + 2qr

The costs of the paths f rom node 1 to L j2 to T, and from node 1 to Ls3 to T are similarly calcu- lated as equal to 2. For any node t ~ L/k, t < Lj3, the cost of a path f rom node 1 to t to T is clearly greater than the cost f rom node 1 to the node for the next literal after node t to T. Thus the cost of the path f rom node 1 to t to T is greater than 2. If node t > L3/ then the cost of this path is greater than the cost of the path from node 1 to T. The cost of a path f rom 1 to T is

0.8 + pH,+jdj, + ( p + q)H,+/d/2

+ (p + q+ r)H,+/d/ ,

0 . 4 ( p + 2 q ) ( p + q + r ) + 0 . 4 r ( p + q) = 0 . 8 +

pq + q2 + pr + 2qr

0 . 4 ( p + q ) ( p + q + r ) = 1 . 2 +

pq + q2 + pr + 2qr

Since we require this last number to be greater than 2, we require

p2 + 2 pq + pr + q2 + qr > 2q 2 + 2 pq + 2 pr + 4qr

or

p2 _ p r > q2 + 3qr.

This is where the M d u m m y periods are used. No te that p > / M + 1, and q + r ~< 2n, so the in- equality will always have a solution. To find it we note that

q2 + 3qr= q ( q + r) + 2qr

<~ 2qn + 2qr

~< 2 q ( 3 n - q) .

This quadratic form is maximized at q = 3n/2, and we have

q2 + 3qr <~ 9n2/2.

Again, using the fact that r ~< 2n, we have p~- - pr >~p2-2pn, so it suffices to have p 2 - 2 p n > 9n2/2 . Equality holds for

p = n 4 - ½ Y / 4 n 2 + 1 8 n 2,

so for our purpose we can set M = [(1 + ½ 2 ~ ) n ] . For a pa th for a clause produc t with three or

more orders, the cost is greater than 3Kn+j = 2.4 > 2 as required. It is easy to see that this construc- tion can be carried out in polynomial time; thus our reduct ion is valid. []

3. Multi-echelon production-distribution systems

The single p roduc t One-Warehouse -N-Re ta i l e r problem ( O W N R ) is a two level distr ibution sys- tem in which a single warehouse supplies N re- tailers. External demand is satisfied by the re- tailers in each time period with no backorders. Over a finite discrete time horizon of length T, retailer n in time period t has to satisfy an exter- nal demand of dnt, n = l . . . . . N, t = l . . . . . T without backorders. Each retailer n has an order- ing cost of K n which is incurred whenever it places an order on the warehouse, and a tradi- tional linear inventory holding cost rate of H" per unit per unit time. The warehouse has an ordering cost of K 0 and an inventory holding cost rate of H 0. The objective is to determine the ordering policy at the warehouse and at each retailer in order to minimize the total ordering and inventory holding costs over the entire time horizon.

We show here that the JRP is a special case of O W N R . First note that in the JRP the demands of each i tem can be scaled so that the holding cost rate of each item is indentical. Let this c o m m o n holding cost rate be denoted by H 0 (thus H n - H 0 V n = l . . . . . N) . Consider then the O W N R in which there is a retailer corresponding to each item in JRP. The retailer n has an ordering cost of K n and an inventory holding cost rate of H 0. The warehouse has an ordering cost of K 0 and an inventory holding cost rate of H 0. Demands for the items in J R P correspond to external demands at the cor responding retailers in O W N R .

In the solution to the O W N R , the warehouse will place an order in time period t only if some retailer also orders in that time period (Schwarz [7]). In the special case of the O W N R described above, which is derived from the JRP, the con-

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verse is also true. To see this, let a retailer n place an order in a time period t in which the warehouse does not place an order. Let the last order of the warehouse prior to t be in time period s. There are two possible cases:

- If retailer n does not order at s, eliminate the order of retailer n in time period t and insert one at s. Order quantities are adjusted accordingly to confi rm to the zero order rule. This does not change the total ordering cost. Further, it just increases the inventory at the retailer in time periods s through t - 1 with a corresponding de- crease in the inventory at the warehouse. Since the inventory holding cost rates at the warehouse and the retailers are identical, this does not change the total inventory holding costs either. Thus the total cost of the policy is not affected.

- I f retailer n has an order in period s, eliminate its order in period t, and correspond- ingly increase the order quant i ty in time period s to cover the demand till the next order. This decreases the total ordering cost by K n, while the inventory holding cost is not affected as explained above.

In either case, we get an opt imal policy to this O W N R in which the warehouse orders whenever any retailer does, and no inventory is ever stored at the warehouse level. Inventory is only stored at the retailers. Thus this O W N R solves the corre- sponding JRP.

Theorem 2. The above problem O W N R is NP-com-

plete.

Proof. Since JRP is a special case of O W N R , and J R P is NP-complete , the theorem follows. []

Corollary 3. The lot sizing problems for the multi- echelon distribution system, and for the general multi-echelon production-distribution system in dis- crete time over a f in i te t ime horizon are NP-com-

plete.

Proof. By restriction to O W N R . []

4. Summary

The known results about the computa t iona l complexity of the commonly studied uncapaci-

tated lot sizing problems can then be summarized as follows:

- The jo in t replenishment problem: NP-com- plete. Can be solved in polynomial time for fixed N or T (see, for example [8] and [10]).

- Serial system: Polynomial , with an O ( N T 3) algori thm in [5].

- Assembly system: Polynomial ; the algori thm uses linear p rog ramming on a formulat ion with O ( N T 2) constraints and O ( N T ( T + 2)) variables. N o specialized algori thm is known. See [6].

- One-warehouse-N-retai ler system: NP-com- plete.

-Genera l distribution system: NP-comple te . O ( N T 3) algori thms exist to find the optimal nested policy for such systems (see [8]).

- General production-distribution system: NP- complete.

References

[1] G. Bitran and H. Matsuo, "Computational complexity of capacitated lot sizing problem", Management Science 28, 1174-1185 (1982).

[2] M. Florian, J.K. Lenstra and A.H.G. Rirmooy Kan, "De- terministic production planning: Algorithms and complex- ity", Management Science 26, 669-679 (1980).

[3] M.R. Garey and D.S. Johnson, Computers and Intractabil- ity, W.H. Freeman and Co., New York, 1979.

[4] E.P.C. Kao, "'A multi-product dynamic lot size model with individual and joint setup costs", Operations Res. 26, 279-289 (1979).

[5] S.F. Love, "A facilities in series inventory model with nested schedules", Managemant Science 18, 327-338 (1971).

[6] S. Rajagopalan and G. Cornuejols, "Dynamic lot size models in multi stage assembly systems", Technical Re- port, GSIA, Carnegie Mellon Univ., Pittsburgh, 1988.

[7] L.B. Schwarz, "A simple continuous review deterministic one-warehouse N-retailer inventory problem", Manage- ment Science 19, 555-566 (1973).

[8] A.F. Veinott, "Minimum concave cost solutions of Leontief substitution models of multi-facility inventory systems", Operations Res. 17, 262-291 (1969).

[9] H.M. Wagner and T.M. Whitin, "Dynamic version of the economic lot size model", Management Science 5/1, 89-96 (1958).

[10] W.I. Zangwill, "A deterministic multi-product multi-facil- ity production and inventory model", Operations Res. 14, 486-507 (1966).

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