The effect of lot-sizing on workload variability

Lot-sizing models which group demand requirements for one or more consecutive time periods into a single production run have received considerable attention in recent years. Material Requirements Planning (MRP) systems must, for instance, make a lot-size decision for each planned order release. Existing decision models attempt to minimize the sum of setup plus inventory holding costs. However, lot-sizing tends to increase the work center load variability, and, consequently, the costs associated with changing production levels from period to period should be incorporated into the economic analysis. This study is concerned, first of all, with analytically describing the relationship between dynamic lot-sizing models and workload variability. Secondly, in order to account for production level change costs we propose a simple modification to existing heuristic models. Lastly, we employ a simulation model to empirically extend these results to a typical MRP multiechelon production environment. An example is included to show clearly that with cost premiums for overtime and severance or guaranteed minimum costs for undertime the traditional lot-sizing techniques significantly underestimate actual costs and can lead to very costly policies.Mean, variance and coefficient of variation of period work time requirements are derived as a function of several algorithm characteristics. Average cycle time (number of periods covered by a single batch) is found to be the most influential factor in determining workload variability. Variance grows approximately in proportion to this cycle time with the proportionality constant being the square of average period workload. Cycle time and demand variability also contribute to workload variability. Results indicate that for a given average cycle time, the EOQ method will minimize workload variability. When N products utilize the same work center, the coefficient of load variation will be reduced by a factor of N^{?$\text{1}\text{2}$} unless requirements are positively correlated. Positive correlation would result when products have similar seasons or parent items. In this case grouping such products cannot help reduce variability.In order to incorporate production level change costs into existing heuristics we may simply introduce a term consisting of a penalty factor times average cycle time. The penalty factor represents the costs of period by period production level changes. Several popular heuristics are extended in this fashion, and it is found that solutions are still readily obtainable, requiring only modifications to setup or holding cost parameters.The effects of level change costs are examined via simulation for a specific yet typical environment. It is found that when setup costs are significant, traditional lot-sizing heuristics can provide cost savings and service level improvements as compared to lot-for-lot production. However, whereas for our model the obtainable profit improvement from lot-sizing was 25% in the case of freely variable capacity, actual improvements were only one half as large when reasonable hiring and firing practices and overtime and undertime costs were considered. Consequently, management needs to consider carefully labor costs and work center product relationships when determining a production scheduling method.     

Computer simulation of lot-sizing algorithms in three-stage multi-echelon inventory systems

This paper evaluates conventional lot-sizing rules in a multi-echelon coalescence MRP system. A part explosion diagram of three levels and twenty-one nodes is simulated using FORTRAN IV Level G. Nine separate lot-sizing methods were evaluated in this analysis. These methods included Lot for Lot, Economic Order Quantity, Periodic Order Quantity, Least Total Cost, Least Unit Cost, Part-Period Balancing, the Silver-Meal Algorithm, and the Wagner-Whitin Algorithm. A hybrid rule using both the Economic Order and the Economic Production Quantity rules was also evaluated.The performance of each lot sizing rule was simulated over nine different sets of market requirements patterns over a twelve month period. The types of demand variation included a constant rate, three different patterns of normally distributed demand, a random pattern, and two cyclic patterns. A hybrid pattern was used which equally weighted components of constant, random, normal, and cyclic demand. Finally, the ninth pattern consisted of actual data obtained from a job shop manufacturing facility.Within the part explosion diagram, ratios of setup cost to carrying cost, “goes into” quantities, and lead times were assigned for each node. Assigned values were selected from uniform distributions with prespecified ranges.A computer model was developed to perform the simulation. It consisted of an executive program, a routine for data generation, and separate routines to exercise each of the different lot-sizing rules. The simulations were conducted under three operational rules. The first rule allowed for the establishment of initial inventories just large enough to “cover” those gross requirements that occurred prior to the time the first order arrived. Carrying costs for this stock were included in the computation of total costs per node. The second rule provided for the delay of application of each lot sizing rule. This avoided receiving an order in a period of zero demand. The third rule addressed the computation of costs. The total cost was computed on the basis of average inventory level and the number of required setups.The analysis required the completion of 1701 separate simulation runs (9 rules X 9 demand patterns X 21 nodes). The performance of each rule was evaluated on the basis of total annual inventory cost. The Periodic Order Quantity (POQ) rule performed best in six of the nine demand patterns analyzed. In two of the remaining three cases, it ranked second on the basis of minimizing costs.The Least Unit Cost (LUC), Least Total Cost (LTC), and Pan-Period Balancing (PPB) algorithms demonstrated identical performance in four of the demand patterns analyzed. Generally, they ranked in the upper half of the rules evaluated. The Economic Order Quantity (EOQ) and the Economic Order/Production Quantity hybrid rules performed only moderately well. On the basis of cost, the consistent worst performers were the Wagner-Whitin (WW), Silver-Meal (SM), and Lot-for-Lot (LFL) rules.It was found that gross requirements tend to occur sporadically in different levels of the system. Order policies of parent nodes often cause the policies in higher level nodes to resemble the lot-for-lot order philosophy, regardless of the rule being used. Because of this phenomenon, those rules that generate fewer orders over the planning horizon for parent nodes often tend to perform better on the basis of total inventory cost.     

An efficient implementation of the Wagner-Whitin algorithm for dynamic lot-sizing

We consider an N-period planning horizon with known demands D_t ordering cost A_t, procurement cost, C_t and holding cost H_t in period t. The dynamic lot-sizing problem is one of scheduling procurement Q_t in each period in order to meet demand and minimize cost.The Wagner-Whitin algorithm for dynamic lot sizing has often been misunderstood as requiring inordinate computational time and storage requirements. We present an efficient computer implementation of the algorithm which requires low core storage, thus enabling it to be potentially useful on microcomputers.The recursive computations can be stated as follows:

$\text{M}{}_{\mathrm{jk}}\text{=A}{}_{\mathrm{j}}\text{+C}{}_{\mathrm{j}}\text{Q}{}_{\mathrm{j}}\text{+}\text{∑}\text{k?1}\text{t=j}\text{H}{}_{\mathrm{t}}\text{∑}\text{k}\text{r=t+1}\text{D}{}_{\mathrm{r}}$

$\text{F}{}_{\mathrm{k}}\text{=}\text{min}\text{1?j?k}\text{[F}{}_{\mathrm{j}}\text{+M}{}_{\mathrm{jk}}\text{];}\text{F}{}_0\text{=0}$

where M_jk is the cost incurred by procuring in period j for all periods j through k, and F_k is the minimal cost for periods 1 through k. Our implementation relies on the following observations regarding these computations:

$\text{M}{}_{\mathrm{j,k}}\text{=A}{}_{\mathrm{j}}\text{+C}{}_{\mathrm{j}}\text{D}{}_{\mathrm{j}}$

$\text{M}{}_{\mathrm{j,k+1}}\text{=M}{}_{\mathrm{jk}}\text{+D}{}_{\mathrm{k+1}}\text{(C}{}_{\mathrm{j}}\text{+}\text{∑}\text{k}\text{t=j}\text{H}{}_{\mathrm{t}}\text{,}\text{k?j}$

Using this recursive relationship, the number of computations can be greatly reduced. Specifically, $\text{3}\text{2}$N² ? $\text{1}\text{2}$N² additions and $\text{1}\text{2}$N² + $\text{1}\text{2}$N multiplications are required. This is insensitive to the data.A FORTRAN implementation on an Amdahl 470 yielded computation times (in 10^?3 seconds) of T = ?.249 + .0239N + .00446N². Problems with N = 500 were solved in under two seconds.     

A multi-objective model for lot-sizing with supplier selection for an assembly system

Lot-sizing with supplier selection (LS-SS) is a fast-growing offspring of two major problem parents in logistics and supply chain management (‘lot-sizing’ and ‘supplier selection’). The model proposed in this paper is an attempt to extend it to an assembly system, by formulating a multi-objective model for an integrative problem of LS-SS for assembly items. The total costs of the system, consisting of purchasing, ordering, transportation, assembly, and holding, is considered the first objective function, while the total reliability of the finished products is considered the second objective function. The decision-maker aims to minimise the total costs while maximising the total reliability. Several constraints of the system (e.g. storage capacity, supplier production capacity) are taken into account. Given the complexity of the model, a heuristic evolutionary algorithm is proposed to solve the model. The results indicate which assembly items to order in which quantities, from which suppliers and in which time periods.     

A heuristic solution procedure for the multi-item,single-level,limited capacity,lot-sizing problem

The problem to be considered is that of determining lot-sizes for a group of products which are produced at a single workcentre. It is assumed that the requirements for each product are known, period by period, out to the end of some common time horizon. (A reasonable assumption in a Material Requirements Planning context when we are dealing with components of one or more other items already scheduled.) For each product there is a fixed setup cost incurred each time production takes place. Unit production and holding costs are linear. The time required to set up the machine is assumed to be negligible. All costs and production rates can vary from product to product but not with respect to time. In each period there is a finite amount of machine time available that can vary from period to period. The objective is to determine lot-sizes so that 1) costs are minimized, 2) no backlogging occurs and 3) capacity is not exceeded.An exact solution to this complex problem is out of the question. Therefore, a simple heuristic has been developed which guarantees a feasible solution, if one exists. Results of a large number of test problems, including three supplied by industrial sources, are presented. The results indicate that the heuristic will usually generate a very good solution with a relatively small amount of computational effort.     

Time-dependent demand in requirements planning: An exploratory assessment of the effects of serially correlated demand sequences on lot-sizing performa

The problem of determining the appropriate stock replenishment quantity within a time-phased requirements planning environment has received considerable research attention in recent years. Relative performance characteristics of lot-sizing policies have been assessed as a function of the cost structure, the demand pattern, the product structure, forecast error, the length of the planning horizon, and the interaction between replenishment quantities and sequencing decisions. In particular, the relationship between lot sizing behavior and variability in the requirements profile has been intensely investigated. However, despite these efforts, the empirical evidence linking lotsizing performance with demand variability remains inconclusive. This article suggests that, in part, some of the ambiguity in the literature may be an artifact of a failure to adequately control for other important dimensions of simulated demand sequences. The features that have been thought to describe “lumpy” requirements profiles are discussed and the characteristic of periodicity or time-dependency in the demand entries is identified as a variable that has been insufficiently controlled in prior work. A reanalysis of the demand sequences originally published by Kaimann, and subsequently used in a number of comparative lot-sizing studies, reveals that the patterns differ not only in variability as measured by the coefficient of variation, but also in terms of correlation structure as described by the autocorrelation function. Alternative methods for simulating demand sequences are reviewed and a correlation transfer technique, which has the capability to simultaneously control both the degree of variability and correlation, is suggested as an improved method for the generation of synthetic sequences of “lumpy” demand. Using this technique, five of Kaimann's original sequences are rearranged, resulting in three sets of sequences differing only in the strength of serial correlation. Four lot-sizing procedures are applied to each of these sets to discern if the correlation structure has any appreciable effect on lot-sizing performance. Results indicate that, on average, higher total inventory costs are experienced when the demand environment is characterized by randomness. Economic order quantity and part-period balancing achieve lowest average costs when confronted with highly autocorrelated demand or patterns of few runs; conversely, minimum cost per period and Wagner-Whitin perform best under conditions of many runs. Both economic order quantity and part-period balancing perform most favorably in comparison to Wagner-Whitin when runs are few. In addition, there appears to be a potential interaction between the level of demand variability and the degree of serial correlation. This finding is somewhat disconcerting since high variability demand sequences used in some prior research were also characterized by relatively high levels of autocorrelation; hence it becomes most difficult to identify and decompose the unique influences of each demand pattern dimension on lot-sizing behavior. Because of this phenomenon, it is suggested that future studies direct greater attention to the demand simulating methodology than has heretofore been accorded.     

The part-period balancing algorithm and its look ahead-look back feature: A theoretical and experimental analysis of a single stage lot-sizing procedure

This paper investigates the discrete Part-Period Balancing (PPB) lot-sizing algorithm and its optional feature, the Look Ahead-Look Back tests. PPB is the most commonly used dynamic lot-sizing procedure in practice and it has also been tested extensively in simulation experiments. Although its overall cost performance, relative to other heuristics, have been fairly good, a fundamental flaw with the model has been noted in the literature. This deficiency leads to poor performance under certain conditions.In this paper a simple adjustment to the main algorithm is analytically derived under the assumptions of a constant demand rate and an infinite planning horizon. The adjustment leads to an optimal behavior for the PPB heuristic under the stated conditions. Subsequent experimental analysis through simulation of lot-sizing performance in environments with time-varying, discrete demand shows that the proposed adjustment leads to significant cost reductions.This paper also analyzes the Look Ahead-Look Back tests which is the distinguishing feature between the PPB procedure and the Least Total Cost algorithm. The tests were devised to improve the cost performance of the PPB heuristic by marginally adjusting each tentative lot-size. The effect of the Look Ahead-Look Back tests have, however, never been verified in the literature. The tests have undergone some changes over time, when they have been included in commercial software packages for inventory management. We suggest yet another modified version in this paper.In the last portion of the paper, the cost effectiveness of the Look Ahead-Look Back tests is confirmed through simulation. That is, when used together with the original PPB procedure, they lead to an improved cost performance. It is also shown that a combination of these tests and the adjustment to the PPB procedure mentioned earlier leads to an even lower average total cost. All cost improvements are statistically significant. It is finally noted that the Look Ahead-Look Back tests perform poorly in certain constant demand situations. Additional analytic and experimental analysis shows that these results stem from a dominance of the Look Back test over the Look Ahead test, leading to the former test being performed more often. This can easily be corrected, however, by checking for sufficient variability in the data before the Look Back test is employed.