Project procurement and disposal decisions: An inventory management model

Transcription

1 Int. J. Production Economics 71 (2001) 467}472 Project procurement and disposal decisions: An inventory management model Keith A. Willoughby* Department of Management, Bucknell University, Lewisburg, PA 17837, USA Abstract In this paper, we develop a two-stage mathematical model to examine an important inventory management problem within a large-scale project context. Speci"cally, we analyze the procurement and disposal of an important, expensive item (e.g. pipeline). There is uncertainty surrounding total requirements of this item during a project. Surplus stock on-hand at the conclusion of the project may be disposed for revenue. However, it may be retained to satisfy requirements during a subsequent project. The time between projects follows a discrete probability distribution. The key decision variables involve the procurement quantity in the initial project as well as the best disposal quantity (should a surplus exist) at the conclusion of the project Elsevier Science B.V. All rights reserved. Keywords: Excess stock disposal; Stochastic inventory models; Large-scale projects 1. Introduction `Except in the midst of a battle"eld, nowhere must men coordinate the movement of other men and all materials in the midst of such chaos and with such limited certainty of present facts and future occurrences as in a huge construction projecta. (Blake Construction Co. v. C.J. Oakley Co. [1], emphasis added) Clearly, inventory management within largescale projects is subject to a tremendous amount of uncertainty. Delays in vendor shipments, quality * Corresponding author. Tel.: # (Res), # (work); fax: # address: kwilloug@bucknell.edu (K.A. Willoughby). problems, mid-stream engineering design changes (even project cancellation), environmental conditions (e.g. unpredictable project `windowsa in remote, harsh climates) and labour disruptions, among others, contribute to the extreme di$culty in e!ectively managing project inventories. This paper examines critical inventory management decisions encountered in large-scale projects. Speci"cally, we develop a mathematical model to analyze the procurement and disposal of an important, expensive item (e.g. pipeline). Due to the inherent uncertainty of subsurface work, total requirements of this item during construction are not known with certainty. Surplus units on-hand after project completion may be salvaged (disposed for revenue), or retained to satisfy requirements during a subsequent project (retention may prove especially bene"cial if procurement prices in the subsequent project increase /01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S ( 0 0 )

2 468 K.A. Willoughby / Int. J. Production Economics 71(2001) 467}472 signi"cantly, and the cost of holding stock is not prohibitive). The time until the next project is modelled as a random variable following a discrete probability distribution. Although one could incorporate continuous probability distributions to represent inter-project time, our selection of a discrete distribution is practitioner-motivated. Representative parameter values in our study were obtained via interviews with materials management and logistics personnel. We felt it was more reasonable to elicit discrete data from these practitioners (i.e. a certain likelihood of a subsequent project occurring within a speci"c time interval), rather than querying them for continuous inter-project distributions. In essence, a greater understanding (and appreciation) of model development may lead to improved prospects of model implementation. The vital decision variables include the appropriate quantity to procure at the beginning of the initial project and the proper amount to dispose, in the event of on-hand surplus, after construction completion. The procurement and disposal quantities are to be selected so as to provide lowest overall costs. We note here that we are not taking into account the total costs associated with a second (and any additional) project(s). We are only concerned with how the presence of various cost and revenue values a!ect initial project procurement and subsequent disposal decisions. Silver [2,3] conducted a survey of senior inventory management personnel involved in large-scale oil and gas projects in the Province of Alberta (Canada). Critical decision areas faced by these professionals involved coping with uncertainty surrounding total project requirements, and the disposal of project surplus. Hence, there would appear to be a practitioner-oriented motivation for a systematic method to analyze the procurement and disposal problem. To the best of our knowledge, no previous attempt has been made to jointly investigate procurement and disposal decisions in a project context. Only a handful of articles have jointly considered procurement (acquisition) and disposal decisions in any context, let alone within a project environment. Fukuda [4] examined ordering and disposal policies in a multiechelon, multiperiod inventory situation. Considering such factors as ordering costs, disposal values, shortage penalties and holding costs, he was able to determine optimal policies for an entire planning horizon. Teisberg [5] developed a multi-period, stochastic dynamic programming tool to guide the ongoing acquisitions and disposals (releases) of the United States strategic petroleum reserve. For each entering stockpile size and each possible oil market state, and using the present value of all relevant costs, he was able to determine optimal stockpile acquisition or release rates for a speci"c time period. Several models have been developed to examine the disposal of excess stock, given that an organization is currently in a surplus inventory situation. A fewof the more signi"cant contributions will be cited. Simpson [6] was an early contributor to the excess stock literature. Basing his analysis on inventories held at Naval supply stores, he found an economic retention quantity by developing a tradeo! between storage and obsolescence costs versus the expenses of repurchasing the material in the future (if and when required). Tersine and Toelle [7] generated relationships for the economic time supply of an item, under the existence or non-existence of present value and in#ation considerations. Incorporating stochastic (Poisson) demand, Rosen"eld [8] developed optimal quantities for excess stock disposal. He applied his methodology to an actual distributor of durable goods faced with excessive amounts of slow-moving items. The model showed that substantial savings could be earned by the judicious disposal of surplus stock. Some researchers have examined various inventory management issues within a project management context. In particular, Smith-Daniels and Smith-Daniels [9] examined the performance of several heuristics for determining lot sizes in projects. Inventory management was especially complex in projects due to uncertainty surrounding the quantity and timing of requirements, as well as the `lumpya nature of those requirements. The format of this paper is as follows. Section 2, by analyzing the inter-project period costs, determines optimal disposal quantities at the conclusion of the initial project. Section 3 then develops expressions for initial project costs. Numerical examples are provided in both sections 2 and 3. Conclusions are included in the "nal section.

3 K.A. Willoughby / Int. J. Production Economics 71(2001) 467} Inter-project analysis We shall introduce the development of our model by analyzing inter-project period costs. The following notation is required: t p time interval i associated with the subsequent project (i"1, 2, 2, n) probability of the subsequent project beginning within time interval (t, t ] Common probability rules maintain that each p is non-negative and that the sum of the respective probabilities over all time intervals is equal to one. We assume that the subsequent project is equally likely to occur any time within the time interval (t, t ] and that t "0. Surplus stock may be retained after completion of the initial project to satisfy subsequent project requirements. Maintaining available on-hand stock, then, means that those speci"c units will not need to be procured during the future project. This represents a cost savings. However, one must pay holding charges to carry these units in inventory. We shall use continuous discounting (see Gurnani [10]) in determining these holding costs. Let us introduce some further notation: h out-of-pocket inventory carrying charges ($ per unit of inventory per unit time) α continuous discount rate (i.e. a cost of x incurred at time t has a present value of x exp(!αt)) v unit procurement cost in the subsequent project (due to varying market conditions, this future unit procurement cost may be greater than, the same as or less than the unit cost in the initial project) h h/α The present value of the inter-project costs may be modelled as f (t )[R(t )] dt, (1) where t represents the largest t value under consideration and R(t )" h exp(!αz)dz!v exp(!αt ) (2) and, for t )t )t, f (t )" p. t!t Evaluation of (2) gives h(1!exp(αt ))!v exp(αt ). (3) Since we are dealing with a discrete probability distribution, we may rewrite (1) as p R(t ) dt t!t which is p t!t R(t )dt. (4) Substituting (3) into (4) gives p t!t [h(1!exp(!αt ))!v exp(!αt )] dt. (5) Evaluation of the integral in (5) gives ht (h exp(!αt )#v exp(!αt )) dt which can be expressed as h(t!t )! h#v α [exp(!αt )-exp(αt )]. (6) Thus, the expected present value of the interproject period costs per surplus unit retained (EIPC) becomes EIPC" p t!t h(t!t )! h#v α (7) [exp(!αt )!exp(!αt )]). Note that the unit inter-project period costs are linear with respect to the quantity of retained stock. In essence, they are independent of the on-hand surplus after initial project completion (assuming that all units retained are needed in the subsequent project, a rather reasonable assumption).

4 470 K.A. Willoughby / Int. J. Production Economics 71(2001) 467}472 Disposing surplus stock upon completion of the initial project generates immediate revenues. The following notation is used: g unit salvage value for surplus stock disposal w disposal quantity (W* represents the optimal disposal quantity) I on-hand surplus stock after completion of the initial project (but before any disposal decision) EPV*(I) expected present value of completing the initial project with I units of inventory on-hand, and proceeding in an optimal fashion from thereon (with respect to disposal decisions) Since we have constant marginal inter-project period costs and disposal revenues, the determination of optimal disposal quantities upon completion of the initial project is quite straightforward. We simply compare the associated costs and revenues in the following fashion: If EIPC#g)0 (this implies that EIPC)!g), then =*"0. No surplus stock is disposed (equivalently, all units are retained) since the bene"t (cost reduction) of retention is higher than the bene"t of disposal. If EIPC#g'0 (this implies that EIPC'!g), then =*"I. All surplus stock is disposed (equivalently, no units are retained) since the bene"t (cost reduction) of retention is less than the bene"t of disposal. The disposal choice after the initial project is, essentially, an `all-or-nothinga decision. We either dispose everything on-hand, or retain all of it to satisfy future project requirements. One could suggest that, if one were indi!erent between surplus stock retention or disposal (i.e. EIPC"!g), then it could be attractive to dispose a portion of the on-hand surplus. However, in this extreme case, observe that total costs would be equivalent under any disposal strategy. The key "nding is that an organization could never be better o! by disposing a portion of the on-hand surplus. The EPV*(I) values are quite important to our analysis, for they `linka the two stages of our model. Since disposal decisions are independent of on-hand stock, the EPV*(I) values are either: EIPC*I, ifw*"0 (all surplus stock is retained),!g*i, if=*"i (all surplus stock is disposed) As a numerical example, assume the following values: p : 0.60 t : 1.0 p : 0.30 t : 2.0 p : 0.10 t : 3.5 h: $13 per unit of inventory per year α: 0.10 v : $100 Using (7), we determine that EIPC"! Consequently, one would need to obtain a unit salvage value of more than $78.22 in order to make disposal attractive. Since the unit acquisition cost is $100, this indicates that (given the parameter values under consideration) the retention of excess stock is quite bene"cial. Decision-makers in our inventory management context would rarely observe a salvage value in excess of 3/4 of the unit acquisition cost! 3. Initial project analysis We shall nowmodel those costs incurred during the initial project. Without loss of generality, we assume that on-hand inventory at the beginning of this project is zero. The following notation is required: Q initial project procurement quantity ETC(Q) expected total discounted costs (initial project costs plus costs relating to postproject disposal decisions) as a function of the initial project procurement quantity The best procurement quantity, Q*, is the one that minimizes ETC(Q). As mentioned earlier, there exists uncertainty surrounding total requirements during the initial project. We shall model this uncertainty with a discrete probability distribution. Speci"cally, let D total requirements in the initial project P (D ) probability of observing a speci"c requirements value

5 K.A. Willoughby / Int. J. Production Economics 71(2001) 467} The following additional notation will be required to determine ETC(Q): v T B B unit procurement cost in the initial project duration of initial project (in years) "xed cost per stockout occasion penalty (expressed as a fraction of the unit value) per unit short The respective cost components of ETC(Q) will nowbe described. Since procurement costs occur at the beginning of the initial project, there is no need to discount them. Quite simply, they are Qv. (8) For any procurement quantity Q, we have the following expression for carrying charges: P (D ) Q!D ¹ t exp!(αt)dt, h which is evaluated as h P (D ) Q α (1!exp(!α¹)) # D ¹ exp(!α¹) ¹ α # 1 α! 1 α. We also recognize any carrying charges incurred prior to the occurrence of a stockout. Since a stockout, should it occur, would happen at time Q¹/D, these carrying charges are expressed as h P (D ) which becomes Q!D ¹ t exp(!αt)dt, h P (D ) α Q 1!exp!αQ¹ # D ¹ exp!αq¹ D Q¹ D # 1 αd α! α 1. This expression can be simpli"ed to yield h P (D ) Q α #D (9) ¹ exp!αq¹ D α 1! α 1. (10) Although we explicitly recognize holding costs prior to a stockout, we shall ignore any carrying charges incurred subsequent to the receipt of expedited stock. In all likelihood, the relatively large stockout penalties would dominate these holding costs. To determine the present value of stockout penalties, these costs must be discounted from the moment at which they occur. The B stockout penalty is given as B P (D )exp!αq¹ (11) D while the B stockout penalty is (1#B )v (D!Q)P (D )exp!αq¹ D. (12) In the previous section, we derived the expression for EPV*(I). We note here that the quantity of on-hand surplus at the conclusion of the initial project, I, is equivalent to maxq!d,0. For D *Q, one begins the inter-project period with zero units of stock on-hand. Since no disposal revenue or holding charges will be incurred, there are no discounted inter-project period costs. However, for D (Q, these discounted costs are exp(!α¹) EPVH(Q!D )P (D ). (13) Combining expressions (8)}(13) yields our mathematical model for determining the expected total discounted costs as a function of the procurement quantity in the initial project. We note here that the "nal model is of a news-vendor type. In other words, a one-time decision procurement decision is made in order to minimize both initial project and inter-project costs. Let us expand on the numerical example presented earlier. In particular, let: v : $100 (same value as the unit acquisition cost in the subsequent project) T : 1 year D : 200 P : 0.10 D : 300 P : 0.20 D : 400 P : 0.40

6 472 K.A. Willoughby / Int. J. Production Economics 71(2001) 467}472 D : 500 P : 0.20 D : 600 P : 0.10 B : $3000 B : 0.5 We "nd that Q*"400 units and that ETC(400)"$46, Moreover, we note that (due to the EIPC value found in Section 2) any on-hand stock after the conclusion of the initial project will be retained to satisfy usage during the subsequent project. The disposal of surplus stock is not warranted. 4. Conclusions This paper has developed a mathematical decision-making tool to assist practitioners in managing project inventories. There is considerable opportunity for further exploration in this vital decision-making area. A possible extension would be to model an important, expensive item (e.g. compressors, valves, electrical control devices) that is required during the initial project, but also has operational usage (as a spare part) during the ongoing operations of the constructed facility. In this case, retaining surplus stock after project construction could help to satisfy both subsequent project requirements and ongoing operational usage. High unit costs may be incurred to replenish small lots of this item to satisfy ongoing usage (there would be no `quantity discountsa), so retention of surplus stock after the initial project may become more attractive. We have considered a situation in which there existed only a single procurement opportunity at the beginning of the initial project. Obviously, one could extend this to the case of multiple procurement opportunities during the project. Acknowledgements The research leading to this paper was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A1485 and by the Carma Chair at the University of Calgary. References [1] Blake Construction Co. v. C.J. Oakley Co., 431 A.2d 569 (D.C. 1981). Transactions of the American Association of Cost Engineers, C [2] E.A. Silver, Policy and procedural issues in procurement and logistics for large-scale projects in the oil and gas industry, Project Management Journal 18 (1) (1987) 57}62. [3] E.A. Silver, Materials management in large-scale construction projects: Some concerns and research issues, Engineering Costs and Production Economics 15 (1989) 223}229. [4] Y. Fukuda, Optimal disposal policies, Naval Research Logistics Quarterly 8 (1961) 221}227. [5] T.J. Teisberg, A dynamic programming model of the U.S. strategic petroleum reserve, Bell Journal of Economics 12 (1981) 526}546. [6] J. Simpson, A formula for decisions on retention or disposal of excess stock, Naval Research Logistics Quarterly 2 (1955) 145}155. [7] R.J. Tersine, R.A. Toelle, Optimum stock levels for excess inventory items, Journal of Operations Management 4 (1984) 245}258. [8] D.B. Rosen"eld, Disposal of excess inventory, Operations Research 37 (1989) 404}409. [9] D.E. Smith-Daniels, V.L. Smith-Daniels, Finding lot sizes for materials used in projects, Production and Inventory Management Journal 27 (1986) 61}71. [10] C. Gurnani, Economic analysis of inventory systems, International Journal of Production Research 21 (1983) 261}277.