Production, Manufacturing and Logistics
Optimal control of a dual service rate M/M/1 production-inventory model

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Abstract

We analyse a dual-source, production-inventory model in which the processing times at a primary manufacturing resource and a second, contingent resource are exponentially distributed. We interpret the contingent source to be a subcontractor, although it could also be overtime production. We treat the inventory and contingent sourcing policies as decision variables in an analytical study and, additionally, allow the primary manufacturing capacity to be a decision variable in a subsequent numerical study. Our goal is to gain insight into the use of subcontracting as a contingent source of goods and whether it can fulfill real-world managers' expectations for improved performance. We prove that a stationary, non-randomised inventory and subcontracting policy is optimal for our M/M/1 dual-source model and, moreover, that a dual base-stock policy is optimal. We then derive an exact closed-form expression for one of the optimal base stocks, which to our knowledge is the first closed-form solution for a dual-source model. We use that closed-form result to advantage in a numerical study from which we gain insight into how optimal capacity, subcontracting, and inventory policies are set, and how effectively a contingent source can reduce total cost, capacity cost, and inventory cost. We find that (i) the contingent source can reduce total cost effectively even when contingent sourcing is expensive and (ii) contingent sourcing reduces capacity cost more effectively than it does inventory cost.

Introduction

Manufacturing companies cope with variability in demand and supply by maintaining buffers of capacity or inventory, or by having longer lead times. A second source of goods in the form of subcontracting is increasingly being used to provide a capacity buffer. Subcontracting in the circuit board assembly industry, for example, is growing rapidly. The total electronics industry revenue has grown recently on the order of 6% while subcontractors' revenues have grown on the order of 20% (see Gordon, 1999). Managers' motivations for subcontracting in the circuit board assembly industry are

  • 1.

    Leveraging subcontractor's lower operating costs.

  • 2.

    Responding to peak demand and/or balancing manufacturing loads.

  • 3.

    Deferring or avoiding capital expenditures for facilities, equipment and employees.

  • 4.

    Reducing inventory.


(See Gordon, 1995; Winter, 1995; Kenton, 1995 and Prasad, 1995.) We build a model of a manufacturer and its subcontractor in this paper based on these motivations. Our model's structure allows the subcontractor to have a lower total cost structure and the manufacturer can subcontract intermittently to relieve demand spikes. The last two points indicate an orientation toward cost savings and, in particular, reducing capacity and inventory cost. Accordingly, our objective function is constructed to assess the effectiveness of subcontracting in reducing these costs and, in addition, includes the cost of subcontracting and the manufacturer's in-house variable cost of manufacture. These latter two costs are relevant because the premium paid in the subcontractor's unit price over and above the manufacturer's unit variable cost for each unit procured from the subcontractor is the cost of whatever capacity and inventory reduction might be attained.

We develop and analyse an integrated production-inventory model with two sources of goods in which the interarrival times between orders are exponentially distributed, as are the processing times at the manufacturer's in-house facility and the second source. We interpret the second, or contingent source, as a subcontractor, although it could also be interpreted as the manufacturer's overtime production. (When the secondary source is overtime production, our model gives guidance in constructing overtime work schedules, negotiating the maximum allowable overtime schedule with the workforce, which is often times part of union-management agreements, and sizing the workforce and other determinants of production capacity.) The importance of integrating production and inventory in our model is that the replenishment lead time depends on the order quantity––high demand causes congestion and subsequent delay in replenishment in our model whereas replenishment lead times are often represented as being independent of the demand level. We prove that the optimal policy is stationary and, subsequently, that the optimal stationary control of inventory and the flow of subcontracted goods is a dual base-stock policy, in which one base stock governs the maximum level of goods and the scheduling of the manufacturer's in-house production capacity, and the second base stock controls the flow of goods from the subcontractor. We compute a closed-form optimal solution for one of these base-stock parameters. That closed-form optimal base-stock solution facilitates a numerical investigation in which we compute the manufacturer's optimal policies for in-house capacity and the two base-stock parameters as well as the resulting performance measures. Thus we assess the effectiveness of subcontracting in controlling capacity and inventory costs, and how subcontracting is optimally used to attain these results.

Of our four contributions, three are technical in nature, which facilitate the final contribution toward managerial insight. Specifically, for our integrated production-inventory model (i) we prove that from within a broad class of policies the optimal policy for managing inventory and a contingent source is stationary, (ii) we prove as an extension of Crabill's (1972) work that the optimal stationary policy is a dual-base stock policy, (iii) we compute the aforementioned exact optimal closed-form base-stock solution, and (iv) we develop insight into the joint management of in-house capacity, inventory, and subcontracting. The first two technical results extend the literature on the optimal control of two sources of goods to another setting. However, because much research has focused on identifying the optimal policy structure for similar problems, perhaps our greatest contribution is the computation of an exact closed-form expression for the base-stock that is to our knowledge the first closed-form solution for a dual-source model, which enables a numerical analysis in which the control policy values are computed. Little emphasis has been placed in the literature on the computation of optimal policy values, which in this paper gives a detailed managerial perspective on the mechanics of optimal control and the effectiveness of subcontracting in reducing capacity and inventory.

Our work is related to others who have analysed two-source replenishment models, such as Daniel (1963) and Fukuda (1964) who prove the structure of the optimal inventory policy in periodic models, each for a different characterization of the response time from the second source. Although knowing this structure is useful, two limitations of these analyses relative to ours is that they do not address the issue of setting the quantitative policy parameters and they consider neither the manufacturer's nor the subcontractor's capacity as a decision variable. Furthermore, the replenishment lead times are independent of order quantities in Daniel's and Fukuda's, whereas our integrated production-inventory models imposes replenishment delay when demand is high, which is a realistic dynamic in many settings.

Many other papers that do provide for numerical analysis also assume unlimited overtime capacity. Dellaert and Melo (1998) analyse the multi-item stochastic lot-sizing problem with overtime, but overtime production is not limited. Duenyas et al. (1997) and Hopp et al. (1993) analyse a discrete-time pull manufacturing model with overtime. The probability of not attaining a production quota with overtime in each period in these models is negligible, and so the sum of primary plus overtime capacity is practically unlimited. Our inclusion of a finite production rate for both the manufacturer's capacity and the second source reflect the real-world limitations on production rates, so that our model is capable of ascertaining the effect of a more, or less responsive alternate source of goods.

Karmarkar et al. (1987) do vary both capacity and overtime in their model, but no results are provided in closed form. Instead, point solutions are computed using an optimisation routine.

The research of Arslan et al. (2001) is closest to ours. In a parallel work, they prove the optimal inventory policy structure for both continuous and discrete-time M/G/1 and G/M/1 models with an alternate source of goods and make-to-order production. They also provide an expression from which inventory costs can be calculated for an M/M/1 model, although no closed-form expression for the optimal policies is possible. We generalize some aspects of the M/M/1 model by Arslan et al. by allowing make-to-stock production (make-to-order production can be represented in our model by setting the base stock to a non-positive value), providing for the possibility of an unlimited number of backorders, and limiting the capacity of the second source. The model of Arslan et al. has some features that we do not incorporate including a fixed lead time from the alternate production source (lead time in our model is due strictly to the effects of congestion) and a fixed cost for each order from the alternate source.

A stream of related work on dynamic control of an M/M/1 model minimises the cost of entities in the queue by selecting optimal service rates, which may vary with the system state (Crabill, 1972; Stidham and Weber, 1989; Weber and Stidham, 1987; and George and Harrison, 2001). Cost accumulates at a rate that is non-decreasing in the queue length in this literature. The cost rate in our model, in contrast, is first decreasing and then increasing in the queue length, which is necessary to model finished goods inventory cost in the traditional manner where the cost rate first decreases as on-hand inventory decreases and, then, after inventory is depleted, increases linearly as the number of backorders increase. In addition, the aforementioned literature does not consider an inventory integrated with the production resource. Thus we extend Crabill's (1972) result to an integrated production-inventory system with a different cost structure.

A general model of the subcontracting problem is introduced in Section 2. We introduce an M/M/1 subcontracting model in Section 3, prove that a stationary control policy is optimal, and, moreover, prove that a stationary dual base-stock policy is optimal. A closed-form expression for one optimal base stock is presented in Section 4. Our numerical investigation of optimal policies is described in Section 5. We analyse the effect of capacity that can be increased only in discrete increments that decrease in marginal cost, i.e., that exhibit economies of scale, in Section 6. We conclude in Section 7.

Section snippets

A general dual-source production-inventory model

We model a single-stage, single-server, make-to-stock manufacturing system in which a single product is produced. Orders are fulfilled from a finished-goods inventory. All demand that cannot be immediately satisfied is backlogged and subsequently fulfilled on a first-come, first-served basis when product becomes available. Goods are supplied to inventory from either an in-house production facility or a second source, both of which have limited capacity. We will refer to the second source as the

The manufacturer's optimal policy is stationary

Hereafter, we assume that U=(Un:n⩾1) and V=(Vn:n⩾1) are each a sequence of independent and identically distributed exponential random variables. Furthermore, U and V are independent of each other. Then the processing times are exponentially distributed with a mean of v/β when only the manufacturer's in-house capacity is used and v/(β+γ) when the contingent source is used. Note that this is a birth-death process in which the death rate depends on the production mode––the death rate is faster

The optimal base stock

To compute the optimal base stock values we compute the expected cost as a function of s1 and b from the steady-state shortfall probability mass function. Subsequently, we compute the optimal base stock s1 in closed form. Define the utilization rates ρ1=λv/β and ρ2=λv/(β+γ) for when only in-house capacity is used and when both resources are used, respectively. Then the steady-state shortfall probability mass distribution for the M/M/1 dual-source model using a dual base-stock policy ispβ,γ,b(y)=

An analysis of optimal capacity, inventory, and subcontracting policies

We have established the optimality of a stationary dual base-stock policy and computed a closed-form expression for one of the optimal base-stocks for the M/M/1 production-inventory model, which enables the investigation upon which we embark in this section. Returning to the motivation for this paper, we investigate whether managers can attain the goals that motivate them to subcontract if they used optimal base-stock and capacity policies. Specifically, we evaluate how in-house capacity,

“Lumpy” capacity and economies of scale

We have considered in previous sections a continuous capacity cost that increases linearly in the capacity level. Two realities not reflected in this basic model are “lumpy” capacity and economies of scale. Capacity is “lumpy” when it can be increased only in significant discrete increments. For example, in the commodity paper business, where firms must offer goods at the competitive market price, the largest scale paper mill may be the only economically feasible investment, the output of which

Conclusions

We have made both theoretical and practical contributions in this paper. Theoretically, we have shown that a stationary dual base-stock policy is optimal for the M/M/1 integrated production-inventory model with two production sources. This required one proof to show that from a class of possibly randomised non-stationary policies that a non-random stationary policy is optimal, and another proof to show the optimality of a dual base-stock policy. The latter is an extension of Crabill's results

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