Sales and operations planning in systems with order configuration uncertainty

European Journal of Operational Research 205 (2010) 604–614

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Sales and operations planning in systems with order configuration uncertainty Ching-Hua Chen-Ritzo a,*, Tom Ervolina a, Terry P. Harrison b,1, Barun Gupta c,2 a

IBM, T.J. Watson Research Center, Yorktown Heights, NY 10598, United States Smeal College of Business, The Pennsylvania State University, University Park 16802, United States c IBM, Integrated Supply Chain, Hopewell Junction, NY 12533, United States b

a r t i c l e

i n f o

Article history: Received 31 January 2008 Accepted 17 January 2010 Available online 25 January 2010 Keywords: Sales and operations planning Configure-to-order Stochastic programming

a b s t r a c t This paper addresses the problem of aligning demand and supply in configure-to-order systems. Using stochastic programming methods, this study demonstrates the value of accounting for the uncertainty associated with how orders are configured. We also demonstrate the value of component supply flexibility in the presence of order configuration uncertainty. We present two stochastic models: an explosion problem model and an implosion problem model. These models are positioned sequentially within a popular business process called sales and operations planning. Both models are formulated as two-stage stochastic programs with recourse and are solved using the sample average approximation method. Computational analyses were performed using data obtained from IBM System and Technology Group. The problem sets used in our analysis are created from actual industry data and our results show that significant improvements in revenue and serviceability can be achieved by appropriately accounting for the uncertainty associated with order configurations. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction ‘‘You can have it in any color, as long as it’s black,” is a phrase attributed to Henry Ford when Ford Motor Company began mass producing the Ford Model T in 1908. A century later, ‘mass customization’ has emerged as a manufacturing/marketing paradigm in which products are customized, rather than standardized, in large quantities and at low cost. Due in part to this new paradigm, traditional ‘make-to-stock’ supply chains are in many cases being replaced by ‘make-to-order’ supply chains. In make-to-order supply chains, the final manufacture of products is performed after customer orders are received. For this reason, the make-to-order system is particularly suited to the production of customized products. This paper addresses the problem of aligning sales targets with available resources in a specific type of a make-to-order environment called a ‘configure-to-order’ (CTO) system. In a CTO system, products are assembled from several modular components. Unlike what is traditionally thought of as an assemble-to-order (ATO) system, in a CTO system, customers explicitly configure their product by selecting the type and quantity of components to include in their order. Since the set of feasible configurations may be extremely large, similar configurations are often

* Corresponding author. Tel.: +1 (914) 945 1893; fax: +1 (914) 945 3434. E-mail addresses: [email protected] (C.-H. Chen-Ritzo), [email protected]. com (T. Ervolina), [email protected] (T.P. Harrison), [email protected] (B. Gupta). 1 Tel.: +1 (814) 863 3357; fax: +1 (814) 863 2381. 2 Tel.: +1 (203) 888 0499. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.029

aggregated into groups for planning (e.g., forecasting and marketing) purposes. In this paper, a CTO product refers to a group of similar configurations. Perhaps the most obvious examples of CTO products are computers and automobiles. In the case of a mainframe computer, a customer can select the number and speed of the processors to be included in her order. While one customer may choose to include four units of 2.6 GHz central processing units in her order for product A, another customer also ordering product A may select to include eight units of the same processor. Therefore, one may consider a CTO product to possess a bill-ofmaterials (BOM) that is uncertain. That is, the usage quantities of resources required to configure a CTO product varies by order. We refer to this type of uncertainty in the BOM as ‘order configuration uncertainty’. In practice, materials requirements planning (MRP) decision support tools typically assume a fixed bill-of-materials. As a result, when defining the BOM for CTO products, manufacturers may substitute fixed usage quantities for what are in reality variable usage quantities. The impact that this simplification has on the quality of the alignment between supply and demand is studied in this paper, in addition to the value of incorporating order configuration uncertainty into the process of matching demand and supply. To this end, we present and implement practical optimization models and methods for dealing with order configuration uncertainty in the context of a practical business process called sales and operations planning (S&OP). S&OP is a business process for managing product availability that is very relevant in practice, as evidenced by an industry roundtable

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(Center for Digital Strategies, 2004) and white papers on the subject (Manugistics, 2001; Aberdeen Group, 2005). The APICS Dictionary (Cox and Blackstone, 2002) defines S&OP as follows: ‘‘A process to develop tactical plans that provide management the ability to strategically direct its businesses to achieve competitive advantage on a continuous basis by integrating customer focused marketing plans for new and existing products with the management of the supply chain. The process brings together all the plans for the business (sales, marketing, development, manufacturing, sourcing and financial) into one integrated set of plans.” Therefore, S&OP is a process that enables alignment between front-end sales and marketing plans with back-end operational plans. While there is no standard for the sequence of decisions made in the S&OP process, a general representation of such a sequence may be given as follows: (1) demand planning, (2) supply planning and (3) demand/supply review. This sequence of decisions is typically repeated every few weeks to a month using a rolling planning horizon. The demand planning step produces an initial demand plan, which comprises a sales target for each product and each period in the planning horizon. The supply planning step takes the initial demand plan as input and produces a supply plan, which comprises the corresponding component supply quantities required to support the demand plan over the planning horizon. This initial supply plan is derived from the initial demand plan and the BOM, assuming that supplier capacity is unconstrained. After communicating component needs to suppliers, it may be the case that suppliers are unable to meet the quantities and/or leadtimes expressed in the initial supply plan. Therefore, a review of the initial supply and demand plans is needed. In the review step, adjustments to the supply and demand plans may be made to improve their alignment with each other. Supply adjustments may be justified by supplier flexibility, while demand adjustments may be justified by flexibility in marketing strategy. The adjusted demand plan produced in the review step of the S&OP process is called the ‘commitment-to-sales’ (CTS). The CTS informs sales managers of the quantity of each product that can be supported by the supply chain over the planning horizon. It not only guides sales and marketing strategy, but influences a company’s revenue projections as well. For publicly-owned companies, the latter plays an important role in conveying its financial health and growth potential to investors. Unreliable or inaccurate financial forecasts can have detrimental effects on a company’s stock value and hence its market capitalization. Thus, it is apparent that S&OP integrates the financial, marketing and supply chain decisions at a company and the CTS reflects the preferences of all three groups. Assuming that the initial demand plan is given, this paper addresses the decision problems faced in the supply planning and demand/supply review steps in S&OP. We refer to these problems as the ‘explosion’ and ‘implosion’ problems, respectively. The explosion problem represents the supply planning step, and it determines the initial supply plan. The implosion problem represents the demand/supply review step, an it determines the adjusted supply plan and the CTS. Both problems are represented as two-stage stochastic programs that are solved using the sample average approximation approach. The problem scenarios studied are defined using data obtained from the Systems and Technology Group at International Business Machines (IBM). IBM’s interest in this problem stems from the fact that their mid-to-high-end server and mainframe computers are very complex configure-to-order products with significant order configuration uncertainty. Despite this fact, IBM currently performs its S&OP assuming a fixed BOM. This research was motivated by IBM’s interest in understanding how models that account for order configuration uncertainty could provide value over current deterministic methods. The key contributions of this research are as follows: First, we present practical models that deal explicitly with order configura-

605

tion uncertainty and component commonality, a prevalent phenomenon in both goods and services industries that has been largely overlooked in previous studies. Second, the models developed here are well-aligned to a very relevant business process called sales and operations planning. Third, problem instances generated from industry data obtained from IBM are effectively solved for realistic problem sizes. We present results which demonstrate the value of the stochastic models and compute effective bounds on the optimal solution to these problems. We also demonstrate how our models can be used to provide insights to decision makers in operations, marketing/sales and finance. This paper is organized as follows: In Section 2, we provide a review of the literature. In Section 3, we present the formulations for explosion and implosion problems. A presentation of the solution methodology is provided in Section 4 and our computational results are provided in Section 5. Finally, we conclude with a discussion in Section 6. 2. Related literature The related literature includes research in the areas of stochastic inventory control in ATO systems, available-to-promise (ATP) planning and flexible supply contracts. Song and Zipkin (2003) provide a comprehensive review of the research that has been done in the area of stochastic inventory control in ATO systems. The literature in this area relate most to our explosion problem, which computes the initial component supply plan from a given demand plan. ATO systems are complex to analyze, and most models in the literature are designed to deliver insights regarding the structure of optimal, typically stationary, inventory ordering policies or the performance of a given policy when the optimal policy is too difficult to express analytically. Therefore, the literature typically considers problem sizes of no more than 10 products and 20 components. Thus, the objectives of such models differ from the model for the explosion problem in this paper, which is intended to determine specific order quantities over a planning horizon, for realistically sized problems. The problem sets analyzed in this paper include up to 51 products and up to 569 components. Swaminathan and Tayur (1998) use a series of nested stochastic programs to solve a multi-period inventory ordering and production problem for an ATO system with non-stationary demand. Srinivasan et al. (1992) analyze a similar problem, but include probabilistic service level constraints. The problems addressed by Srinivasan et al. (1992) and Swaminathan and Tayur (1998) treat product demand as uncertain and order configurations as fixed. The reverse is true for the problem setting that we consider for the following reasons: First, in practice, product demand uncertainty is often accounted for in the demand planning step using a combination of quantitative methods and expert judgment. Typically, this results in a set of point sales forecasts which are then assumed to be deterministic in the subsequent steps of the S&OP process. Second, the data required to accurately represent probability distributions for product demand are not readily available in practice. The reasons for this include the changing competitive landscape, non-stationary demand, the inability to record lost sales and the challenge of quantifying expert opinion. In contrast, product configurations are considerably less affected by the aforementioned factors, and a relatively reliable and direct representation of product configuration uncertainty can be readily generated from the shipped order history. Therefore, our models are more practical for a CTO system than existing models. Only a small portion of the ATO literature explicitly models uncertainty in product configurations. In particular, Cheng et al. (2002) study the inventory-service tradeoff in a CTO setting with

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multiple demand classes. Lu et al. (2003) study the impact of product structure, demand and lead time variability, and advance demand information on system performance. However, both Cheng et al. (2002) and Lu et al. (2003) assume a stationary base stock policy and stationary demand. The implosion problem studied in this paper is concerned with generating the commitment-to-sales. It relates most to the ATP scheduling literature. An ATP schedule provides the quantities of products that are available to promise to customer orders over time. Ball et al. (2004) provide an excellent overview and review of the literature in this area. They classify ATP models as being either ‘push based’ or ‘pull based’. Push based ATP models allocate resources to products or demand classes prior to receiving orders, while pull based ATP models perform the allocation in response to receiving orders. The primary advantages of push based scheduling over pull based scheduling are that order promising decisions can incorporate long term objectives and can be provided to customers immediately. Our implosion problem relates to push based ATP scheduling since the commitment-to-sales quantities are determined prior to observing orders. Chen et al. (2002) consider a deterministic, rolling horizon, pull based ATP problem for a configurable product. In their model, orders are batched over some prespecified period of time, after which order commitment dates and the production schedule are obtained by solving a multi-period mixed-integer program. Ervolina and Dietrich (2001) study deterministic push based ATP scheduling models for CTO systems with multiple products, components and time periods. While they acknowledge that product configurations are uncertain, they use an ‘average box’ BOM to represent them. To the best of our knowledge, there are no studies that deal specifically with stochastic push based ATP scheduling in CTO or other systems. Ettl et al. (2006a) address the problem of recommending profitable product configurations for sale in ATO supply chains. Assuming a fixed bills-of-materials, they formulate this problem as a deterministic nonlinear program and solve it using a decomposition and column generation method. An integrated framework for this ‘available-to-sell’ model, which also borrows from the concept of S&OP, is presented in Ettl et al. (2006b). Dietrich et al. (2005) formulate a deterministic ‘available-to-sell’ problem that also minimizes the cost of purchasing additional components in order to fulfill the recommended product configurations. Since our model for the implosion problem allows for the case where component suppliers are flexible, it is also related to the literature on flexible supply contracting. Tsay and Lovejoy (1999) study quantity flexible supply contracts for a multi-echelon supply chain in a rolling horizon setting. In their model, supply flexibility is a function of the time difference between the current period and period in which supply is being requested. In general, flexibility is greater than the time difference. We make a similar assumption in the problems studied in this paper. A more thorough review of the supply contracting literature is provided by Tsay et al. (1998). However, to the best of our knowledge, none of the flexible supply contracting literature addresses problems in a CTO setting.

3. Mathematical models 3.1. Order configuration uncertainty In this paper, we consider a manufacturer who produces a set of P, CTO products, indexed by p. These products are configured from a set of C common components, indexed by c. Let ncp be the random usage quantity of component c in a unit of product p. A realization of an order for one unit of product p is given by the vector np ðxÞ ¼ ½n1p ðxÞ; n2p ðxÞ; . . . ; nCp ðxÞ with ncp ðxÞ 2 Zþ and x 2 X,

where X is the set of all random events. The ncp are assumed to be independent random variables. We refer to the uncertainty associated with these random variables as ‘order configuration uncertainty’. 3.2. The explosion problem The explosion problem is used to represent the decision problem faced in the supply planning step of the S&OP process. Given an initial demand plan containing an initial set of sales targets, the explosion problem is used to determine the quantity of each component to request from suppliers in each period so as to maximize the manufacturer’s total expected profit. The planning horizon is T periods long. The explosion problem is formulated as a two-stage stochastic program with complete recourse. In the first stage of the explosion problem, the manufacturer is given the initial demand plan, dt , which is the vector of product sales targets in period t, for t ¼ 1; 2; . . . ; T. In the presence of order configuration uncertainty, the manufacturer makes a decision regarding component order quantities, ut , which is the vector of component quantities to request from suppliers, for delivery in period t. Let Pdpt nicp be the average quantity of component c used hcpt ¼ 1=dpt  i¼1 to configure an order for product p in period t, where dpt 2 dt and nicp is an i.i.d. copy of ncp . Let Ht be the C  P matrix of random order configurations in period t, given by

h11t 6h 6 21t Ht ¼ 6 6 .. 4 .

h12t



h1Pt

3

 .. .



h2Pt .. .

7 7 7: 7 5

hC1t

hC2t

   hCPt :

2

ð1Þ

At the end of the first stage, sales targets are met and random order configurations for all product sales targets are realized in all T periods. In the second stage, the manufacturer decides how to allocate the requested component supply, ut , to the products that are committed to sales so as to maximize his profit over the planning horizon. This decision is given by v t0 t , a vector of the number of orders for every product p that were received in period t0 and fulfilled in period t, where t 0 6 t 6 T. It is necessary to distinguish the period in which an order is received from the period in which it is fulfilled since realizations of order configurations depend on the period in which the orders originate and not the period in which they are fulfilled. We assume that all required components must be available in order for an order to be fulfilled and assembly capacity is assumed to be infinite (i.e., assembly lead times are assumed to be negligible). Let It be the vector of component inventory at the end of period t, with initial component inventory levels given by I0 . The number of products that are ordered in period t0 and which remain backordered at the end of period t P t 0 is given by the vector bt0 t . At the end of the second stage, revenue is earned and costs are incurred for product backorders, component orders and component inventory. The per unit revenue for each product is contained in the vector r. Per unit component ordering costs are given by the vector o. Product backorder costs are given by the vector q. Additionally, let h be the vector of component holding costs Finally, a discount factor, c, is applied over the planning horizon to account for the time value of money. For a given set of component supply requests, fu1 ; . . . ; uT g and the realization x 2 X, the profit over the planning horizon is given by the objective value to the linear program (2). In this formulation, the dependence of v, b and I on x 2 X is fundamentally different from the dependence of H on x. While HðxÞ is a realization of

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the random parameter H, the dependence of v, b and I on x is not a functional one. Instead, it reflects the fact that these decisions are typically not the same under different realizations of x.

Q ðu1 ; . . . ; uT ; H1 ðxÞ; . . . ; HT ðxÞÞ " # T T X X t1 c hIt ðxÞ þ ðrvt0 t ðxÞ  qbt0 t ðxÞÞ ¼ max bðxÞ;IðxÞ;v ðxÞ

ð2Þ ð3Þ

t 0 ¼t

t¼1

s:t: It ðxÞ ¼ It1 ðxÞ þ ut 

t X

Ht0 ðxÞv t0 t ðxÞ 8t ¼ 1; 2; . . . ; T; ð4Þ

beginning of period t. The details of how supplier flexibility is modeled are provided in Section 3.3.1. At the end of the first stage, it is assumed that the CTS quantities are achieved and product order configurations are realized for all periods. To capture the order configuration uncertainty for product P P Pdpt nicp , where nicp is an i.i.d. p, we define kcp ¼ ð1= Tt¼1 dpt Þ  Tt¼1 i¼1 copy of ncp . kcp approximates the average usage of component c in product p over the planning horizon. Let K be the C  P matrix of random order configurations given by

2

t0 ¼1

k12

   k1P

ð7Þ

6k 6 21 K¼6 6 .. 4 .

 .. .

   k2P 7 7 7: ... 7 5

ð8Þ

kC1

kC2

   kCP

ð5Þ

btt ðxÞ ¼ dt  v tt ðxÞ 8t ¼ 1; 2; . . . ; T;

ð6Þ

It ðxÞ P 0 8t; bt0 t ðxÞ; v t0 t ðxÞ P 0 8t; t0 6 t:

The objective function (3) captures the total discounted profit, not including the ordering costs. The flow of component inventory from one period to the next is constrained by (4). Backorder constraints are given by (5). Since the demand plan is assumed to be deterministic (i.e., demand is equal to sales targets) we use the constraints (6) to initialize the number of orders that are demanded at the start of period t. Constraints (7) and (8) are non-negativity constraints, where 0 is a vector of zeroes of length C in (7) and length P in (8). However, recall that the component supply quantities, ut , are first-stage decision variables and must be decided prior to the realization of any orders. Therefore, the explosion problem is formulated as a two-stage stochastic program with recourse (Birge and Louveaux, 1997) as follows:

Maximize Z EXP ðuÞ ¼

T X

ct1 out þ Qðu1 ; . . . ; uT Þ;

3

k11

bt0 t ðxÞ ¼ bt0 t1 ðxÞ  v t0 t ðxÞ 8t ¼ 2; 3; . . . ; T; t0 < t;

ð9Þ

t¼1

where Qðu1 ; . . . ; uT Þ ¼ E½Q ðu1 ; . . . ; uT ; H1 ; . . . ; HT Þ, and E½x generally denotes the expectation of x. Q is referred to as the recourse function. Let Z EXP be the optimal, total expected discounted profit for the explosion problem. 3.3. The implosion problem After the explosion problem is solved and the component supply request is presented to suppliers, the manufacturer receives a response from its suppliers. Since suppliers may be constrained, their response regarding component availability may differ from the manufacturer’s request in regards to the quantities or delivery dates of components. This leads to the third step of the S&OP process. In this step, decision makers review the initial demand plan and supply plans. Given the initial demand plan and the suppliers’ responses, the objective of the implosion problem presented in this section is to determine the optimal quantity of each product that the company should include in its commitment-to-sales (CTS). At the same time, the solution to this problem can also suggest how supplier flexibility should be leveraged to obtain better alignment between the CTS and the component supply. The implosion problem is also formulated as a two-stage stochastic program. As before, dt is the initial demand plan. Let st be the vector of component quantities that will be delivered in period t based on the suppliers’ response. Given dt and st , the manufacturer’s firststage decision is to determine the vector of CTS quantities for all products, given by xt for all t ¼ 1; 2; . . . ; T. A penalty cost is incurred if the CTS quantity for a product p deviates from its targeted sales quantity (i.e., if xpt – dpt , where xpt 2 xt and dpt 2 dt ). The details of how the penalty costs are structured are provided in Section 3.3.2. In the case where suppliers are flexible, the manufacturer faces an additional first-stage decision, yt , which is the vector of component quantities that the manufacturer will expect to be available at the

607

ð10Þ

The reason that kcp is an approximation is that the sampling distribution for the average usage of component c in product p should be dependent on the commitment-to-sales for product p. However, since the CTS is an output variable in our formulation, we use the P targeted sales quantity t dpt to approximate the CTS and hence to parameterize the sampling distribution, resulting in an approximation. In the second stage, the planned quantity of available components in period t; yt , are allocated to the realized orders so as to maximize profit over the planning horizon. This allocation decision is captured by the vector wt , which contains number of orders fulfilled for each product in period t. The vector gt contains the number of units of each product that are backordered at the end of period t. As in the explosion problem, r is the product revenue vector, and q, h and o are the component backorder, holding, and ordering costs, respectively. We use the following notation to represent quantities that are P 0 ¼ 0. Hence, x  t is the t ¼ ti¼1 ai and a accumulated over time: a vector of cumulative CTS quantities for all products up to period  t is the vector of cumulative planned supply quantit. Similarly, y ties for all components up to period t. For a given set of first-stage T g and fy 1 ; . . . ; y  T g and realization x 2 X, the 1 ; . . . ; x decisions fx sum of the revenue, less holding and backorder cost is given by the objective value of the following linear program:

1 ; . . . ; x T ; y 1 ; . . . ; y  T ; KðxÞÞ Q 0 ðx ¼

max

It ðxÞ;gt ðxÞ;wt ðxÞ8t

T X

ct1 ðrwt ðxÞ  qgt ðxÞ  hIt ðxÞÞ

ð11Þ ð12Þ

t¼1

t  y t1  KðxÞwt ðxÞ 8t; s:t: It ðxÞ ¼ It1 ðxÞ þ y t X t  x  t1  wt0 ðxÞ 8t; gt ðxÞ ¼ x

ð13Þ ð14Þ

t 0 ¼1

It ðxÞ P 0 8t;

ð15Þ

gt ðxÞ; wt ðxÞ P 0 8t:

ð16Þ

The objective function (12) captures the total discounted profit for a particular event x 2 X, not including the ordering costs or penalty costs associated with deviations of the CTS from the sales targets. The flow of component inventory from one period to the next is constrained by (13). Backorder constraints are given by (14). Constraints (15) and (16) are non-negativity constraints, where 0 is an vector of zeroes of length C in (15) and length P in (16). 1 ; . . . ; x  T g and fy 1 ; . . . ; y T g Note that in (12)–(16), the variables fx are not dependent on x because they are decisions made in the first-stage, and are therefore considered fixed in the formulation above. Before we can specify the two-stage stochastic program with recourse that characterizes the full implosion problem, it is necessary to first describe how supplier flexibility constrains the  T g and penalty costs may be incurred for certain 1 ; . . . ; y values of fy  T g. 1 ; . . . ; x values of fx

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3.3.1. Modeling supplier flexibility In the implosion problem, we consider the case where the supply of components over the planning horizon is flexible. In the context of sales and operations planning, requests for supplies are communicated by the manufacturer to the supplier in a rolling fashion. That is, for example, the manufacturer may communicate a supply request every 2 weeks, which contains a request for supply volumes to be delivered over the next 13 weeks. Every 2 weeks, some of these volumes may change slightly from what was requested 2 weeks earlier. Often, depending on the relationship between the manufacturer and supplier, these changes can be accommodated by the supplier, within reason. Supply flexibility may exist when suppliers have excess capacity and are willing to rebalance their capacity between customers, or to otherwise accommodate their customers requests. In some cases, suppliers may be contractually obligated to be flexible, within limits. At the same time, the manufacturer may also be contractually obligated to order a minimum quantity over a given period of time. In a rolling planning horizon setting, suppliers may be more willing to accommodate changes in supply for periods later in the planning horizon. Therefore, in our model, we assume that supply flexibility is greater for periods farther out in the planning horizon. The benefit of upward supply flexibility is that it may help reduce or eliminate backorders. Meanwhile, downward supply flexibility may also be beneficial when hard constraints on the supply of one component reduce the need for complementary components, or when customer orders are cancelled or otherwise lower than expected. The supply adjustments may be manifested as changes in the supply lead times (i.e., requesting a component later, or earlier) or a change in the actual supply quantities. It should be noted that supply uncertainty is not modeled in this paper. That is, while suppliers may accommodate changes in requests from the manufacturer, the supply quantities supplied are assumed to the quantities most recently requested. To model component supply flexibility, the cumulative planned  t , is bounded to lie availability of components up to period t; y within limited range around the cumulative supply response up to period t; st . These limits are expressed by

 t 6 ð1 þ dt Þst 8t; y  t P ð1  dt Þst 8t; y

ð17Þ ð18Þ

where the vector 0 6 d 6 1 specifies degree of supply flexibility for components, expressed as a fraction of the cumulative component supply response in period t. 0 is a vector of zeroes, 1 is a vector of ones, and both are of length C. To ensure that the cumulative component supply is non-negative and non-decreasing with time, it is required that

t  y t1 P 0 8t: y

ð19Þ

3.3.2. Penalizing deviations of commitment-to-sales from sales targets A certain degree of similarity between the optimal CTS quantities, xt , and the initial sales targets, dt , should be maintained since the sales targets in the initial demand plan capture the qualitative judgments of decision makers from various parts of the business (e.g., accounting and finance, operations, sales and marketing). For example, sales targets may be influenced by the desire to maintain market share for certain products, or by the need to meet quarterly revenue targets. At the same time, allowing the model to freely make trade-offs in the CTS between high and low revenue products, and between products with high and low order configuration uncertainty can help to improve profits. Additionally, allowing such limited, penalty-free deviations is an acknowledgement that there is uncertainty associated with the sales targets set in the initial demand plan. Therefore, the model for the implosion problem allows for limited, penalty-free, deviations between the

CTS and the initial demand plan. Deviations exceeding certain limits will, however, incur linear penalty costs. These costs are applied for modeling purposes and are not actual supply chain costs. t be the vector of cumulative CTS As previously defined, let x quantities for all products in period t. To ensure that the cumulative CTS quantities are non-negative and non-decreasing with time, we require that

t  x t1 P 0 8t: x

ð20Þ

 t , are bounded from above and beThe cumulative CTS quantities, x  t , as exlow with respect to the initial cumulative sales targets, d pressed by the following constraints:

 t 8t;  t Þd t P ð1  at  b x  t 8t;  t Þd  t 6 ð1 þ at þ b x   at ; at ; bt ; bt P 0 8t:

ð21Þ ð22Þ ð23Þ

In constraints (21)–(23),  at and  at are vectors of length P, containing  pt , which are measures of the  pt and a auxiliary decision variables a penalty-free deviations of the cumulative CTS from the cumulative sales targets. While  at reflects deviations in the negative direction,  at represents deviations in the positive direction. They are expressed as a proportion of the cumulative sales target. For example,  t is a vector containing, for each product, the un-penalized por at d tion of the cumulative CTS quantity that exceeds the cumulative t are measures t and b sales target in period t. In a similar fashion, b of the penalized deviations of the cumulative CTS from the cumulative sales targets. We limit the penalty-free deviations by bounding  at as follows: at and 

at 6 l t t at 6 l

8t; 8t;

ð24Þ ð25Þ

where l t and l t are vectors of length P. The limits of the penalty-free region may be chosen so as to reflect any flexibility that the manufacturer is willing to allow with respect to the initial sales targets. The total penalty cost paid in period t is given by the function pt b pt b pt þ q pt Þ, where q t ; b t Þ ¼ PP ðq  pt d  pt d  pt and q  pt are the H t ðb p¼1 per unit penalty costs associated with positive and negative deviations of the CTS for product p in period t, respectively. Meanwhile, pt 2 d t. pt 2 b pt 2 b t ; b t and d b 3.3.3. Formulation of the implosion problem Recall that in the implosion problem, the manufacturer decides t , and the cumulative response to the cumulative CTS quantities, x  t , in the first stage. Then in the second stage, after the suppliers, y order configurations are realized, the manufacturer allocates available supply to the committed quantities. The implosion problem is formulated as a two-stage stochastic program with recourse, as follows:

Maximize

; y Þ Z IMP ðx ¼

T X t ; b t Þ  oðy t  y t1 Þg fct1 ½Ht ðb t¼1

ð26Þ

1 ; . . . ; x T ; y 1 ; . . . ; y T Þ þ Q0 ðx s:t:

ð17Þ—ð25Þ

1 ; . . . ; x T ; y 1 ; . . . ; y T Þ ¼ E½Q 0 ðx ; . . . ; x T ; y 1 ; . . . ; y  T ; KÞ, is the where Q ðx recourse function. The terms in the summand represent the discounted penalty costs and ordering costs, respectively. Let Z IMP be the optimal, total discounted profit for the implosion problem. 0

4. Solution approach The solution approach used to solve both the explosion and implosion problems is called the sample average approximation (SAA) method. This method essentially solves several deterministic

C.-H. Chen-Ritzo et al. / European Journal of Operational Research 205 (2010) 604–614

approximations of each problem and then estimates bounds on the solution to the original, stochastic problem. In Section 4.1, we present the deterministic equivalent formulations of the explosion and implosion problems. The SAA method is described in more detail in Section 4.2. 4.1. Deterministic equivalent formulations A two-stage stochastic program with a discrete, finite number of scenarios, or realizations, may be expressed as its ‘deterministic equivalent’ by explicitly representing all possible outcomes, or scenarios, associated with the stochastic elements of the problem (Birge and Louveaux, 1997). For linear stochastic programs, this deterministic equivalent formulation is a linear program. Each scenario, k ¼ 1; 2; . . . ; K is assumed to occur with probability pk . These probabilities may be estimated from historical records or based on expert judgment. In the explosion problem, each scenario k corresponds to an independently generated collection of realizations of product configurations, fH1 ðxk Þ; . . . ; HT ðxk Þg. Let Z EXP ðKÞ be objective function to the K-scenario explosion problem. The K-scenario deterministic formulation of the explosion problem is given by (27), where the recourse function Q in (9) has been replaced by its equivalent, multi-scenario, linear program representation.

Maximize Z EXP ðKÞ ¼

T X

(

c

t1

out þ

t¼1

K X k¼1

þ

T X

pk ½hIt ðxk Þ #)

ð27Þ

ðrvt0 t ðxk Þ  qbt0 t ðxk ÞÞ

t 0 ¼t

s:t:

ð4Þ—ð8Þ for every scenario; k ¼ 1; 2; . . . ; K:

When K is equal to the full set of possible scenarios, then (27) is the deterministic equivalent to (9) and Z EXP ðKÞ ¼ Z EXP . In a K-scenario deterministic formulation of the implosion model, each scenario k corresponds to an independently generated set of realizations of product configurations, Kðxk Þ. Let Z IMP ðKÞ be the objective function to the K-scenario implosion problem. A deterministic formulation of the implosion problem is given in (28), where the recourse function Q0 in (26) has been replaced by its equivalent, multi-scenario, linear program representation.

Maximize Z IMP ðKÞ ¼

T X



K X

#

pk ðrwt ðxk Þ  qgt ðxk Þ  hIt ðxk ÞÞ

k¼1

s:t:

large sets of scenarios (Kleywegt et al., 2001; Verweij et al., 2003; Akçay and Xu, 2004; Linderoth et al., 2006). In this method, several independently drawn subsets of the set of all possible scenarios are used as a basis for identifying a good solution to the original problem. Before we describe the general procedure for this method, we first present the following concept: Let K be the number of all possible scenarios. Consider a subproblem of the deterministic equivalent problem that includes M  K scenarios from the complete set of K scenarios. We refer to this as the M-scenario problem. For example, the M-scenario explosion problem approximates the true explosion problem by replacing K by M in (27). Likewise, the M-scenario implosion problem approximates the true implosion problem by replacing K by M in (28). As is typical in the SAA method, a probability of pM ¼ 1=M is associated with each of the sampled scenarios. Note that while there is only one instance of the K-scenario problem, there may be many possible instances of the M-scenario problem. For each of the explosion and implosion problems, we derive a solution as follows: (i) Generate N candidate solutions by solving N instances of the M-scenario problem, where M  K; (ii) The ‘best’ candidate solution is selected by evaluating each candidate solution in an independently generated instance of the L-scenario problem, where M < L  K. Much of the ‘art’ in this approach lies in selecting the number of candidate solutions, N, and the size of the subproblems, M, from which the candidate solutions are generated. In step (i), the larger N is, the more candidate solutions there are to choose from, but at the same time, the more subproblems need to be optimized. The larger M is, the better the approximation the M-scenario problem to the original problem is likely to be. However, increasing M will also increase the computational effort required to solve the optimization problem. Since step (ii) involves the evaluation of an objective function rather than optimization, we can typically choose L to be much larger than M. The larger L is, the more confident we can be regarding the true performance of the identified ‘best’ solution. Since this procedure is not deterministic (i.e., one generally will not obtain identical results if the procedure is repeated), one may compute a confidence interval for the optimality gap associated with the identified solution. In the appendix, we describe how a confidence interval for the optimality gap is computed for the explosion problem. A similar approach was applied to the implosion problem.



ct1 Ht bt ; bþt  oðy t  yt1 Þ

t¼1

þ



609

ð17Þ—ð25Þ; and ð13Þ—ð16Þ for every scenario; k ¼ 1; 2; . . . ; K: ð28Þ

When K is equal to the full set of possible scenarios, then (28) is the deterministic equivalent to (26) and Z IMP ðKÞ ¼ Z IMP . Since the exact deterministic equivalent formulation can represent a very large linear programming problem when K is very large, a method called sample average approximation is used to approximate the problem. This method is also used to identify a good solution and compute statistical estimates of the optimality gap of the identified solution. In Section 4.2, we explain how the sample average approximation method is used to approximate and solve the explosion problem. This method can be similarly applied to the implosion problem.

4.2.1. Myopic approximation Even though the SAA method reduces the number of scenarios that need to be considered in any given linear program, it is important not to select a value for M that is too small, so that acceptable confidence intervals for the optimality gap are obtained. In our case, we found that in order to obtain acceptable confidence intervals, M needed to be set to a value large enough that the M-scenario problem could not be efficiently solved. Therefore, we considered a myopic approximation approach, which substitutes the M-scenario, T-period problem with T; M-scenario, single period problems. In this approach, for all t > 1, the inventory and backorders that remain at the end of the ðt  1Þst period are used to initialize the problem for the tth period. The myopic solution is the collection of T, single period solutions. In Section 5, we compare the performance of the myopic solution against the T-period solution. We provide the expression for the (1-a)-level confidence interval for optimality gap associated with the myopic solution in Appendix 1.

4.2. Sample average approximation

5. Computational analysis

The sample average approximation (SAA) method is a statistical approach for dealing with stochastic programs with extremely

We performed computational analyses, primarily to understand the value of using stochastic models for aligning supply and

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demand in CTO systems. The 10 problem sets used in this study were created from actual planning data obtained from IBM Systems and Technology Group in 2005. The product sets studied belong to IBM’s line of mid-to-high-end server computers. The first three columns of Table 1 display the name of the problem set, the number of products in the problem set, and the total number of components used to configure these products, respectively. The bill-of-materials (BOM) ratio displayed in the fourth column refers to the number of unique components that are used to configure a product, on average. The average usage quantity coefficient of variation (CoV) is displayed in the last column of Table 1. It is a measure of the degree of order configuration uncertainty in a data set and is obtained by computing the average of the coefficient of variation of ncp for all product–component pairs in the data set. Product configuration data was provided by IBM in the form of a 12-month historical log of orders shipped. Each entry in this log provided the order number, the name of the product ordered, the name of a component configured in the product, and the quantity of the component used to configure the product. Empirical distributions for all product–component pairs were derived from this historical log. These empirical distributions are used to generate the various problem scenarios via the alias method (Law and David Kelton, 2000). Per unit component ordering costs and per unit product revenues were obtained from IBM. The backorder cost for each unit of product per unit time was assumed to be 5% of the revenue of the product. The penalty cost for CTS quantities lying outside the penalty-free region was set lower than the backorder cost. These cost assumptions were determined to be appropriate by IBM subject matter experts. Telescoping time buckets were used to condense IBM’s 32-week planning horizon into 20 time buckets (i.e., T ¼ 20), where the degree of aggregation of time buckets increases further out in the planning horizon. To analyze the impact of supplier flexibility, three levels of supply flexibility dt are considered: 2%, 5% and 10%. The flexibility factors are kept constant over time. However, since these factors are applied to the cumulative supply response quantities, supply flexibility is greater for periods lying closer to the end of the planning horizon. For the purpose of comparison, common random numbers were used across supplier flexibility settings. The penalty-free region includes deviations of the CTS quantities that are no more than 50% below and no more than 5% above the cumulative initial sales targets. All computations were run on a Sun v40z server equipped with four, 2.6 GHz, 64-bit Opteron processors and 32 GB of random access memory. CPLEX v9 was used as the linear programming solver in the SAA method. 5.1. Results In this section, three main sets of results are reported. First, we report the quality of the myopic approximation in both the explosion and implosion problems. Second, we report on the value of the

stochastic solution for both problems, which compares the quality of solution produced when accounting for order configuration uncertainty with the quality of that produced when uncertain quantities are replaced with their expected values. Third, we show how the implosion problem can be used to understand how supplier flexibility affects expected revenue. 5.1.1. Assessing the quality of the myopic approximation To assess the quality of the myopic approximation for both the explosion and implosion problems, we compare the optimality gap obtained when using the myopic solution as opposed to that obtained using the T-period solution. Optimality gap comparisons are performed for the first three problem sets (i.e., PS-1, PS-2 and PS-3). For the remaining problem sets, it is assumed that the quality of the approximation is similar. For the purpose of comparison, common random numbers are used to compute both the myopic solution and the T-period solution. For the results presented in this section, N ¼ 20; L ¼ 1000, and M is as indicated in the tabled results. Table 1 presents the estimates of the optimality gap for the explosion problem, as well as solution times. The value representing the optimality gap is the upper confidence limit of the 90% confidence interval for the optimality gap. The reader is referred to Appendix 1 for expressions for the confidence interval. In Table 2, when comparing the optimality gap estimates for the T-period solution (with M ¼ 100) and the myopic solution (with M ¼ 100), we observe that while the optimality gaps increase slightly for the myopic approach, the solution times are considerably shorter. Due to the increased tractability of the myopic solution approach, we increased the number of scenarios used to generated the candidate solutions from M ¼ 100 to M ¼ 500. Notice that when M ¼ 500, the optimality gap estimates, which have a range between 0.67% and 1.91% are considerably smaller than both the Tperiod solution (with M ¼ 100) and the myopic solution (with M ¼ 100). Additionally, solution times are reasonable at 2–4 hours. We perform a similar analysis of the quality of the myopic approach for the implosion problem. In this analysis, the baseline supply around which supply is allowed to be flexible is computed by solving the explosion problem. Table 3 presents the optimality gap estimates (assuming 90% confidence level) and solution times for the implosion problem. In Table 3, results are reported for various degrees of supplier flexibility. Similar to what was observed for the explosion problem, the myopic SAA approach with M ¼ 500 produces improved optimality gaps over both the T-period approach and the myopic approach with M ¼ 100. The extent of these improvements increases with the degree of supplier flexibility. Under fixed supply conditions, there is an approximately 25% reduction, on average, in the optimality gap when the myopic approach with M ¼ 500 is used as opposed to the T-period approach. When there is 10% supplier flexibility, the average reduction in the optimality gap improves to approximately 87%. Table 3 also shows that the time to solve the implosion problem using the myopic

Table 1 Summary of characteristics for data sets used in analysis. Problem set

No. of products

No. of components

BOM ratio

Usage quantity Avg. CoV

PS-1 PS-2 PS-3 PS-4 PS-5 PS-6 PS-7 PS-8 PS-9 PS-10

4 6 15 10 51 19 8 2 39 21

181 421 319 265 55 256 434 129 74 569

80.25 147.17 56.27 59.60 4.73 36.95 116.38 94.50 8.18 41.00

3.55 14.49 4.04 4.27 7.61 3.74 13.03 2.54 5.72 6.76

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C.-H. Chen-Ritzo et al. / European Journal of Operational Research 205 (2010) 604–614 Table 2 Comparison of optimality gap estimate and solution time (in CPU hours) using T-period solution vs. myopic solutions. Problem set

PS-1 PS-2 PS-3 a

T-period solution with M ¼ 100 a

Myopic solution with M ¼ 100

Myopic solution with M ¼ 500

a

Gap (%)

Time

Gap (%)

Time

2.44 4.93 1.42

3.64/2.08 31.23/5.11 16.15/7.16

2.64 5.13 1.42

0.07/1.30 0.19/2.18 0.17/2.13

Gap (%)

Timea

1.09 1.91 0.67

0.48/1.29 1.84/2.03 1.32/2.15

Times to the left of the slash are for the generation of candidate solutions while times to the right of the slash are for the selection of the ‘best’ candidate solutions.

Table 3 Comparison of optimality gap estimates and solution times (in CPU hours) using T-period solution vs. myopic solutions. Problem set

T-period solution with M ¼ 100 Gap (%)

Myopic solution with M ¼ 100

Myopic solution with M ¼ 500

Gap (%)

Timea

Gap (%)

Timea

PS-1 (no flex.) PS-2 (no flex.) PS-3 (no flex.)

0.47 0.55 0.24

10.97/2.69 11.88/4.36 29.89/9.72

0.49 1.51 0.22

0.22/2.90 0.27/5.49 0.26/4.48

0.37 0.42 0.16

1.24/2.38 1.37/5.74 0.30/4.31

PS-1 (2% flex.) PS-2 (2% flex.) PS-3 (2% flex.)

0.64 2.17 0.56

22.45/2.26 35.69/4.72 78.67/9.17

1.07 4.08 0.47

0.21/2.46 0.44/5.04 0.42/4.48

0.35 1.69 0.06

4.63/2.38 11.56/5.76 11.20/4.69

PS-1 (5% flex.) PS-2 (5% flex.) PS-3 (5% flex.)

1.72 3.51 0.99

27.30/2.40 48.58/5.37 169.01/10.78

1.88 6.62 0.97

0.22/2.51 0.55/5.52 0.46/4.40

0.15 0.38 0.07

5.58/2.40 16.42/5.30 9.93/4.62

PS-1 (10% flex.) PS-2 (10% flex.) PS-3 (10% flex.)

2.84 6.28 1.28

32.42/2.53 74.41/5.60 290.80/11.42

2.95 9.35 1.26

0.26/2.56 0.70/5.07 0.49/4.26

0.15 1.76 0.07

6.45/2.37 22.48/3.04 11.71/4.29

Times to the left of the slash are for the generation of candidate solutions while times to the right of the slash are for the selection of the ‘best’ candidate solutions.

approach with M ¼ 500 is roughly 25% less than that required by the T-period approach. Using the myopic approach with M ¼ 500, the solution times for the test problems with no supplier flexibility, 2%, 5% and 10% supplier flexibility are 5, 13, 15 and 18 hours, on average, respectively. However, note that by considering that the implosion problem would typically be solved every 2 weeks, these solution times are still reasonable. 5.1.2. Value of the stochastic solution Having established the quality of the myopic solution, the myopic approximation method is used for all subsequently reported results. The value of the stochastic solution (VSS) is measured by the relative improvement in the objective value that the stochastic solutions (SS) yields over the expected value solution (EVS). In this paper, the stochastic solution is the solution obtained using the sample average approximation method. Therefore, it is a solution that accounts for the uncertainty in the problem. Meanwhile, the expected value solution is obtained by solving a single scenario version of the explosion problem, wherein the random variables are fixed to their expected values. In general, the value of the stochastic solution (VSS) for a given stochastic problem is given by the difference between the objective value obtained when implementing the stochastic solution (SS), and the objective value obtained when implementing the expected value solution (EVS). Note that both the SS and EVS are being evaluated for the same stochastic problem, with the same objective function. The reader is referred to Birge and Louveaux, 1997 for further details on this concept. Fig. 1 compares the expected profit associated with the stochastic solution and the expected value solution for the stochastic explosion problem, for each of the 10 problem sets analyzed. Only relative profit (i.e., profit divided by a constant) is reported, for reasons of confidentiality. From Fig. 1 we observe that the median improvement in expected profit as a result of accounting for order configuration uncertainty is approximately 25%. We exclude backorder and penalty costs from the profit calculation, since they are not actual costs in the supply chain. To provide a better under-

20 Expected Value Solution Stochastic Solution

15

Relative Profit

a

Timea

10

5

0 PS-1

PS-2 PS-3

PS-4

PS-5

PS-6

PS-7

PS-8

PS-9

PS-10

Problem Set Fig. 1. Expected profit associated with expected value solution and stochastic solution for the explosion problem (N ¼ 20; M ¼ 500 and L ¼ 2000).

standing of the factors contributing to the expected profit, Table 4 presents the percentage changes in revenue, ordering and inventory costs when using the stochastic solution as opposed to the expected value solution. The average relative improvement in revenue is roughly 24%. This is the result of higher fill-rates when using the stochastic solution as compared to the expected value solution. A comparison of the fill-rate performance is reported in Table 5. Here, fill-rate is defined as the proportion of orders that are fulfilled in the period in which they were ordered. Comparing the results for the component ordering and inventory costs in Table 4, it is observed that the stochastic solution incurs higher component ordering costs than the expected value solution. However, while ordering costs are roughly 8% higher on average, the expected savings in inventory holding costs is roughly 51%, on average. We now examine the value of the stochastic solution for the implosion problem, for each of the 10 problem sets. Fig. 2 compares the profit achieved by the stochastic solution and the expected value solution. In this comparison, component supply

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Table 4 Comparison of relative changes in revenue and costs associated with stochastic solution vs. expected value solution for the explosion problem (N ¼ 20; M ¼ 500 and L ¼ 2000). Problem set

Revenue (%)

Ordering cost (%)

Inventory cost (%)

PS-1 PS-2 PS-3 PS-4 PS-5 PS-6 PS-7 PS-8 PS-9 PS-10

33.85 40.51 22.33 21.31 5.75 30.77 50.15 12.21 1.66 21.79

10.47 6.13 9.83 10.90 4.91 9.96 7.74 3.80 1.57 17.31

57.06 80.54 61.22 53.67 8.31 58.29 75.39 32.26 41.96 14.24

Fraction of Total Orders Filled

612

Expected Value Solution Stochastic Solution

1.0

0.5

0.0 PS-1

PS-2

PS-3

PS-4

PS-5 PS-6 PS-7

PS-8

PS-9

PS-10

Problem Set Table 5 Expected fill-rates associated with expected value solution and stochastic solution for the explosion problem. Problem set

Expected value solution

Stochastic solution

PS-1 PS-2 PS-3 PS-4 PS-5 PS-6 PS-7 PS-8 PS-9 PS-10

0.54 0.52 0.72 0.68 0.78 0.86 0.50 0.61 0.82 0.71

0.99 0.94 0.95 0.86 0.87 0.93 0.94 0.91 0.90 0.94

quantities are given by the expected value solution to the explosion problem. Note that the performance of the stochastic solution and the expected value solution are comparable with respect to expected profit. For some problem sets, the stochastic solution may result in lower expected profit than the expected value solution since the SAA method is an approximation method (i.e., not all possible scenarios are considered). Referring to Fig. 3 we found that, while the stochastic implosion solution may not improve profit, it is able to fulfill roughly 20% more of the orders that were committed to sales compared to the expected value solution. Therefore, for the IBM data sets studied, the primary benefit of the stochastic implosion model is in generating more reliable commitment-tosales quantities. Fig. 4 provides a more detailed illustration of this point. It shows the CTS quantities and fill-rates for problem set PS-3. These

30 Expected Value Solution

25

Relative Profit

Stochastic Solution

20 15 10 5 0 PS-1

PS-2 PS-3

PS-4 PS-5

PS-6

PS-7 PS-8

PS-9 PS-10

Fig. 3. Expected fraction of total orders filled with expected value solution and stochastic solution for the implosion problem with fixed supply (N ¼ 50; M ¼ 500 and L ¼ 2000).

results were generated by simulating the performance of the optimal implosion solution over 2000 realizations of the order sequences and order configurations. A first-come-first-served allocation of components to orders is used. The vertical bars in this figure provide a comparison of the total CTS quantities under the stochastic solution and the expected value solution, for all 20 periods. A comparison of the expected fill-rate in each period in this same figure reveals that the stochastic solution provides a more conservative (i.e., lower CTS quantities), but more reliable CTS over the expected value solution. Using the stochastic solution, a company can provide more reliable revenue forecasts to investors. Quarterly revenue estimates are strong financial signals that are interpreted by the market as to anticipated profits and overall performance. When actual quarterly revenue falls short of estimates, the value per share of the firms stock can drop, often quite precipitously. There are many examples where small misses of quarterly revenue estimates lead to large market capitalization losses for the firm. By referring to the stochastic implosion solution, a company can have a more reasonable expectation of what can be sold, and therefore what level of revenue can be expected. The CTS can also be used to guide marketing plans. For example, if the CTS indicates that supply constraints will limit the company’s ability to meet initial sales targets for a certain product configuration, then the company may consider promoting alternative product configurations based on what is available to sell. While the overall CTS quantities for problem set PS-3 are lower than the initial sales targets, the CTS quantities for individual products may be lower or higher than the initial sales targets. To see this, we partition the products in this problem set into two groups. Products in Group 1 possess cumulative CTS quantities that were consistently greater than or equal to their initial sales targets. Products in Group 2 possess cumulative CTS quantities that were consistently less than their initial sales targets. Fig. 5 shows the deviation of the cumulative CTS quantities from the cumulative initial sales targets, expressed as a fraction of the cumulative initial sales targets. Note that these deviations lie within the penalty-free region described in Section 3.3.2. We observe from Fig. 5 that the optimal implosion solution suggests that the sales targets for products in Group 1 should be increased with respect to the initial demand plan. Meanwhile, the results suggest that sales targets for products in Group 2 should be reduced.

Problem Set Fig. 2. Expected profit associated with expected value solution and stochastic solution for the implosion problem with fixed supply (N ¼ 50, M ¼ 500 and L ¼ 2000).

5.1.3. Impact of supplier flexibility In addition to guiding marketing efforts, the commitment-tosales produced by the implosion problem is used to help planners

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1

0.8 150 0.6 100 0.4

Fill Rate

Total Commitment-To-Sales

200

50 0.2

0

0 1

2

3

4

5

6

7

8

9 10 11 12

13 14 15 16 17 18

19

20

Period CTS (SS)

CTS (EVS)

Fill Rate (SS)

Fill Rate (EVS)

Fraction of Cumulative Initial Sales Target

Fig. 4. Comparison of commitment-to-sales quantities and fill-rates for PS-3.

0.1 0.05 0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

-0.05 -0.1 -0.15

Product Group 1

-0.2

Product Group 2

-0.25 -0.3 -0.35

cases. In Case 1, supply and demand are not well-aligned prior to solving the commitment-to-sales problem, which explains why we observe significant increases in the average expected revenue as supplier flexibility increases. However, in Case 2, demand and supply are already well-aligned, so the incremental value of supplier flexibility with respect to revenue is small. This sort of analysis can be used to guide procurement negotiations with suppliers by providing a better understanding of the potential value of supplier flexibility. These results can also provide insight into the degree of ‘supply risk’ associated with making more aggressive revenue projections. For example, in Case 1, an 8% increase in the projected expected revenue is, all other things being equal, contingent upon the ability to adjust supply by up to 10% over the planning horizon.

-0.4

Period Fig. 5. Deviation of cumulative commitment-to-sales quantities from cumulative initial sales targets for PS-3.

Change in Expected Revenue

10% Case 1

8%

Case 2

6%

4%

2%

0% 0%

2%

4%

6%

8%

10%

12%

Supplier Flexibility Fig. 6. Change in expected revenue for various levels of supplier flexibility.

estimate the future revenue that can be achieved through product sales for the purposes of reporting revenue projections to investors. Fig. 6 plots the average increase expected revenue with respect to supplier flexibility, taken over all problem sets, for two different

6. Conclusion The problems addressed in this paper represent common challenges being faced by managers of configure-to-order supply chains. Despite the prevalence of CTO systems, practical stochastic optimization models which deal explicitly with configuration uncertainty are not common, either in practice or in the academic literature. This research provides practical optimization models and methods for dealing with order configuration uncertainty and situates them within a sales and operations planning process. Based on a computational analysis of product data obtained from IBM, we have shown that significant benefits in profit and revenue can be expected from utilizing stochastic models that account for order configuration uncertainty. Additionally, we have shown that, using the SAA method, the myopic solution can provide tighter bounds than the T-period solution for the same computational effort. The improved tractability of the myopic problem allows for a larger number of scenarios to be considered when generating candidate solutions, resulting in candidate solutions of a higher quality. Although this research was implemented with manufacturing applications in mind, many services, such as business process outsourcing, business transformation and consulting services must deal with uncertainty in human resource utilization not unlike the order configuration uncertainty encountered in a CTO system. Therefore, this approach may be worth exploring in services settings.

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Acknowledgement

(29). For the myopic approximation, the ð1  aÞ-level confidence interval for the optimality gap is given by

The authors would like to thank IBM Integrated Supply Chain and the Center for Supply Chain Research at Penn State University for their generous support of this research. We would also like to extend our gratitude to the Penn State High Performance Computing Group for providing us with excellent computing resources.

  ^ u tL1;a=2 r ~‘ tN1;a=2 r pffiffiffiffi pffiffiffi C:I:ða; GAPmyopic Þ ¼ 0; ½GAP myopic þ þ þ ; N L

Appendix 1. Bound and confidence interval estimation for explosion problem

References

The M-scenario explosion problem approximates the explosion problem by replacing K by M  K in (28). A probability of pM ¼ 1=M is associated with each of the sampled scenarios. Let the objective value for this problem be given by Z EXP ðMÞ. The gap between the objective value derived using SAA and the optimal solution to the exact explosion problem, (9), can be statistically estimated using the approach described by Mak et al. (1999). To estimate an upper bound on Z EXP , one first solves N, independent instances of the M-scenario explosion problem optimally to ^ 1n ; u ^ 2n ; . . . ; u ^ Tn  ^ n ¼ ½u obtain N independent candidate solutions: u for n ¼ 1; 2; . . . ; N. Let the optimal objective value corresponding b EXP;n ðMÞ for n ¼ 1; 2; . . . ; N. ^ n , be given by Z to each candidate, u Then, an unbiased estimate of an upper bound for Z EXP is given by

UB ¼

N 1X b EXP;n ðMÞ: Z N n¼1

ð29Þ

In order to identify the ‘best’ solution from among the candidates, each of the candidate solutions is evaluated with respect to an independently generated L-scenario problem, where M < L  K. We assume that the probability associated with each ^  2 fu ^ n gNn¼1 be the candidate solution that scenario is pL ¼ 1=L. Let u yields the highest objective value for this L-scenario problem. ^  is a feasible solution to the exact explosion problem, it Since u can be used to estimate a lower bound, LB, on Z EXP . To ensure that LB is an unbiased estimator, the sample used to estimate the lower bound must be independent of the sample that was used to iden^  . Therefore, we generate yet another L-scenario problem and tify u ^. estimate the lower bound by evaluating its objective function at u ^  Þ. The optimality gap is estimated by That is, LB ¼ Z EXP ðL; u GAP ¼ UB  LB. As noted by Linderoth et al. (2006), GAP overestimates the true optimality gap and is therefore a conservative estimate. In general, let ½aþ ¼ maxf0; ag. An approximate ð1  aÞ-level ^  is given by confidence interval for the optimality gap at u

  ^ u tL1;a=2 r ^‘ tN1;a=2 r pffiffiffiffi pffiffiffi C:I:ða; GAPÞ ¼ 0; ½GAPþ þ ; þ N L

ð30Þ

where t n1;a=2 is the t-value for a two-tailed Student’s t-distribution ^ 2u and with n  1 degrees of freedom and tail probability of a, and r r^ 2‘ are the sample variance estimators for Var½Z EXP ðMÞ and ^  Þ, respectively. Var½Z EXP ðK; u Similarly, we can compute a confidence interval for the optimality gap of a myopic solution. Suppose that the myopic solution ~ 1 ; u ~ 2 ; . . . ; u ~ T . An unbiased estimate of a lower ~  ¼ ½u is given by u bound on Z EXP is computed by evaluating the objective function, ~  , for an independently generated L-scenario problem. That is, at u ~  Þ. Since the myopic solution is feasible to the LBmyopic ¼ Z EXP ðL; u T-period problem, this approximation is only applied to the lower bound estimate. The optimality gap obtained using the myopic solution is GAPmyopic ¼ UB  LBmyopic , where UB is as defined in

ð31Þ

~  Þ. ~ 2‘ is the sample variance estimator for Var½Z EXP ðK; u where r

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