Capacity planning, contract selection, and financial diversification

Integrated strategy for risk control: capacity planning, contract selection, and financial diversification

H. Steve Peng


A model is proposed to help a contract manufacturer (CM) identify the best mix among competing contract opportunities and the optimal allocation of resources to balance between profitability and risk exposure. The decisions are made to maximize the probability of meeting or exceeding a given profit target. We show that the objective of maximizing expected earnings can be justified only when there is less than 50% chance of reaching the profit target. When the target is more achievable (more than 50% chance), pursuing the goal of profit maximization will generally be too risky for the manager. Under conditions of constrained resources, our model highlights the important role of risk control and makes explicit the linkage between overall financial performance and the operational decisions of capacity planning, capital allocation, and contract selection. We also compare the effectiveness of controlling risk through operational decisions with that of a financial approach using financial securities to form a portfolio that can compensate for the financial uncertainty from operations. Our results show that when the unit return of the portfolio is significantly lower than that of investing on the production line, and when the portfolio cannot closely track the cash flow uncertainty from operations, it is more effective to enhance the probability of surpassing the target through operational decisions.


In contrast to the supply relationships with standardized off-the-shelf goods, when a contract In contrast to the supply relationships with standardized off-the-shelf goods, when a contract manufacturer (CM) commits to a relationship of Early-Supplier-Involvement (ESI) in new product development, this usually requires the CM to make commitments at a stage when there is still a great deal of uncertainty about demand and product design details. Since a competitive CM may have multiple contract opportunities, the selection among contract opportunities is analogous to optimizing a portfolio. So our decision model for a supplier/CM captures the portfolio effects that arise when choosing the best mix among contract opportunities to optimally balance between two classes of criteria: profitability and risk. While we choose the aviation industry as inspiration for our modeling, the conclusions and insights are broadly applicable to other industries, such as electronics and semiconductor, where ESI plays an important role in new product development.

Our model is inspired by the situation of a Japanese CM of aircraft frames who is actively involved in numerous aircraft projects worldwide which include designing and producing partial fuselage or wings for regional aircrafts, business jets, helicopters, and large airliners. For each project or contract, the CM needs to make a “relationship-specific investment”, such as design, prototype, molds, tooling, and dedicated assembly lines before the actual size of the order is known. Most capacity planning and contract selection models in the field of operations management are aimed at maximizing the expected profit without considering the uncertainty of future cash flow. As a result, the motivation of our model is to explore the potential of using operational decisions to control risk. The specific objective function in our model is to help a CM maximize the probability of achieving a given monetary target within the planning period. This objective function is common for evaluating management performance by outsiders or shareholders (Browne 2000).

Gardner and Buzacott (1999) consider the problem of investing in major facility expansions under technology uncertainty. They find that a whole spectrum of decisions such as facility sizes and construction sequences can change the pattern of the project outcome and can be used to hedge the risk faced by the firm. To evaluate the uncertainty associated with a single newsvendor problem, Singhal (1988) proposes a model that uses the capacity decision to determine the optimal mean and variance of the profit faced by a newsvendor. Birge and Zhang (1999) reach the same result as Singhal’s but with a simpler expression. They use an option pricing approach that assumes a project’s future demand is a tradable asset and use this underlying asset to derive the optimal capacity investment. Carr and Lovejoy (2000) propose a model for forming a portfolio among all potential contracts that will maximize the firm’s expected profit. Like most operations management models, however, all the papers above focus only on the objective function of maximizing the expected earnings and do not describe explicitly how to implement risk control. Our model explicitly considers the risk and sets the decision maker’s objective as maximizing the probability of surpassing a given monetary target. In addition, we use a hybrid approach to control the operational risk. Therefore, the interaction among contract selection, capital allocation, and capacity planning better reflects the reality in decision-making.

While there are a number of financial models for valuing or selecting risky opportunities, most of them consider a contract or project as a known cash flow stream and imply that the management may select but have no control over the pay off associated with each project or contract. Devinney and Stewart (1988) directly apply financial portfolio models and optimize the product line decision through fixed profit and risk matrixes. This overlooks the fact that for many industrial firms, the monetary outcome is a complicated mapping from a set of operational decisions. Smith and Nau (1995) discuss the situation where the uncertainties of potential projects cannot be fully traced to derive a precise option price by using unlimited funding resource based on the theory of cost of capital (cf. Huang and Litzenberger 1988; Ross et al 1996). However, in operations the available capital for most industrial firms is considered to be limited and usually the firm cannot access a capital market freely, i.e., unlimited borrowing or lending, under the risk-free interest rate. That means few firms, except some financial institutes, can take full advantage of the market mechanism to diversify the operational risk.


Consider a CM who has a given set of potential contracts, i = 1,2, .., M. The CM has to make two choices based on his best interest. The first decision [s.sub.i] indicates the portion of job that the CM commits to produce for the contract proposal i. For example, a contract proposal of fuselage system contains Y sections, where Y can be tens or even over one hundred. The CM may decide to produce y sections (i.e. a portion [s.sub.i] = y/Y of the whole system in the contract), which depends on the contract proposed by an aircraft producer and the CM’s current operational constraints. That means, after reviewing all the contract proposals, the CM can decide either to carry the whole contract ([s.sub.i] = 1), a portion (0 < [s.sub.i] < 1), or not to participate at all (s, = 0). We consider s, as a continuous variable for simplicity.

The second decision the CM needs to make for each contract job i with participation portion [s.sub.i] is to set a quantity cap [x.sub.i] which is the maximum quantity of the subsystem that can be delivered for the contract i by the end of the period. In making these two decisions, the CM needs to assign a dedicated capacity [k.sub.i] = [[phi].sub.i]([s.sub.i] [x.sub.i]) to the contract job i, where [[phi].sub.i] is the function of required capacity. At the time when the CM decides on production capacity and participation level, the demand [D.sub.i] of the contract job i is not known, only its probability distribution. Also, the CM knows that contract i identifies a payment of [[alpha].sub.i]([s.sub.i]) for each subsystem delivered, and there is a required penalty payment by the CM of [[beta].sub.i]([s.sub.i]) for each subsystem demanded and not delivered. To deliver each subsystem for the contract i, the CM incurs a cost [c.sub.di]([s.sub.i]). To concentrate on the issue of risk control, in the following analysis, we consider a special case that all functions are linear with the CM’s decision variables [s.sub.i] and [x.sub.i]. In specific, [k.sub.i] = [s.sub.i][x.sub.i], [[alpha].sub.i]([s.sub.i]) = [[alpha].sub.i][s.sub.i], [[beta].sub.i]([s.sub.i]) = [[beta].sub.i]([s.sub.i]) and [c.sub.di]([s.sub.i]) = [c.sub.di]([s.sub.i]). All the coefficients are non-negative and the generalization to include nonlinear cost structure is straightforward.

Total production infrastructure K of the CM is limited and is shared by all contracts selected. This limit could be in terms of floor space, engineering man-hours, or some other indicator of the bottleneck resource. In the beginning of the period, the production infrastructure can be configured into the dedicated production capacity for the contract i with a unit configuration cost []. This may represents the cost of ordering special ground tools or changing bay layout. For producing commercial aircraft frames, the time for configuring a general production capacity into a dedicated production capacity is relatively long. As a result, in our case, it is appropriate to assume that a production capacity dedicated to contract i cannot be switched to any other contract j within the relevant time horizon, for example, one year. In the beginning of the period, the CM has a total available working capital of W. Unused capital can be committed to a safe contract 0 with a unit return but there is no need for physical production capacity to be allocated to the contract 0.

Let [[pi].sub.i] denote the profit from contract i. We define the random variable [N.sub.i] = min([D.sub.i], [x.sub.i]) as the actual number of subsystems delivered under the contract i. Then it follows that

[[pi].sub.i] + -[] [s.sub.i] [x.sub.i] + ([[alpha].sub.i] – [c.sub.di]) [S.sub.i][N.sub.i] – [[beta].sub.i] [s.sub.i] ([D.sub.i] – [N.sub.i])

Define [pi] as the total profit of the CM for given choices of [s.sub.i], [x.sub.i].

[pi] = [m.summation over (i=0)] [[pi].sub.i] ([s.sub.i], [x.sub.i])

The CM may want to consider strategies other than maximizing expected profit if the associated risk can be reduced. For a given expected profit[mu] where 0 [less than or equal to] [mu] < [??], the efficient hedging strategy is defined as the one which leads to the minimum variance of the CM's total profit among all feasible strategies. We only need to pay attention to the efficient hedging strategies because for any [mu], the CM generally would like to keep the variance to be as small as possible. A strategy [rho]([mu]) = {[s.sub.i], [x.sub.i]}, i = 1, 2 ,.., M will be an efficient hedging strategy if and only if it is a solution to the program of variance minimization (1).


(1b) [summation over(i[not equal to])] [s.sub.i][x.sub.i] [less than or equal to] L

(1c) [SIGMA] [][s.sub.i][x.sub.i] [less than or equal to] W

[s.sub.i] [member of] [0,1], [x.sub.i] [greater than or equal to] 0


This section analyzes and evaluates the potential benefits of including financial securities in the hedging strategy to increase the probability of the CM reaching a given profit target. Besides using working capital to set up production capacities, we now assume the firm can also choose to invest the money in securities markets at the beginning of the period and sell them at the end. It is common for a firm to invest some short-term surplus of working capital in the money market to earn higher interest rates. However, when working capital is a scarce resource we would like to know under what conditions the manager would include financial securities in the hedging strategy to enhance risk management.

3.1. The Generalized Efficient Frontier

The generalized function of the efficient frontier [[sigma].sup.2]([mu]) is stated in (2) to include the financial diversification. The new inputs to this procedure are the expected return for each security and the co-variances between all securities and the outcomes of contract opportunities. We follow the assumption that covariance can be neglected among the outcomes of all contract opportunities due to the distortion of mapping from demands to cash flows.


Because of the linear nature of the financial portfolio, it is easy to show that Propositions 1 and 2 will still hold when the financial securities are included in the optimization.

3.2. Boundary Condition for Acquiring Financial Securities

Here we compare the effectiveness of controlling risk through operational decisions with that of using a portfolio of financial securities to compensate for the financial uncertainty from operations. In the rest of this section we will identify the boundary condition under which financial hedging can be justified for enhancing the achievement of a stated profit target.

In order to concentrate on comparing financial hedging with contract selection and capacity planning, we assume that any tracking portfolio can be formed with zero transactional cost. When there are significant tracking costs, the hedging efficiency will certainly drop and make financial hedging less attractive. Let [psi] be one unit of an arbitrary portfolio.

[psi] = {[S.sup.F.sub.1], [S.sup.F.sub.2], …, [S.sup.F.sub.N]}/[SIGMA][[S.sup.F.sub.i]

We define R as this portfolio’s unit return in the end of period, V is the aggregated variance of [psi](a unit of the portfolio), and COVs([[pi].sub.i]) is the covariance between [psi] and the contract [??]s outcome [[pi].sub.i]. It is worth noting that for any financial portfolio, R = 1 when the systematic risk is zero and R > 1 when the systematic risk is greater than zero. This is based on the fact that in an efficient asset market the expected return of any risky asset must be greater than that of the safe asset.

Given a set of shadow prices [[lambda].sub.a], [[lambda].sub.b] and [[lambda].sub.c], it follows that the first order conditions are necessary and sufficient for global optimization. As a result, for this portfolio, the optimal purchase unit S needs to meet the following condition

(3) [partial derivative]/[[partial derivative]S [[S.sup.2]V + S [m.summation over(i=1) [cov.sub.s] ([[pi].sub.i]) – [[lampda].sub.a] S (R-1)+ [[lambda].sub.c] S] = 2SV + [SIGMA] [COV.sub.s] ([[pi].sub.i]) – [[lampda].sub.a] (R-1)+ [[lambda].sub.c] = 0

For any risky (i.e. V > 0 and R > 1) portfolio to be included in the frontier strategy, the sufficient and necessary condition is that there exists an S > 0 such that (3) holds. This is equal to

[SIGMA] [COV.sub.S] ([[pi].sub.i]) – [[lampda].sub.a] (R-1)+ [[lambda].sub.c] = 0

PROPOSITION 3: The CM will include any financial portfolio in the frontier strategy if and only if

(4) [SIGMA] [COV.sub.s] ([[pi].sub.i]) – [[lampda].sub.c] /R-1 + [[lambda].sub.a]


Our model considers a firm selecting contracts from a set of competing contracts and making appropriate capacity allocations. We have explained the importance of risk control for a resource-constrained firm that wishes to optimize the decisions of allocating resources, selecting contract mix, or forming security portfolio to increase the possibility of reaching the organizational objectives. We have demonstrated how the profitability and risk exposure can be balanced among different opportunities. We have showed that when the unit return of the financial portfolio is significantly lower than that of investing on the production line, and when the portfolio cannot closely track the cash flow uncertainty from operations, it is more effective to enhance the probability of surpassing the target through operational decisions. Since we have constructed the efficient frontier of the firm’s aggregated profit, it is convenient to apply our model to objectives with different risk measures. This model can be applied to analyze a variety of product line decisions such as which existing goods and services will be offered and in which new products and markets the firm will participate in the future. The answer to any of these questions requires that the firm commit its scarce and valuable capital only after thorough and comprehensive evaluation of all risks. Consequently, all these strategic issues fall under the general heading of risk control.


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Author Profile:

Dr. Steve Peng earned his Ph.D. at York University, Canada, in 2001. Currently he is an assistant professor of operations management at California State University, Hayward.

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