Dynamic Lot Size Problem with Multiple Customers:Customer-Specific Shipping and Backlogging Costs

Suresh ChandKrannert Graduate School of Management,

Purdue University,West Lafayette, Indiana 47907

Email: suresh@mgmt.purdue.edu

Vernon Ning HsuSchool of Management,

George Mason University,Fairfax, Virginia 22030Email: vhsu@gmu.edu

Suresh SethiSchool of Management,

University of Texas at Dallas,Richardson, Texas 75083Email: sethi@utdallas.edu

Vinayak DeshpandeKrannert Graduate School of Management,

Purdue University,West Lafayette, Indiana 47907

Email: deshpandev@mgmt.purdue.edu

February 16, 2004

Dynamic Lot Size Problem with Multiple Customers:Customer-Specific Shipping and Backlogging Costs

Abstract

This paper considers a dynamic lot size problem faced by a producer who supplies

a single product to multiple customers. Characterized by their backorder costs as well

as shipping costs, the customer with a high backorder cost is the one whose need for the

product is more critical than the need of a customer with a low backorder cost. We show

that the general problem with time-varying, customer-dependent backlogging and shipping

costs is NP-hard in the strong sense. We then develop an efficient dynamic programming

algorithm for an important instance of the problem when there is no speculative motive for

backlogging. We also establish forecast-horizon results for the case of stationary production

and shipping costs, which help the decision maker determine proper forecast horizon in a

rolling-horizon planning process.

Subject classifications: dynamic lot size model, dynamic programming, forecast horizon,

multiple customers.

1. Introduction

We consider a lot sizing problem for a producer who supplies a single product to

multiple customers. Characterized by their backorder costs as well as shipping costs, the

customer with a high backorder cost is the one whose need for the product is more critical

than the need of a customer with a low backorder cost. The demands from m customers

are deterministic but time-varying over an n-period horizon. In addition to a production

function, which consists of a fixed cost and a variable cost, and a unit inventory holding

cost in each period, we also consider time-varying backlogging and shipping costs that may

be different for different customers. It is assumed that a customer expects shipment for

one period at a time except in those periods when demands have been backlogged. As a

result, the shipment in a period is either equal to the demand in that period or equal to

the sum of demands in periods for which demands have been backlogged. This assumption

is not unrealistic; most customers nowadays are reluctant to carry any excess inventories.

In addition, we assume that the producer is responsible for shipping charges.

With customer-dependent backlogging and shipping costs, our model is a significant

generalization of the traditional Wagner-Whitin (WW) model (Wagner and Whitin 1958).

Since the seminal work of WW, there has been a rich body of literature on various ex-

tensions and generalizations of the WW model, collectively known as the dynamic lot size

(DLS) (or economic lot size) problems. These extensions include DLS problems with back-

logging, production capacity limitation, learning in set-ups, perishability, and lost sales.

We refer readers to Aggarwal and Park (1993) and Hsu (2000) for lists of references on

some of the DLS problems.

Vast majority of single-product DLS problems assume an aggregate demand in each

period, making them unsuitable for the situation where the producer directly markets and

distributes his product to the customers – a supply chain strategy that has increasingly

gained popularity in the recent decade. With direct shipping, the costs to satisfy de-

mands from different customers may differ significantly due to different shipping charges.

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In addition, if backordering is allowed, different customers may have varying degree of

tolerance for temporary product shortage, resulting in different backlogging costs. To our

knowledge, there has not been a DLS model in the literature that explicitly considers

customer-dependent backlogging and shipping costs as proposed in this study.

A production and inventory planning problem that is related to our problem is the

one warehouse, multi-retailer (OWMR) problem studied in Roundy (1985), Muckstadt

and Roundy (1987), and Arkin, Joneja and Roundy (1989). In this problem, production

orders at the warehouse (producer) are used to satisfy demands from multiple retailers

(customers). However, the OWMR problem is different from ours in the following as-

pects. Firstly, the former includes warehouse and retailers in a two-echelon system where

inventory can be held at both warehouse and retailer levels. The OWMR problem there-

fore needs to derive inventory replenishment policies at both echelons of the system. Our

problem, however, considers only the producer’s optimal inventory replenishment policy as

customers are treated as external entities. Secondly, backlogging is typically not included

in the OWMR problem, while it is considered explicitly in our model.

Our problem is also related to the area of providing differentiated service to different

customer classes (e.g. yield management). Deshpande et al. (2003a, 2003b) provide exam-

ples from the military where customers have different service (e.g. fillrate) requirements

or differing backorder costs. Kleijn and Dekker (1998) provide an overview of inventory

systems with several customer demand classes, including examples ranging from airlines to

petrochemical companies. This area of research has primarily focussed on analyzing nested

threshold rationing policies under stationary stochastic demand and stationary backorder

costs. In contrast, we focus on the deterministic demand case, but with non-stationary

demand, non-stationary and differing backorder costs and shipment costs.

In this paper we present a DLS problem with customer-dependent, time-varying back-

logging and shipping costs. We show that the general problem is NP-hard in strong sense.

We then develop a polynomial-time dynamic programming (DP) algorithm to solve an

2

important instance of the problem when there is no speculative motive for backlogging.

We also establish forecast-horizon results for an instance of the problem with stationary

variable production and shipping costs.

Forecast-horizon results have been established for many DLS problems as well as for

other planning problems in the literature. We refer readers to a recent survey paper by

Chand, Hsu and Sethi (2002) for a fairly comprehensive list of research papers on the

subject. In many planning situations, the management is given the demand and cost pa-

rameters for the next n periods. The demands in period n + 1 and beyond are unknown.

In solving such multi-period problems, management usually implements decisions on a

rolling-horizon basis. That is, the management will solve the n-period problem, but imple-

ment only the first decision. After one period, the management will update the problem

by dropping the first period and adding period n+1. The management will solve this new

n-period problem and again implement the first decision.

In such an environment, the management would like to choose the problem horizon n

in such a way that the first decision is insensitive to any demands and costs in periods n+1

and beyond. The problem horizon n such that the first decision is optimal for any problem

with horizon longer than n for any demand and costs in periods n+1 and beyond is called

the forecast horizon. The forecast-horizon results help the management in determining if

period n is a forecast horizon.

The rest of the paper is organized as follows. Section 2 presents the general model

and some of its properties (including its NP-hardness). Section 3 develops an O(mn2)-time

DP algorithm to solve the problem with no speculative motive for backlogging. Section 4

establishes the forecast horizon results. Section 5 concludes the paper.

2. The General Model and Some Properties

2.1. Problem Formulation

The following notation is used in our model:

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m = total number of different customers, which are indexed as i = 1, . . . ,m;

n = total number of time periods in the planning horizon, which are indexed as

j = 1, . . . , n;

dij = demand from customer i in period j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n;

fj = the setup cost for the production in period j, where 1 ≤ j ≤ n;

pj = unit production cost in period j, where 1 ≤ j ≤ n;

hj = unit holding cost in period j, where 1 ≤ j ≤ n;

sij = unit cost to ship the product in period j to customer i, where 1 ≤ i ≤ m and

1 ≤ j ≤ n;

bij = unit backlogging cost in period j for customer i, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

The decision variables of the problem are:

Xj = units of the product produced in period j, where 1 ≤ j ≤ n;

Ij = units of inventory held in period j, where 1 ≤ j ≤ n;

Zij = units of backorder in period j for customer i, where 1 ≤ j ≤ n and 1 ≤ i ≤ m;

Yij = units of product shipped in period j to satisfy demands from customer i, where

1 ≤ i ≤ m and 1 ≤ j ≤ n.

A period j is called a production period if Xj > 0.

Without loss of generality, we assume that the lead-time to satisfy a demand is zero. In

each period, production and/or shipments, if any, take place at the beginning of the period.

We also assume zero inventory at the beginning of period 1 (I0 ≡ 0) and no backorder at

the begin of period 1 and at the end of period n (i.e., Zi0 = Zin ≡ 0, 1 ≤ i ≤ m). Defining

δ(x) ={

1, if x > 0,0, otherwise,

our DLS problem with multiple customers can be formulated as the following mathematical

program:

minimizen∑

j=1

[fjδ(Xj) + pjXj + hjIj +

m∑

i=1

bijZij +m∑

i=1

sijYij

](1)

4

subject to : Ij = Ij−1 + Xj −m∑

i=1

Yij , 1 ≤ j ≤ n (2)

Yij + Zij − Zi,j−1 − dij = 0, 1 ≤ i ≤ m, 1 ≤ j ≤ n (3)

Ij , Xj , Yij , Zij ≥ 0 1 ≤ i ≤ m, 1 ≤ j ≤ n. (4)

In each period j, the total supply of the product is given by Ij−1 + Xj . Constraint

(2) shows that by shipping∑m

i=1 Yij units in period j to meet customer demands, the

remaining amount of inventory (Ij) is carried to the next period. For each customer i, the

unfilled demand at the beginning of period j is Zi,j−1 + dij . Constraint (3) says that this

unfilled demand should either be filled by a shipment (Yij) to customer i in period j, or

by backordering to the next period (Zij). Constraint (4) is the non-negativity constraint.

The objective of our problem is to minimize the total production, inventory, backordering,

and shipping costs. We will denote this problem as MP.

It is easy to show that MP is equivalent to a minimum concave cost network flow

problem on a specially constructed network G with a single source node and multiple

destination nodes, each corresponding to a demand dij from customer i in period j. An

example of the network G that corresponds to a 4-period, 2-customer problem is shown in

Figure 1.

————————————–

Insert Figure 1 about here

————————————–

We will later use this network transformation to derive some properties of optimal solution

for MP.

With customer-dependent backlogging and shipping costs, MP is a significant gen-

eralization of the classical Wagner-Whitin DLS model that captures various management

realities from the real-world applications. We will demonstrate this in the following exam-

ple.

Example 1. Consider an instance of MP with 5 periods and 2 customers. Customers

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1 and 2 have demands of 2 and 5 in each period, respectively. The costs are as follows:

pj = hj = 1, 1 ≤ j ≤ 5; (fj) = (1, 10, 10, 4, 1); (b1j) = (1, 1, 1, 2); (b2j) = (3, 3, 3, 3);

(s1j) = (1, 4, 1, 4, 1); and (s2j) = (4, 2, 4, 2, 4). The optimal solution to this problem is

to setup production in periods 1, 4 and 5. The production in period 1 is used to satisfy

demands d11, d12, d13; and d21, d22. The production in period 4 is used to satisfy demands

d23 and d24. The production in period 5 is used to satisfy demands d14, d15; and d25. The

total costs is 108. This optimal solution is shown in Figure 2 with flows on dotted lines in

the network.

————————————–

Insert Figure 2 about here

————————————–

We observe at least two phenomena from the above solution that do not exist in

classical DLS models. First, note that the demand d12 is met not in period 2 where

inventory is available; instead, it is backordered and met in period 3 by the inventory

which is produced in period 1. This happens because of the non-stationary shipping costs.

This is clearly realistic in the real world applications. For example, a firm may contract a

third party shipper who provides a fixed outbound shipping schedule. In another example,

a firm may ship its product through a transportation service provider who charges varying

shipping rates in different periods depending on its transportation demand, and therefore

its ability to consolidate cargos in each period.

Secondly, note that in period 4, the demand d14 is backordered, instead of being filled

from production in period 4. This could happen because of two reasons: the first, which

has been discussed earlier, is that the shipping cost to customer 1 may be too expensive

in that period; the second is due to the fact that different customers have different levels

of tolerance for backordering. In the solution above, unlike customer 1 whose demand in

period 4 is backordered, customer 2 has no patience to wait for one more period. As a

result, customer 2’s demand is filled in period 4 from the production in period 4. Thus,

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the producer has to setup a production just for customer 2, though it is clear that period

4 is not the cheapest period to produce.

We note that none of the above two phenomena occurs in any optimal solution to the

WW model, where a demand is always satisfied whenever the product is available.

2.2. Some Properties of MP

The following theorem implies that it is unlikely that there is an algorithm to solve

MP efficiently with a complexity that is a polynomial function of both n and m. It

is also unlikely that even a pseudo-polynomial time algorithm exists. (All proofs are in

Appendix.)

Theorem 1. MP is NP-hard in the strong sense.

Recall that MP is equivalent to a minimum cost network flow problem on G. It

is well-known that there is an optimal solution to the single-source minimum concave

cost network flow problem with at most one positive incoming flow into each node (see

Zangwill 1968). Applying this result to our problem, we obtain the following properties to

the optimal solution of MP:

Property 1. There exists an optimal solution {X∗, Y ∗, Z∗, I∗} to problem MP such that

(a) If there is a positive production in period t (X∗t > 0), then the inventory at the end of

period t− 1 is zero (I∗t−1 = 0);

(b) For fixed i and j, 1 ≤ i ≤ m, 1 ≤ j ≤ n, the demand dij is either satisfied entirely by

a shipment in period j, or completely backordered to the next period j + 1;

(c) For fixed i, 1 ≤ i ≤ m, if demand dik, 1 ≤ k ≤ n− 1, is satisfied by a shipment in a

future period t, where k < t, then all demands dir, k ≤ r ≤ t, are satisfied by the same

shipment in period t.

3. No Speculative Motive in Backlogging

Before we consider the problem with no speculative motive in backlogging, first note

7

that if backlogging is not allowed, we can drop the shipping cost from the objective func-

tion as it does not vary with decisions. We can then aggregate demands from m cus-

tomers in each period into a single demand. This will transform the problem into the WW

model, which is solvable in O(n log n)-time by algorithms developed by Aggarwal and Park

(1993), Federgruen and Tzur (1991), and Wagelmans et al. (1992). The overall compu-

tational complexity to solve the instance of MP with no backlogging allowed, including

pre-processing and transformation efforts discussed above, is O(mn + n log n).

It is clear that the difficulty of solving the general MP is mainly due to the customer-

dependent backlogging and shipping cost structure. However, we observe that in real

world applications, very often the penalty cost of backordering customer demand is so

undesirable that there is no economic incentive for the producer to carry backorders if

they can be filled immediately from current production. This is a situation known in the

DLS literature as no speculative motive for backlogging. In the context of our problem, it

implies the following assumptions on the cost parameters: For any i, 1 ≤ i ≤ m, and t,

1 ≤ t ≤ n− 1,

pt + sit ≤ pt+1 + si,t+1 + bit (5)

and

sit ≤ si,t+1 + ht + bit. (6)

Assumption (5) says that the variable production and shipping costs to satisfy a demand

in period t is no more than the variable production, shipping and backlogging costs to

satisfy the same demand, which is backordered, in the next period t + 1. Assumption (6)

means that the producer does not have any economic incentive to backorder a demand in

a period t if there is on-hand inventory to fill it.

Denote BP as the instance of MP that satisfies conditions (5) and (6). We will show

that BP can be efficiently solved by a polynomial-time DP algorithm. First, we state the

following two lemmas.

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Lemma 1. Under condition (5), if there is production in period t, then there is a shipment

in period t to meet all unfilled demands in period t.

Lemma 2. Under condition (6), if there is inventory available in period t, then there is a

shipment in period t to meet all customer demands in period t.

By these two lemmas and Property 1, we state the following property for BP:

Property 2. There is an optimal solution to BP that satisfies Property 1 and the following

property: Given two consecutively indexed production periods s and t, s < t, for any

customer i, 1 ≤ i ≤ m, there is an index ki, s ≤ ki < t, such that demands dis, . . .,

di,ki are satisfied by production in period s; and demands di,ki+1, . . ., dit are satisfied by

production in period t. In particular, demands in a production period are met from the

production in that period.

Based on Property 2, we now give the following DP recursion to solve problem BP.

First, we define a new period n + 1 with di,n+1 = 1, 1 ≤ i ≤ m, fn+1 = pn+1 = 0, and

hn+1 = bn+1 = +∞. Period n + 1 will always be a production period that serves demands

in that period only and contributes 0 value to the objective function value of problem BP.

For any j, 1 ≤ j ≤ n + 1, let V (j) be the minimum cost to satisfy demands from

periods 1 to j, provided that period j is a production period. It is clear that the optimal

objective value for BP is given by V (n + 1). For each pair of consecutive production

periods k and j, 0 ≤ k < j ≤ n+1, denote Ti(k, j) as the minimum variable cost to satisfy

demands from customer i in periods k +1 through j. We have the following DP recursion:

V (0) ≡ 0 and for 1 ≤ j ≤ n + 1,

V (j) = min0≤k<j

{V (k) + fj +m∑

i=1

Ti(k, j)}. (7)

We now discuss the computation of Ti(k, j) values. For notational simplicity, we will

drop the index i (the index for customers) and therefore dir, bir, and sir become dr, br, and

sr, respectively. Given three indices k, l and j, where k ≤ l < j, suppose that demands

9

in l + 1 through j are satisfied by production in period j (with a total variable costs of

B(l, j)); and if l > k, demands in periods k + 1 through l are satisfied by production in

period k (with a total variable costs of H(k, l)). Note that by Property 2, demands in every

period r, k + 1 ≤ r ≤ l, are satisfied by shipments in the same period. Denote S(k, l, j) as

the total variable cost to satisfy the demand (from customer i) in periods k + 1 through

j. Defining for h ≤ t, Diht = Dht =

∑tr=h dr, where

∑tr=h(·) ≡ 0 if t < h, we have

H(k, l) =l∑

r=k+1

srdr + pkDk+1,l +l−1∑

r=k

hrDr+1,l, (8)

B(l, j) = sjDl+1,j + pjDl+1,j +j−1∑

r=l+1

brDl+1,r, (9)

and

S(k, l, j) = H(k, l) + B(l, j).

For a fixed customer index i, 1 ≤ i ≤ m, we have

Ti(k, j) = T (k, j) = mink≤l<j

S(k, l, j). (10)

The following example illustrates the above DP algorithm.

Example 2. Consider the 5-period, 2-customer problem in Example 1. We will keep all

parameters of the problem the same except for the shipping costs for customer 1, which

are changed to (s1j) = (2, 1, 2, 1, 2). It is easy to verify that the modified problem satisfies

(5) and (6) and, therefore, it is an instance of BP. Starting with V (0) = 0, we have

V (1) = V (0) + 1 + 31 = 32;

V (2) = 10 + min{V (0) + 55, V (1) + 19} = V (1) + 10 + 19 = 61;

V (3) = 10 + min{V (0) + 144, V (1) + 57, V (2) + 31} = V (1) + 10 + 57 = 99;

V (4) = 4 + min{V (0) + 158, V (1) + 81, V (2) + 55, V (3) + 19} = V (1) + 4 + 81 = 117;

10

V (5) = 1 + min{V (0) + 333, V (1) + 142, V (2) + 102, V (3) + 59, V (4) + 31}

= V (4) + 1 + 31 = 149.

Finally, it is obvious that

V (6) = V (5) + 0 = 149.

The optimal solution is to produce in periods 1, 4 and 5. The production in period 1 covers

demands from both customers in periods 1 and 2; the production in period 4 covers all

demands in periods 3 and 4; and the production in period 5 satisfies demands in the same

period.

We now analyze the computational complexity of the DP recursion (7). First note

that for 1 ≤ k ≤ l − 1 ≤ n + 1,

H(k, l) = H(k, l − 1) + (l−1∑r=1

hr −k−1∑r=1

hr)dl + (pk + sl)dl;

and for n + 1 ≥ j > l + 1 ≥ 1,

B(l, j) = B(l + 1, j) + (j−1∑r=1

br −l∑

r=1

br)dl+1 + (pj + sj)dl+1.

Hence after spending O(n)-time to pre-compute quantities {∑tr=1 br | 1 ≤ t ≤ n + 1} and

{∑tr=1 hr | 1 ≤ t ≤ n + 1}, all {H(k, l) | k ≤ l ≤ n + 1} for fixed k can be obtained in

O(n)-time. Similarly, for fixed j, all {B(l, j) | 1 ≤ l < j} can be obtained in O(n)-time.

The overall effort to pre-compute {H(k, l)} and {B(l, j)} for all 1 ≤ k ≤ l < j ≤ n + 1,

and for all customers is therefore bounded by O(mn2). After this pre-computation step,

each S(k, l, j) can be evaluated in constant time when needed. Each Ti(k, j) can then be

obtained in O(n)-time via (10).

We conclude that it takes a total of O(mn3)-time to compute all {Ti(k, j) | 1 ≤ i ≤m, 1 ≤ k < j ≤ n+1}. Once these quantities are available, each V (j), 1 ≤ j ≤ n+1, in (7)

can be evaluated in O(mn)-time. Therefore, the overall complexity of the DP algorithm,

if implemented in the above straight-forward way, is bounded by O(mn3).

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For m = 1, we get O(n3), which is the same as in Blackburn and Kunreuther (1974)

for the classical Wagner-Whitin DLS problem with backlogging. Morton (1978) improves

Blackburn and Kunreuther’s complexity to O(n2). We now use a similar approach to

reduce the complexity of our DP algorithm to O(mn2).

Note that in the straight-forward approach outlined earlier, the bottlneck of the com-

putational complexity occurs at computing quantities {Ti(k, j)} via (10), which requires

O(mn3)-time. We now show that this computation effort can be reduced to O(mn2)-time.

For fixed k and j, where 1 ≤ k < j ≤ n + 1, define

lkj = arg mink≤l<j

S(k, l, j).

It is easy to verify that lkj is the smallest index l, k ≤ l < j, such that the variable cost

(pj + sj +∑j−1

r=l br) of satisfying dl from production in j (via a shipment in period j) is

no larger than the variable cost (pk + sl +∑l−1

r=k hr) of satisfying dl from production in k

(via a shipment in period l).

Lemma 3. For a fixed k, 1 ≤ k ≤ n, lkj ≤ lk,j+1.

For a fixed k, 1 ≤ k < n+1, by Lemma 3, we can obtain all indices lkj , k+1 ≤ j ≤ n+1,

in O(n)-time by scanning from k to n + 1. The total efforts to find all {lkj | 1 ≤ k < j ≤n + 1} for all customers is therefore bounded by O(mn2). Once these indices are found,

each Ti(k, j) can be evaluated in constant time via

Ti(k, j) = T (k, j) = H(k, lkj) + B(lkj , j).

(Recall from earlier discussions that, after an O(mn)-time pre-processing step, each H(k, l)

and B(l, j) can be computed in constant time.) It follows that the overall complexity of

our DP algorithm is reduced to O(mn2).

4. Forecast Horizon Results

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In this section, we provide a sufficient condition for a period n to be a forecast horizon

as defined in Section 1. To present this result, we assume that the variable production

cost is time-invariant; that is, pt = p, a constant for values of t. We also assume that the

shipping cost is also time-invariant, i.e., sit = si, 1 ≤ i ≤ m. With these assumptions,

the variable production cost and the shipping costs can be dropped from the objective

function.

Define a P (n)-problem as the n-period problem with the following assumptions: (a)

Iin = 0; and (b) Zin ≥ 0, for 1 ≤ i ≤ m. It is easy to see that an optimal solution to

the P (n)-problem is optimal for the first n periods for the (n + 1)-period problem with

a production in period n + 1. Let L(n) denote the last production period in an optimal

solution to the P (n)-problem. We prove that L(n) is monotonically non-decreasing in n.

Lemma 4. For di,n+1 ≥ 0, 1 ≤ i ≤ m, there is an optimal solution to the P (n+1)-problem

such that L(n + 1) ≥ L(n).

Define a C(n)-problem as the n-period problem with In = 0 and Zin = 0 for 1 ≤ i ≤ m,

and let K(n) denote the last production period in an optimal solution to C(n). The

difference between problems C(n) and P (n) is that in C(n), demands in periods K(n) + 1

through n are met by the production in K(n); while in P (n), demands in periods L(n)+1

through n are satisfied either by production in period L(n) or by backordering to period

n + 1, whichever is cheaper. Using this observation, we show that K(n) ≥ L(n).

Lemma 5. Given any solution to P (n) with L(n) being the last production period, there

is an optimal solution to the C(n)-problem such that K(n) ≥ L(n).

The next two lemmas will lead to our forecast-horizon result.

Lemma 6. The P (t)-problem for any t ≥ n + 1 and any demands in periods n + 1 to t

has an optimal solution with a production in the interval {L(n), L(n) + 1, . . . , n}.

Lemma 7. The C(t)-problem for any t ≥ n + 1 and any demands in periods n + 1 to t

has an optimal solution with a production in the interval {L(n), L(n) + 1, . . . , n}.

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The Forecast Horizon Theorem. If there is a first decision Θ such that Θ is optimal

for all P (k)-problems with k in the set {L(n)− 1, L(n), . . . , n− 1}, then Θ is optimal for

all C(t)-problems with t ≥ n + 1 and arbitrary demands in periods n + 1 and beyond.

Note that the period n that satisfies the conditions of the Forecast Horizon Theorem

is called a forecast horizon. Furthermore, each P (k)-problem can be efficiently solved by

the DP algorithm developed in the last section.

5. Conclusion

This paper presents a DLS problem in which a producer needs to meet demands from

multiple customers with customer-dependent backlogging and delivery costs. We show

that the general problem is NP-hard in strong sense, and propose an efficient polynomial

time DP solution for an important version of the problem with no speculative motive

for backlogging. We also establish forecast horizon results that will help a manager to

determine proper forecast horizon in a rolling-horizon planning process.

Possible future research on the subject include heuristic solutions to solve the general

problem. It would also be of interest to solve the problem under more general cost settings,

for example, considering the transportation cost structures studied in Chan et al. (2000)

and Li, Hsu and Xiao (2003).

Appendix

Proof of Theorems 1:

We will transform the Exact Cover by 3-Sets (X3C) problem to MP. Given a set

A = {a1, a2, . . . , a3q} and a collection C = {A1, A2, . . . , Ar} of 3-element subsets of A, the

X3C problem asks whether there exists a subcollection C ′ ⊆ C such that every element of

A occurs in exactly one member of C ′. The X3C problem is known to be NP-hard in the

strong sense (Garey and Johnson 1979).

14

Given an arbitrary instance of X3C, we construct a corresponding instance of MP as

follows: Let m = 3q, n = r + 1, and let

fj = 1, 1 ≤ j ≤ r, and fr+1 = 0;

pj = 0, hj = M, 1 ≤ j ≤ n;

bij = 0, 1 ≤ i ≤ m, 1 ≤ j ≤ n;

di1 = 1, dij = 0, 1 < j ≤ n, 1 ≤ i ≤ m;

sij ={

0, if ai ∈ Aj and j 6= r + 1,M, otherwise, 1 ≤ i ≤ m, 1 ≤ j ≤ n,

where M is any integer greater than |C ′| = q. Note that each customer only has one unit

of demand which occurs in period 1. Obviously, the above construction can be done in

polynomial time. We will show that there exists C ′ ⊆ C such that every element of A

occurs in exactly one member of C ′, if and only if there exists a solution to MP with total

cost no more than q.

Suppose there exists C ′ ⊆ C such that every element of A occurs in exactly one

member of C ′. For each Aj ∈ C ′, let Aj = {al1 , al2 , al3}, where l1 < l2 < l3. Then for

1 ≤ i ≤ m and 1 ≤ j ≤ n, we set Iij = 0 and

Xj ={

3, if Aj ∈ C ′,0, otherwise;

Zit ={

1, if ai ∈ Aj , 1 ≤ t ≤ j − 1 and Aj ∈ C ′,0, otherwise;

Yij ={

1, if ai ∈ Aj and Aj ∈ C ′,0, otherwise.

Because every element of A occurs in exactly one member of C ′, we see that each

demand di1 = 1 is satisfied by production in a period j, where ai ∈ Aj . The total cost

of the above solution is the setup cost incurred in the q periods corresponding to the q

members of C ′.

15

Conversely, if there exists a solution to MP with total cost no more than q, then we

know that each demand di1 is satisfied by production in a period j, where ai ∈ Aj , for

otherwise, the total costs of the problem would be at least M > q. Thus, let

J = {j | Xj > 0; 1 ≤ j ≤ r}

and

C ′ = {Aj | j ∈ J}.

We see that every element of A occurs in one of members of C ′, which implies that |C ′| ≥ q.

On the other hand, it is clear that |J | ≤ q. We therefore conclude that |C ′| = q, and every

element of A occurs in exactly one member of C ′.

This completes the proof of Theorems 1.

Proof of Property 1:

According to Zangwill (1968), there is an optimal solution such that there is at most

one positive incoming flow into a node. The nodes that correspond to production will

be called production nodes (labelled as Pj in Figure 1) and the nodes that correspond to

customers demands will be called customer nodes (labelled as Cij in Figure 1). Note that

inventory can be held only on the production nodes.

If (a) does not hold, there is a production node t with X∗t > 0 and I∗t−1 > 0. Then

node t has two positive inflows, which violates the Zangwill’s property.

If (b) does not hold, then a customer node would have two positive inflows, one from

shipment in that period and another one from backlogging from a future period.

To show (c), note that if dik is met from production in a future period t, then there

are positive flows in arcs (Pt, Cit) and (Cij, Ci, j-1), for all k + 1 ≤ j ≤ t (see Figure 1).

For any r, k ≤ r ≤ t, we see that dir can only be met by a single positive flow entering

node Cir, that carries the production from period t which is shipped in period t (via arc

(Pt, Cit)).

16

Proof of Lemma 1:

The variable cost of meeting any unfilled demand from a customer i from production

in period t is pt + sit. If this demand is not met from production in period t, then it

must be met from production in a future period. From condition (5), the variable cost of

meeting this demand from production in any future period is not less than pt + sit.

Proof of Lemma 2:

The cost of meeting a demand dit from inventory in period t is sit. Carrying the

inventory to a future period and then meet the demand would not cost less than sit.

Proof of Property 2:

Suppose s < t are two consecutively indexed production periods. First by Lemma 1,

all demands in periods s and t are met from production in periods s and t, respectively.

For an arbitrary customer i, suppose ki is the largest index k, s ≤ k < t, such that dik is

met from production in s. By Lemma 2, demand dir, k ≤ r ≤ ki, is met by a shipment of

the available inventory in period r. Finally, for any r, ki < r ≤ t, dir is backordered and,

by Lemma 1, is met in period t by a shipment in period t.

Proof of Lemma 3:

Suppose lk,j+1 > lkj . By the definition of lk,j+1,

pj+1 + sj+1 +j∑

r=lk,j+1

br ≤ pk + slk,j+1 +lk,j+1−1∑

r=k

hr.

By (5), we have

pj + sj +j−1∑

r=lk,j+1

br ≤ pj+1 + sj+1 +j∑

r=lk,j+1

br ≤ pk + slk,j+1 +lk,j+1−1∑

r=k

hr.

Since by assumption lk,j+1 > lkj , the above is a contradiction to the definition of lkj .

Proof of Lemma 4:

Given two optimal solutions S(P (n)) and S(P (n + 1)) to P (n) and P (n + 1), respec-

tively, let us denote V (P (n)) and V (P (n + 1)) as their corresponding optimal objective

function values. Suppose L(n + 1) < L(n).

17

Let Sn(P (n + 1)) be the first n period decisions in the solution S(P (n + 1)), and

Vn(P (n+1)) be its corresponding objective function value. It is obvious that Sn(P (n+1))

is a feasible solution to P (n) and therefore,

Vn(P (n + 1)) ≥ V (P (n)). (A1)

Furthermore, since L(n + 1) < L(n) ≤ n, we have

V (P (n + 1)) = Vn(P (n + 1)) +m∑

i=1

min{n∑

r=L(n+1)

hr, bi,n+1}di,n+1, (A2)

where the last term is the minimum variable costs to satisfy demands in period n + 1 by

either the production from period L(n + 1) or backordering, whichever is cheaper.

On the other hand, we can construct a feasible solution to P (n + 1) from solution

S(P (n)) by keeping all decisions in periods 1 through n unchanged and satisfying each

di,n+1 by either production in period L(n) or backordering it, whichever is cheaper. The

constructed solution has the objective function value of

Vn+1(P (n)) = V (P (n)) +m∑

i=1

min{n∑

r=L(n)

hr, bi,n+1}di,n+1.

By (A1)-(A2) and L(n + 1) < L(n), we have

Vn+1(P (n)) ≤ Vn(P (n + 1)) +m∑

i=1

min{n∑

r=L(n)

hr, bi,n+1}di,n+1

≤ Vn(P (n + 1)) +m∑

i=1

min{n∑

r=L(n+1)

hr, bi,n+1}di,n+1 = V (P (n + 1)),

which implies that the constructed feasible solution to P (n+1) is also its optimal solution.

Note that this optimal solution has the last production in period L(n). This completes

the proof of the lemma.

Proof of Lemma 5:

18

Given two optimal solutions S(P (n)) and S(C(n)) to P (n) and C(n), respectively,

denote V (P (n)) and V (C(n)) as their corresponding optimal objective function values.

Suppose K(n) < L(n).

We first construct a feasible solution to C(n) from S(P (n)) as follows: (a) Keep

decisions in periods 1 through L(n) unchanged; (b) Satisfy all demands in periods L(n)+1

through n by production in period L(n). Denoting V̂ (P (n)) as the corresponding objective

function value, we have

V̂ (P (n)) ≥ V (C(n)). (A3)

Also, by earlier discussion about the difference between C(n) and P (n), it is easy to verify

that

V̂ (P (n)) = V (P (n)) + ∆(P (n), L(n)), (A4)

where

∆(P (n), L(n)) =m∑

i=1

[ n∑

j=L(n)+1

( j−1∑

r=L(n)

hr −min{j−1∑

r=L(n)

hr,

n∑

r=j

bir})dij

].

Similarly, we can construct a feasible solution to P (n) from S(C(n)) as follows: (a)

Keep decisions in periods 1 through K(n) unchanged; (b) Satisfy each demand in periods

K(n) + 1 through n either by production in period K(n) or by backordering to period

n + 1, whichever is cheaper. Denoting V̂ (C(n)) as the corresponding objective function

value, we can similarly show that

V̂ (C(n)) ≥ V (P (n)) (A5)

and

V̂ (C(n)) = V (C(n))−∆(C(n),K(n)), (A6)

where

∆(C(n),K(n)) =m∑

i=1

[ n∑

j=K(n)+1

( j−1∑

r=K(n)

hr −min{j−1∑

r=K(n)

hr,

n∑

r=j

bir})dij

].

19

Since K(n) < L(n), we can easily show that ∆(C(n),K(n)) ≥ ∆(P (n), L(n)). Thus by

(A3)-(A6),

V̂ (P (n)) ≤ V̂ (C(n)) + ∆(P (n), L(n))

= V (C(n))−∆(C(n),K(n)) + ∆(P (n), L(n)) ≤ V (C(n)).

This implies that the feasible solution to C(n) constructed from S(P (n)) is also optimal.

The last production period in this constructed solution is L(n). This completes the proof.

Proof of Lemma 6:

Consider some t ≥ n + 1; we have L(t) ≥ L(n) from Lemma 2. If L(t) is in the

set {L(n), L(n) + 1, . . . , n}, then we are done. If not, then L(t) > n. The solution to

the P (L(t) − 1)-problem is a partial solution to the P (t)-problem. Now consider the last

production period in the optimal plan for the P (L(T ) − 1)-problem. If that production

period is in the set {L(n), L(n) + 1, . . . , n}, then we are done. Otherwise, it must be that

the production period is larger than n. We can continue this procedure until we reach a

point when the partial solution has a production period in {L(n), L(n) + 1, . . . , n}. This

completes the proof.

Proof of Lemma 7:

Consider some t ≥ n + 1; K(t) ≥ L(n) from Lemma 3. If K(t) is in the set

{L(n), L(n) + 1, . . . , n}, then we are done. If not, then K(t) > n. In this case, solu-

tion to the P (K(t)− 1)-problem constitutes a partial solution to the C(t)-problem. From

Lemma 4, we know that the P (K(t)−1)-problem has an optimal solution with a production

period in the set {L(n), L(n) + 1, . . . , n}. This completes the proof.

Proof of The Forecast Horizon Theorem:

A proof follows from the observation that any C(t)-problem with t ≥ n + 1 has an

optimal solution with a production period in the set {L(n), L(n)+1, . . . , n}. Thus, solution

to at least one of the P (k)-problems with k in the set {L(n)−1, L(n), . . . , n−1} provides a

20

partial solution to the C(t)-problem. If all these partial solutions agree on a first decision,

then the first decision must be optimal for the C(t)-problem. This completes the proof.

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22

f1+p1X1 f2+p2X2 f3+p3X3 f4+p4X4

h1I1 h2I2 h3I3

b11Z11 b12Z12 b13Z13

b21Z21 b22Z22 b23Z23

d11 d12 d13 d14

d21 d22 d23 d24

s11Y11 s12Y12 s13Y13 s14Y14

s21Y21 s22Y22 s23Y23 s24Y24

S

P1 P2 P3 P4

C11 C12 C13 C14

C21 C22 C23 C24

Figure 1. The Constructed Network G

d12d11 d13 d14 d15

d21 d22 d23 d24 d25

Figure 2. The Optimal Solution for Example 1