Dynamic lot sizing for discrete and continuous shipping.

Kim, DaeSoo ; Jun, Minjoon

ABSTRACT

Managing inventories has gained more significance today as managing an entire supply chain has become increasingly important. This study examines the structure of the general Wagner-Whitin (WW) dynamic lot sizing (DLS) model from a practical standpoint. Through mathematical model formulation and inventory curve graphs, we show that the general WW model does not always exactly represent DLS situations in practice. For those cases, we formulate two general DLS models with discrete and continuous shipping. For both discrete and continuous dynamic lot sizing (DDLS, CDLS) models, we analyze optimal properties of the zero inventory production (ZIP) policy related to Wagner-Whitin theorem 1 and 2 (i.e., no simultaneous ordering or production and prior inventory carryover and coverage of integer period demands) and show that the ZIP policy is not always optimal unlike the general WW model. And we develop an optimal dynamic programming approach to the DDLS and CDLS models satisfying ZIP conditions. Through numerical examples, we show a potential problem of blindly applying the WW approach to the DDLS and CDLS situations in practice.

INTRODUCTION

The dynamic lot sizing (DLS) model, often referred to as Wagner-Whitin (WW) model (Wagner & Whitin, 1958) in its basic form, concerns determining undiscounted cost-minimizing lot sizes of a product in a single uncapacitated system with dynamic demands for discrete-time, multiple periods in a finite planning horizon. This problem has received much attention, since it has served as a basis for analyzing more complex production and inventory systems. Particularly, previous studies have focused on 1) extending Wagner-Whitin's planning horizon theorem for more general cases (Blackburn & Kunreuther, 1974; Eppen, Gould, & Pashigian, 1969; Zabel, 1964), 2) developing a forecast horizon concept and procedure (Chand & Morton, 1986; Chand, Sethi, & Proth, 1990), 3) developing efficient optimal algorithms (Evans, 1985; Federgruen & Tzur, 1991; Lundin & Morton, 1975; Wagelmans, Hoesel, & Kolen, 1992), and 4) developing fast heuristic procedures and conducting comparative studies (Berry, 1972; Karni, 1985; Vollmann, Berry, & Whybark, 1992 to name a few]. For a more detailed review of relevant research, readers are referred to Federgruen and Tzur (1991).

This study revisits this classic problem from a different perspective. Unlike the previous studies, its main focus is on problem structuring related to the Wagner-Whitin's theorem 1 and 2 (i.e., no simultaneous ordering or production and inventory carryover from a prior period and integer period coverage of meeting dynamic demands). Specifically, we explore what dynamic lot sizing situations in practice are or are not exactly represented by the general Wagner-Whitin model. Through mathematical model formulation and inventory curve graphs, we show that the general WW model is an exact representation of some DLS situations, but only an approximation of other DLS cases. And for those approximate situations, we develop two general DLS models by explicitly considering the timing of satisfying demand or shipping requirements, i.e., discrete shipping and continuous shipping.

In detail, the general WW model differs from the proposed models in terms of handling demand shipping timing, holding costs, and production rates. In the WW model, inventory holding costs are based on period ending inventories and production rates are implicitly assumed to be infinite with simultaneous shipping or usage at the beginning of the instantaneous ordering or production period (see Blackburn & Kunreuther, 1974; Chand & Morton, 1986; Chand, Sethi, & Proth, 1990; Eppen, Gould, & Pashigian, 1969; Evans, 1985; Federgruen & Tzur, 1991; Lundin & Morton, 1975; Wagelmans, Hoesel, & Kolen, 1992; Wagner & Whitin, 1958; Zabel, 1964). On the contrary, the discrete (shipping version of the) DLS (DDLS) model reflects DLS situations in which inventories are accumulated at finite production rates until demand shipping occurs at the end of each time period and holding costs are captured by the area under the inventory curve. Many production and inventory systems in reality are based on this discrete shipping, including a single stage or flexible assembly system (FAS) such as Ford's tail lamp assembly plant (see Kim & Mabert, 2000; Kim, Mabert, & Pinto, 1993) and a final stage in a multi-stage repetitive manufacturing environment, and an MRP system in a job shop environment (see Vollmann, Berry, & Whybark, 1992). And the continuous (shipping version of the) DLS (CDLS) model represents situations in which production occurs at finite rates and demands are met or shipped out continuously during each period as in the classic economic production quantity (EPQ) model (see Hax & Candea, 1984). Thus, this model can also be viewed as a multiple-period, finite-horizon, dynamic demand extension of the EPQ model, which is also an extension of Schwarz's (1972) single-period, finite-horizon, constant demand EPQ model. This continuous shipping situation is typically found in component manufacturing, subassembly, and just-in-time (JIT) continuous-flow repetitive manufacturing systems in practice (see Vollmann, Berry, & Whybark, 1992). Further, in terms of the problem or model structure, it should be noted that our study examines finite production rates for its generality over infinite rates and even without finiteness, the above three DLS models are different as evident from inventory curves in Figure 1 and model structure presented next.

[FIGURE 1 OMITTED]

Due to these differences, when we directly apply the Wagner-Whitin based approach to these new DLS situations, the optimal solution is no longer guaranteed. Further, this blind application can cause more detrimental impact on an entire supply chain inventory costs. Therefore, we investigate optimal zero inventory production (ZIP) properties of the DDLS and CDLS models. The analysis reveals that in these cases the ZIP policy is not always optimal and instead the non-ZIP policy can be optimal in some situations. We also develop an optimal dynamic programming (DP) algorithm for the special cases satisfying ZIP conditions. Through numerical examples, we exemplify the cost difference among the WW, DDLS, and CDLS models. The rest of the paper is organized as problems and model formulation, optimal properties and solution approach, and numerical examples, followed by conclusions.

PROBLEMS AND MODEL FORMULATION

Both the general Wagner-Whitin (WW) dynamic lot sizing (DLS) model and the proposed discrete and continuous DLS (DDLS, CDLS) models are concerned with finding undiscounted cost-minimizing (production) lot sizes of a single product in a single uncapacitated facility with dynamic demands for discrete-time, multiple periods in a finite planning horizon. In this section, we examine structural differences of the models through mathematical model formulation and inventory curve graphs. To facilitate the discussion, we use the following notation:

t = discrete time period index

[Q.sub.t] = (production or ordering) lot size for period t

[f.sub.t] = binary {0 if [Q.sub.t]=0, 1 if [Q.sub.t]>0} setup variable in t

[I.sub.t] = inventory at the end of t

[d.sub.t] = demand or shipping requirements for t

[p.sub.t] = production rate in t

[a.sub.t] = cost per setup (or order) in t

[c.sub.t] = unit production (or purchase) cost in t

[h.sub.t] = unit inventory holding cost in t

The General Wagner-Whitin (WW) Dynamic Lot Sizing (DLS) Model

Min [[??].sub.t] ([a.sub.t][f.sub.t] + [c.sub.t][Q.sub.t] + [h.sub.t][I.sub.t]) (1)

s.t. [I.sub.t-1] +[Q.sub.t] - [d.sub.t] = [I.sub.t] [for all]t (2)

[f.sub.t] = {0, 1} [for all]t (3)

Note that the general WW model is based on the implicit assumptions that demand for period t ([d.sub.t]) is satisfied either by inventory carried over from a prior period ([I.sub.t-1]) or by order or production lot size received or produced instantaneously at the beginning of t ([Q.sub.t]) and that the period ending inventory [I.sub.t] is used in calculating inventory holding cost for period t, i.e., [h.sub.t][I.sub.t] (see Blackburn & Kunreuther, 1974; Chand & Morton, 1986; Chand, Sethi, & Proth, 1990; Eppen, Gould, & Pashigian, 1969; Evans, 1985; Federgruen & Tzur, 1991; Lundin & Morton, 1975; Wagelmans, Hoesel, & Kolen, 1992; Wagner & Whitin, 1958; Zabel, 1964). Therefore, when we closely examine the general WW model, we can find two important points. First, as easily seen from the inventory curve in Figure 1, this model is an exact representation of the situation where the lot size for period t is instantaneously received or produced at the beginning of t and demands or shipping requirements are met at the same time. Second, this model is an approximate representation of the situations where the production rate for period t is finite and demands or shipping requirements are 1) met at the end of period t as in the discrete DLS (DDLS) model or 2) met continuously during t as in the continuous DLS (CDLS) model presented next as similar to the economic order quantity (EOQ) or economic production quantity (EPQ) models.

The Discrete Dynamic Lot Sizing (DDLS) Model

Min [[summation].sub.t] [[a.sub.t][[phi].sub.t] + [c.sub.t][Q.sub.t] + [h.sub.t] ([I.sub.t-1] + 0.5 [Q.sup.2.sub.t]/[p.sub.t])] (4) s.t. (2) - (3)

This DDLS model accurately describes the first approximate situation of the DLS. That is, in this model, the production for period t occurs during t at a finite rate and demands or shipping requirements are satisfied at the end of period t (see Figure 1). Due to the finiteness of production rate, holding cost is calculated by the area under the inventory curve. Observe the difference in model formulation between [I.sub.t] in (1) of the general WW model and [I.sub.t-1] + 0.5 [Q.sup.2.sub.t]/[p.sub.t] in (4) above. Also, observe that in the ordering or purchasing situation, 0.5 [Q.sup.2.sub.t/[p.sub.t] in (4) approaches 0, as pt approaches infinity, which is still different from the general WW model in terms of the timing of inventory holding cost calculation.

The Continuous Dynamic Lot Sizing (CDLS) Model

Min [[summation].sub.t] [[a.sub.t][[phi].sub.t] + [c.sub.t][Q.sub.t] + 0.5[h.sub.t] ([I.sub.t-1.sup.2] + [Q.sup.2.sub.t](1 - [d.sub.t]/[p.sub.t]) - [I.sup.2.sub.t])/[d.sub.t]] (5) s.t. (2) - (3)

In this model, the production Qt occurs during period t at a finite rate and demands or shipping requirements [d.sub.t] is satisfied continuously during period t (see Figure 1). Given the general assumption of uncapacitated lot sizing, production rate is assumed to be far greater than demand rate and setup is assumed to be done not more than once per period and production is not continued in the next period with the current setup, although the case of multiple setups can also be easily incorporated by redefining t as a non-negative integer (0,1,2,). And holding cost is calculated by the area under the inventory curve. Observe the difference from the general WW and DDLS models, even in the ordering situation where [d.sub.t]/[p.sub.t] in (5) approaches 0 as pt approaches infinity. Further, this model can also be viewed as a dynamic demand extension of the EPQ model for multiple periods in a finite planning horizon (see Hax & Candea, 1984; Schwarz, 1972).

OPTIMAL PROPERTIES AND SOLUTION APPROACH

One of the important findings in Wagner-Whitin (1958) is that the zero inventory ordering or production (ZIP) policy (i.e., no simultaneous ordering or production and inventory carryover from a prior period from theorem 1 and the coverage of integer period requirements from theorem 2) is optimal to the DLS problem. In this section, we analyze whether optimal properties of the ZIP policy hold true for discrete and continuous dynamic lot sizing (DDLS, CDLS) models and develop an optimal dynamic programming (DP) based solution approach satisfying the ZIP policy. To facilitate the discussion, we define the following:

i,j = period indices where i < j

j1,j2 = partitioned partial period indices of j such that [d.sub.j1] + [d.sub.j2] = [d.sub.j] where j1 < j2

[d.sub.i,j] = [d.sub.i] + ... + [d.sub.j]

[h.sub.i,j] = [h.sub.i] + ... + [h.sub.j]

T[C.sub.1] = non-ZIP (NZIP) based total cost from period i to j when there is production in period i and j, covering [d.sub.i,j1] and [d.sub.j2], respectively

T[C.sub.2] = ZIP based total cost from period i to j when there is production in period i and j, covering [d.sub.i,j-1] and [d.sub.j], respectively

T[C.sub.3] = ZIP based total cost from period i to j when there is production in period i only, covering [d.sub.i,j]

[[DELTA].sub.12] = T[C.sub.1] - T[C.sub.2]

[[DELTA].sub.13] = T[C.sub.1] - T[C.sub.3]

Then, for the optimal properties of the ZIP and NZIP policies, we need to make comparisons of only [[DELTA].sub.12] and [[DELTA].sub.13].

Optimal Properties of the DDLS Model

From (4) in the DDLS model, we obtain the following total costs:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, after solving and rearranging the terms of [[DELTA].sub.12] = T[C.sub.1] - T[C.sub.2] and [D.sub.13] = T[C.sub.1] - T[C.sub.3], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where 0 < [d.sub.j1],[d.sub.j2] < [d.sub.j] = [d.sub.j1] + [d.sub.j2], [H.sub.1] = [h.sub.i][d.sub.i,j-1]/ [p.sub.i] - [h.sub.j][d.sub.j]/[p.sub.j] + [h.sub.i+1,j] > 0, [H.sub.2] = [h.sub.i]/[p.sub.i] + [h.sub.j]/[p.sub.j] > 0, H = [H.sub.1] + [H.sub.2][d.sub.j] = [h.sub.i][d.sub.i,j]/[p.sub.i] + [h.sub.i+1,j] > 0, and C = [c.sub.j] - [c.sub.i] [less than or equal to] or [greater than or equal to] 0 for i < j and j1 < j2. (8)

Theorem 1. The ZIP policy is always optimal to the DDLS, if one of the following is true:

a. C = [less than or equal to] [H.sub.1], i.e., ([c.sub.j] - [c.sub.i]) [less than or equal to] [h.sub.i] [d.sub.i,j-1]/[p.sub.i] - [h.sub.j][d.sub.j]/[p.sub.j] + [h.sub.i+1,j],

b. C [less than or equal to] 0, i.e., ([c.sub.j] - [c.sub.i]) [less than or equal to] 0 for i < j (i.e., unit production cost is non-increasing),

c. C [greater than or equal to] H, i.e., ([c.sub.j] - [c.sub.i]) = [h.sub.i][d.sub.i,j]/[p.sub.i] + [h.sub.i+1,j],

d. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Theorem 2. The non-ZIP (NZIP) policy is optimal to the DDLS, if [H.sub.1] < C < H - [(2[H.sub.2][a.sub.j]).sup.0.5], i.e., [h.sub.i][d.sub.i,j-1]/[p.sub.i] - [h.sub.j][d.sub.j]/[p.sub.j] + [h.sub.i+1,j] < ([c.sub.j] - [c.sub.i]) < [h.sub.i][d.sub.i,j]/[p.sub.i] + [h.sub.i+1,j] - [(2([h.sub.i]/[p.sub.i] + [h.sub.j]/[p.sub.j])[a.sub.j]).sup.0.5]. (10)

Proof. By inspection of (6) and (7), [[DELTA].sub.12] > 0 if [H.sub.1] - C [greater than or equal to] 0 (theorem 1a) or C [less than or equal to] 0 (theorem 1b) and [[DELTA].sub.13] > 0 if C - H [greater than or equal to] 0 (theorem 1c). Now, suppose [H.sub.1] < C < H. Let f = [(C - H).sup.2] - 2[H.sub.2][a.sub.j], i.e., the square root term of (7). Then, [[DELTA].sub.13] [greater than or equal to] 0 if f [less than or equal to] 0, i.e., H - [(2[H.sub.2][a.sub.j]).sup.0.5] [pounds sterling] C. If f > 0, i.e., C < H - [(2[H.sub.2][a.sub.j]).sup.0.5], then (7) has two roots, say, ([R.sub.2],[R.sub.1]) = ((H - C) [+ or -] [f.sup.0.5])/[H.sub.2]. Therefore, [[DELTA].sub.13] [greater than or equal to] 0, if H - [(2[H.sub.2][a.sub.j]).sup.0.5] [less than or equal to] [H.sub.1] (theorem 1d). Otherwise (i.e., if H - [(2[H.sub.2][a.sub.j]).sup.0.5] > [H.sub.1]), [[DELTA].sub.13] > 0, if H - [(2[H.sub.2][a.sub.j]).sup.0.5] [less than or equal to] C < H (theorem 1e), which in turn implies that [[DELTA].sub.13] < 0, if [R.sub.1] < [d.sub.j2] < [R.sub.2] or [H.sub.1] < C < H - [(2[H.sub.2][a.sub.j]).sup.0.5] (theorem 2).

Therefore, unlike the Wagner-Whitin DLS model, the ZIP policy is not always optimal to the DDLS. Optimal Properties of the CDLS Model

Similarly from (5) in the CDLS model, the total costs are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, after solving and rearranging the terms of [[DELTA].sub.12] and [[DELTA].sub.13], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Theorem 3. The ZIP policy is always optimal to the CDLS, if one of the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Theorem 4. The non-ZIP (NZIP) policy is optimal to the CDLS, if one of the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Proof. Consider the following four possible cases of [[DELTA].sub.12] in (11). [[DELTA].sub.12] = 0 if [d.sub.j1] = {0, R =: 2(C - [H.sub.1])/[H.sub.2]}. (i) If [H.sub.2] [greater than or equal to] 0 and R [less than or equal to] 0 (i.e., C [less than or equal to] [H.sub.1]), then [[DELTA].sub.12] [greater than or equal to] 0. (ii) If [H.sub.2] [greater than or equal to] 0 and R > 0 (i.e., C > [H.sub.1]), then [DELTA].sub.12] = 0 iff [d.sub.j1] [greater than or equal to] R. However, since min[[DELTA].sub.12] < 0 at [d.sub.j1] = min{R/2, [d.sub.j] - [epsilon]} where a is an infinitesimal value, [[DELTA].sub.12] < 0. (iii) If [H.sub.2] < 0 and R [less than or equal to] 0 (i.e., C [less than or equal to] [H.sub.1]), then [[DELTA].sub.12] < 0. (iv) If [H.sub.2] < 0 and R > 0 (i.e., C > [H.sub.1]), then [[DELTA].sub.12] = 0 iff [d.sub.j1] [less than or equal to] R. Thus, if [d.sub.j] [less than or equal to] R (i.e., C [greater than or equal to] [H.sub.1] + [H.sub.2][d.sub.j]/2), then [[DELTA].sub.12] BOL179\f"Symbol"\s12 0. Otherwise [DELTA].sub.12] < 0.

Now consider the following four possible cases of [[DELTA].sub.13] in (12). [[DELTA].sub.13] = 0 if [d.sub.j2] = {[R.sub.1] =: ((H - C) - [f.sup.0.5])/[H.sub.2], [R.sub.2] =: ((H - C) + [f.sup.0.5])/[H.sub.2]} where f = [(H - C).sup.2] - 2[H.sub.2][a.sub.j] = 0, and [[DELTA].sub.13]. [not equal to] 0 if f < 0. (v) If [H.sub.2] [greater than or equal to] 0 and f [less than or equal to] 0 [(i.e., H - [(2[H.sub.2][a.sub.j]).sup.0.5] [less than or equal to] C [less than or equal to] H + [(2[H.sub.2][a.sub.j]).sup.0.5], then [[DELTA].sub.13] = 0. (vi) If [H.sub.2] [greater than or equal to] 0 and f > 0 (i.e., C < H - [(2[H.sub.2][a.sub.j]).sup.0.5] or C > H + [(2[H.sub.2][a.sub.j]).sup.0.5]), then a) If ([R.sub.1] <) [R.sub.2] < 0 (i.e., C > H), then [[DELTA].sub.13] > 0 and so the ZIP condition for C is C > H + [(2[H.sub.2][a.sub.j]).sup.0.5]. b) If ([R.sub.2] >) [R.sub.1] > 0 (i.e., C < H), then [[DELTA].sub.13] > 0, iff [d.sub.j] [less than or equal to] [R.sub.1] (i.e., C [greater than or equal to] H - ([H.sub.2][d.sub.j]/2 + [a.sub.j]/[d.sub.j]), and so the ZIP condition is H - ([H.sub.2][d.sub.j]/2 + [a.sub.j]/[d.sub.j]) < C [less than or equal] H - [(2[H.sub.2][a.sub.j]).sup.0.5], since ([H.sub.2][d.sub.j]/2 + [a.sub.j]/[d.sub.j]) [(2[H.sub.2][a.sub.j]).sup.0.5] = [([([H.sub.2][d.sub.j]/2).sup.0.5] - ([[a.sub.j]/.sub.d]j).sup.0.5]).sup.2] [greater than or equal to] 0. (vii) The condition of [H.sub.2] < 0 and f < 0 does not exist, since f = [(H - C).sup.2] - 2[H.sub.2][a.sub.j] > 0 if [H.sub.2] < 0. (viii) If [H.sub.2] < 0 and f [greater than or equal to] 0 (i.e., H - [(2[H.sub.2][a.sub.j]).sup.0.5] [less than or equal to] C [less than or equal to] H + [(2[H.sub.2][a.sub.j]).sup.0.5]), then regardless of the two possible cases depending upon the sign of [d.sub.j2] = (H - C)/[H.sub.2] > or < 0 which maximizes [[DELTA].sub.13], [[DELTA].sub.13] [greater than or equal to] 0 iff [d.sub.j] [less than or equal to] [R.sub.2] (i.e., C [less than or equal to] H + (-[H.sub.2])[d.sub.j]/2 - [a.sub.j]/[d.sub.j]), since [R.sub.1] < 0 and [R.sub.2] > 0.

Therefore, we obtain theorem 3a (i.e., 0 [less than or equal to] [H.sub.2] [less than or equal to] 2[a.sub.j]/[d.sub.j.sup.2]) from (i), (v) and (vi), theorem 3b (i.e., [H.sub.2] > 2[a.sub.j]/[d.sub.j.sup.2] and C [less than or equal to] [H.sub.1] or C [greater than or equal to] H - ([H.sub.2][d.sub.j]/2 + [a.sub.j]/[d.sub.j])) from (i), (v) and (vi.b), theorem 3c (i.e., [H.sub.2] < 0 and C [less than or equal to] H + (-[H.sub.2][d.sub.j]/2 - [a.sub.j]/[d.sub.j]) or H - (-[H.sub.2])[d.sub.j]/2 [less than or equal to] C [less than or equal to] [H.sub.1]) from (iv) and (viii), and theorem 4a and 4b from the complementary set of theorem 3b and 3c, respectively.

Therefore, again unlike the Wagner-Whitin DLS, the ZIP policy is not always optimal to the CDLS.

Solution Approach

For the situations where any of the ZIP properties in Theorem 1 and 3 is satisfied, we can obtain the optimal solution to the DDLS and CDLS by readily applying the Wagner-Whitin type dynamic programming (DP) recursion algorithm. For the situation where the non-ZIP policy in Theorem 2 and 4 is optimal, the ZIP policy based DP recursion can serve as an efficient heuristic approach. The optimal algorithm presented below is for the cases satisfying the ZIP conditions, developed by modifying the Evans' (1985) forward DP algorithm. Now let:

[F.sub.j] = minimum cost for periods 1 through j when [I.sub.j] = 0 and

[M.sub.i,j] = cost incurred by producing in period i, covering [d.sub.i,j] for i j.

Then the forward DP recursive relations are constructed as

DP recursive relations for the DDLS model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DP recursive relations for the CDLS model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

NUMERICAL EXAMPLES

In order to illustrate a potential problem of blindly applying the Wagner-Whitin (WW) to the DDLS or CDLS situations, we compared total costs among the WW, the DDLS, and the CDLS approach, using the twelve-period demand data in Wagner and Whitin (1958). For the comparison, we generated 20 replications for non-decreasing unit holding cost per period from the uniform distribution (0, 10] with the constant unit production cost per period for simplicity, hence directly satisfying the optimal properties of the ZIP policy. Table 1 summarizes the data and solution. These examples showed some total cost difference of the DDLS (CDLS) from the WW approach for the average and the worst case being 29.53% (0.29%) and 123.47% (3.33%) deviations, respectively. Also, in 3 (14) out of 20 replications, the WW approach yielded the optimal solutions to the DDLS (CDLS). As is apparent from the computation results, the WW approach performed quite poorly in the DDLS situation than in the CDLS.

CONCLUSIONS

In this study, we investigated the structure of the general Wagner-Whitin (WW) dynamic lot sizing (DLS) model. Through a thorough examination, we showed that the WW model is not always an exact representation of real-world DLS situations with discrete and continuous shipping. For these approximate situations, we presented new discrete and continuous DLS (DDLS, CDLS) models and explored optimal properties of the zero inventory production (ZIP) policy, along with the development of an optimal dynamic programming solution approach. The analysis revealed that the ZIP policy is not always optimal and the non-ZIP policy can be optimal in some situations. And through examples, we illustrated a potential problem of blindly applying the WW approach to the DDLS and CDLS.

This study intended to provide a new insight into the DLS problem in terms of problem structuring, which can serve as a basis for more complex discrete and continuous DLS situations. Due to the practical significance of the developed models in many real production and inventory systems, it is desired that future research focuses on further analysis of developing an efficient algorithm for non-ZIP policy cases and for more complex situations dealing with limited capacity, multiple products, and/or multiple stage.

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DaeSoo Kim, Marquette University

Minjoon Jun, New Mexico State University

Table 1: Numerical Examples and Solutions
Data set:
Number of periods = 12,

Demand rate = {69, 29, 36, 61, 61, 26, 34, 67, 45, 67, 79, 56}
units/period,

Production rate = 280 units/period,

Unit production cost per period = constant,

Unit holding cost per period = generated from the uniform distribution
(0, 10],

Setup cost to unit holding cost ratio = 10, and

Number of replications = 20.

Average and Worst Case Statistics between Wagner-Whitin (WW) and
DDLS Approach:

 Average Statistics

Approach WW DDLS % difference

Setup Cost 526.65 611.32 -13.85%
Holding Cost 692.13 329.63 109.97%
Total Cost 1,218.78 940.95 29.53%

 Worst Case Statistics

Approach WW DDLS % difference

Setup Cost 300.19 513.18 -41.50%
Holding Cost 1,510.67 297.15 408.39%
Total Cost 1,810.86 810.33 123.47%

# same solutions 3 out of 20 replications

For the worst case,

lot size (WW) = {69, 29, 97, 0, 61, 26, 34, 112, 0, 146, 0, 56}

lot size (DDLS) = {69, 29, 36, 61, 61, 26, 34, 67, 45, 67, 79, 56}

Average and Worst Case Statistics between Wagner-Whitin (WW) and
CDLS Approach:

 Average Statistics

Approach WW CDLS % difference

Setup Cost 526.65 534.77 -1.52%
Holding Cost 1,335.70 1,322.27 1.02%
Total Cost 1,862.36 1,857.04 0.29%

 Worst Case Statistics

Approach WW CDLS % difference

Setup Cost 258.32 255.96 0.92%
Holding Cost 700.90 672.31 4.25%
Total Cost 959.22 928.27 3.33%

# same solutions 14 out of 20 replications

For the worst case,

lot size (WW) = {69, 29, 36, 61, 61, 26, 34, 67, 112, 0, 79, 56}

lot size (CDLS) ={98, 0, 36, 61, 61, 26, 34, 67, 45, 67, 79, 56}

Note that solution figures are subject to rounding errors and that
total costs consist of only setup and holding costs since unit
production costs are constant and thus irrelevant for comparison.