# Optimal Control of a Make-to-Stock System with Outsourced Production and Price-Sensitive Demand

Authors: Liuxin Chen; Gang Hao; Huimin Wang
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

We consider a make-to-stock system with controllable demand rate (by varying product selling price) and adjustable service rate (by outsourcing production). With one outsourcing alternative
and a choice of either high or low price, the system decides at any point in time whether to produce or even outsource for additional capacity as well as which price to sell the product at. We show in the paper that the optimal control policy is of dynamic threshold type: all decisions are based on the product inventory position which represents the state of the system; there is a state dependent base stock level to decide on production and a higher level on outsourcing; and there is a state dependent threshold which divides the choice of high and low prices.


---

## Body

## 1. Introduction

A case study of Mattel, the world’s largest toy maker, was done by Johnson [1] with a focus on its production capacity management. In particular, Johnson reported that Mattel owned a state-of-the-art die-cast facility in Penang, Malaysia, that was operating at full capacity to produce die-cast toy vehicles. Due to surge of demand for Hot Wheels, a core line of product, Mattel considered several options to expand production capacity, including one through the Vendor Operations Asia Division to outsource production in Asia-Pacific. VOA added flexibility to Mattel’s in-house manufacturing capability and was one of the company’s most valuable assets. In the meantime Mattel managed demand for the Hot Wheels through a new marketing strategy that changed the assortment mix of cars every two weeks.In this paper, we consider a single-product make-to-stock system that has the option to increase production capacity by outsourcing to external contract manufacturers. The systems can also manage product demand through adjusting its selling price. For the basic setting of one outsourcing alternative and a choice of either high or low price, the system optimal control problem is to decide at any point in time whether to produce at the in-house facility or to outsource for additional capacity as well as which price to sell the product at. We model the production processes at the in-house and external facilities by exponential times of different means and the demand process by a Poisson process with a price-dependent rate. Thus, mathematically, the problem is optimal control of anM
/
M
/
1 make-to-stock queue with discretely adjustable production and demand rates.With the objective to maximize the total discounted profit, we show in the paper that the optimal control policy is of dynamic threshold type: all decisions are based on the product inventory position which represents the state of the system; there is a state dependent base stock level to decide on production and a higher level on outsourcing; and there is a state dependent threshold which divides the choice of high and low prices. Furthermore, we show that, for a given outsourcing production capacity, all three thresholds of the optimal control policy and the associated optimal profit are decreasing in the outsourcing cost. This implies that outsourcing to lower cost facilities will lead to lower inventory holdings and lower selling price but higher profit.There is a rich literature on the optimal control ofM
/
M
/
1 make-to-stock queues, most of which take demand as exogenous and solve for the optimal control of the production rate. Typically, the optimal policy is of base stock type (produce at the maximum rate when the inventory holding falls below certain level, otherwise halt production), which was first proved by Gavish and Graves [2] and Sobel [3] for the single product and single machine case. Later works of Zheng and Zipkin [4], Wein [5], Veach and Wein [6], and Bertsimas and Paschalidis [7] attempt to extend the base stock policy to the multiple product cases. There are also extensions to the case of single product but with multiple demand classes, like Ha [8–10]. Another direction of extension has been to incorporate more detailed modeling of the production facility. For example, Kapuscinski and Tayur [11] model the production process by a tandem queue, and Feng and Yan [12] and Feng and Xiao [13] deal with unreliable production facilities.Li [14] and Chen et al. [15, 16] are three works that incorporate controls on both production and demand processes in a make-to-stock queue optimization problem. Li [14] assumes a continuous spectrum of product selling prices and corresponding demand rates and, hence, manages to derive a concave and differentiable profit function in terms of the production rate and the selling price, which yields a qualitative characterization of the optimal policy. Chen et al. [15] allow discrete choices of prices and derive an efficient algorithm to compute the optimal policy as well as its qualitative characterization. Chen et al. [16] consider a make-to-stock manufacturing system with batch production and discrete choices of price and derive the characterization of the optimal control policy. Similar work on a make-to-order queue is done by Ata and Shneorson [17]. Carr and Duenyas [18] study the optimal control of a mixture of make-to-stock and make-to-order queues. Our work adds in another dimension with the outsourcing option to expand production capacity.The rest of the paper is organized as follows. Section2 describes precisely the system model and defines an optimization problem that solves for the optimal policy. Section 3 characterizes the optimal threshold policy, proves its global optimality amongst all nonanticipative control policies, and discusses its relationship to the cost of outsourcing. Section 4 briefly discusses the extension to multiple price choices. Section 5 concludes the paper with a summary of the results and possible extensions in the future research.To streamline presentation of the paper, we state in the main body of the paper all the results without proofs and collect all the proofs in the appendix.


## 2. Problem Formulation

The make-to-stock system of concern in the paper has an in-house facility with a production rateμ and a unit production cost b. The system can outsource production to an external facility which can produce at a rate a and a per unit cost c. We assume that the existing in-house facility has a lower variable production cost than the external facility, that is, b
<
c, which holds true in the case of Mattel, for example, and is the reason for keeping the in-house facility. We also assume that the production processes at both facilities are random and follow exponential distributions.The demand process for the product is assumed to be a Poisson process with a price-dependent rate. Specifically, there are two selling prices: highp
1 and low p
2, which correspond to two demand rates: λ
1 and λ
2. We assume that λ
1
<
λ
2 and
(1)
λ
2
p
2
-
λ
1
p
1
λ
2
-
λ
1
>
b
,
which indicates that the marginal profit gain from switching price from high to low is greater than the in-house production cost b. Also to ensure system stability, we assume that μ
+
a
>
λ
1.When a demand arrives, it is filled from the finished goods inventory if possible; otherwise, it is added to a waiting queue which is served in first-come-first-serve order. The finished goods inventory carries a holding cost ofh
+ per unit product per unit time, and the backordering cost for demand unmet at arrival is h
- per waiting demand per unit time. Define the inventory cost function h
(
x
)
=
h
+
x
+
+
h
-
x
-, where x
+
=
max
⁡
[
x
,
0
] and x
-
=
max
⁡
[
-
x
,
0
]. We will consider the total discounted profit, assuming a discount factor γ. To ensure that it is more profitable to produce to fill demand than to backlog forever, we assume that b
<
c
<
h
-
/
γ.We specify a dynamic control policy for the system byu
=
{
μ
(
t
)
,
a
(
t
)
,
p
(
t
)
:
t
>
0
}, where μ
(
t
)
=
0 or 1 representing in-house production is off or on; similarly, a
(
t
)
=
0 or 1 representing outsourced production being off or on, and p
(
t
)
=
p
1 or p
2 representing the price charged at time t. A policy u is callednonanticipatory if, at all t
>
0, μ
(
t
), a
(
t
), and p
(
t
) depend only on information prior to t. Let U be the collection of all nonanticipatory control policies. Under a given u
∈
U, denote the total demand sold at price p
i up to time t
>
0 by N
i
u
(
t
)
,
i
=
1,2, the total in-house production by P
I
u
(
t
), and the total outsourced production by P
O
u
(
t
). Then, the product inventory level at time t is given by
(2)
X
u
(
t
)
=
x
+
P
I
U
(
t
)
+
P
O
u
(
t
)
-
N
1
u
(
t
)
-
N
2
u
(
t
)
,
where x is the initial inventory at t
=
0.Consequently, the total discounted profit under policyu is
(3)
V
u
(
x
)
=
E
[
∫
0
∞
e
-
γ
t
(
∑
i
=
1
2
p
i
d
N
i
u
(
t
)
-
b
d
P
I
u
(
t
)
-
c
d
P
O
u
(
t
)
-
h
(
X
u
(
t
)
)
d
t
∑
i
=
1
2
p
i
d
N
i
u
(
t
)
-
b
d
P
I
u
(
t
)
)
]
.A policyu
*
∈
U is said to beoptimal if it solves the following optimization problem:
(4)
V
u
*
(
x
)
=
sup
⁡
u
∈
U
V
u
(
x
)
.The optimal solution to this semi-Markov decision problem can be characterized by the following Hamilton-Jacobi-Bellman (HJB) equation (cf. Chapter 7 of [19]):
(5)
0
=
-
γ
V
(
x
)
-
h
(
x
)
+
max
⁡
α
=
0
,
μ
α
[
V
(
x
+
1
)
-
V
(
x
)
-
b
]
+
max
⁡
β
=
0
,
a
β
[
V
(
x
+
1
)
-
V
(
x
)
-
c
]
+
max
⁡
j
=
1,2
λ
j
[
V
(
x
-
1
)
-
V
(
x
)
+
p
j
]
.Sincec
>
b, we have V
(
x
+
1
)
-
V
(
x
)
-
c
<
0 when V
(
x
+
1
)
-
V
(
x
)
-
b
<
0, and thus, β
=
0 when α
=
0. In essence, we can envisage an effective production process with rates μ
0
=
0, μ
1
=
μ, and μ
2
=
μ
+
a corresponding to the unit production costs b
0
=
0, b
1
=
b, and b
2
=
(
μ
b
+
a
c
)
/
(
μ
+
a
), respectively. As a result, HJB equation (5) can be simplified to
(6)
0
=
-
γ
V
u
(
x
)
-
h
(
x
)
+
max
⁡
i
=
0,1
,
2
μ
i
[
V
(
x
+
1
)
-
V
(
x
)
-
b
i
]
+
max
⁡
j
=
1,2
λ
j
[
V
(
x
-
1
)
-
V
(
x
)
+
p
j
]
.We are especially interested in a class of control policies which are parameterized by three thresholds:R, D, and S, with max
⁡
{
R
,
D
}
<
S. An (
R
,
D
,
S
) policy decides on production, outsourcing, and pricing in the following manner: (1) when the inventory is above or equal to S, there is no production at the in-house facility and no production outsourcing; (2) when it is below S and above or equal to D, production is on at the in-house facility but there is no outsourcing; (3) when it is below D, production is on at the in-house and outsourced to the external facility; (4) the product sale price is set low at p
(
t
)
=
p
2 when it is above R, and the produce selling price is set low at p
2; otherwise, the price is high at p
1. In Section 3 below, we characterize the best (
R
,
D
,
S
) policy and verify that it satisfies the above HJB equation (6) and, thus, is optimal amongst all policies in U.


## 3. Optimality of(
R
,
D
,
S
) Policy

The HJB equation (6) can be made more specific when given an (
R
,
D
,
S
) policy. For example, for an (
R
,
D
,
S
) policy with 0
<
R
<
D
<
S, it can be simplified to the following equations.Forx
≥
S,
(7)
0
=
-
γ
V
(
x
)
-
h
+
x
+
λ
2
[
V
(
x
-
1
)
-
V
(
x
)
+
p
2
]
;
for D
≤
x
<
S,
(8)
0
=
-
γ
V
(
x
)
-
h
+
x
+
λ
2
[
V
(
x
-
1
)
-
V
(
x
)
+
p
2
]
+
μ
[
V
(
x
+
1
)
-
V
(
x
)
-
b
]
;
for R
<
x
<
D,
(9)
0
=
-
γ
V
(
x
)
-
h
+
x
+
λ
2
[
V
(
x
-
1
)
-
V
(
x
)
+
p
2
]
+
(
μ
+
a
)
[
V
(
x
+
1
)
-
V
(
x
)
-
μ
b
+
a
c
μ
+
a
]
;
for 0
≤
x
≤
R,
(10)
0
=
-
γ
V
(
x
)
-
h
+
x
+
λ
1
[
V
(
x
-
1
)
-
V
(
x
)
+
p
1
]
+
(
μ
+
a
)
[
V
(
x
+
1
)
-
V
(
x
)
-
μ
b
+
a
c
μ
+
a
]
;
and for x
<
0,
(11)
0
=
-
γ
V
(
x
)
-
h
-
x
+
λ
1
[
V
(
x
-
1
)
-
V
(
x
)
+
p
1
]
+
(
μ
+
a
)
[
V
(
x
+
1
)
-
V
(
x
)
-
μ
b
+
a
c
μ
+
a
]
.LetV
(
R
,
D
,
S
)
(
x
) be the profit function of a given (
R
,
D
,
S
) policy. The following lemma characterizes some of its limiting behaviors.Lemma 1.
The profit function of an(
R
,
D
,
S
) policy has the following limits: (1)
lim
⁡
x
→
±
∞
[
V
(
R
,
D
,
S
)
(
x
)
-
V
(
R
,
D
,
S
-
1
)
(
x
)
]
=
0;
(2)
lim
⁡
x
→
±
∞
[
V
(
R
,
D
,
S
)
(
x
)
-
V
(
R
,
D
-
1
,
S
)
(
x
)
]
=
0;
(3)
lim
⁡
x
→
±
∞
[
V
(
R
,
D
,
S
)
(
x
)
-
V
(
R
-
1
,
D
,
S
)
(
x
)
]
=
0;
(4)
lim
⁡
x
→
+
∞
D
V
(
R
,
D
,
S
)
(
x
)
=
-
h
+
/
γ;
andlim
⁡
x
→
-
∞
D
V
(
R
,
D
,
S
)
(
x
)
=
h
-
/
γ.Our approach is to first find thebest
(
R
,
D
,
S
) policy and then to prove its global optimality.Definition 2.
An(
R
,
D
,
S
) policy is said to be better than an (
R
′
,
D
′
,
S
′
) policy if V
(
R
,
D
,
S
)
(
x
)
≥
V
(
R
′
,
D
′
,
S
′
)
(
x
) for any initial inventory level x and at least for an x, the inequality holds strictly. It is said to be the best (
R
,
D
,
S
) policy if no other (
R
,
D
,
S
) policies are better.For notational simplicity we also define a first-order difference operatorD:
(12)
D
g
(
x
)
=
g
(
x
)
-
g
(
x
-
1
)
,
for
any
function
g
(
x
)
.The following lemma provides means to compare(
R
,
D
,
S
) policies.Lemma 3.
Comparing with a given(
R
,
D
,
S
) policy, we have the following results: (1)
the(
R
,
D
,
S
+
1
) policy is better if and only if
(13)
D
V
(
R
,
D
,
S
)
(
S
+
1
)
>
b
o
r
D
V
(
R
,
D
,
S
+
1
)
(
S
+
1
)
>
b
;
(2)
the(
R
,
D
+
1
,
S
) policy is better if and only if
(14)
D
V
(
R
,
D
,
S
)
(
D
+
1
)
>
c
o
r
D
V
(
R
,
D
+
1
,
S
)
(
D
+
1
)
>
c
;
(3)
the(
R
+
1
,
D
,
S
) policy is better if and only if
(15)
D
V
(
R
,
D
,
S
)
(
R
+
1
)
>
λ
2
p
2
-
λ
1
p
1
λ
2
-
λ
1
,
o
r
D
V
(
R
+
1
,
D
,
S
)
(
R
+
1
)
>
λ
2
p
2
-
λ
1
p
1
λ
2
-
λ
1
.The first comparison implies that, under either the(
R
,
D
,
S
) policy or the (
R
,
D
,
S
+
1
) policy, if the marginal profit at inventory level x
=
S for producing one more unit of product is higher than the in-house production cost, then the (
R
,
D
,
S
+
1
) policy is better than the (
R
,
D
,
S
) policy. Similar implications can be drawn from the other two comparisons.Lemma3 leads to the following characterization of the best (
R
,
D
,
S
) policy.Theorem 4.
If the(
R
*
,
D
*
,
S
*
) policy is the best (
R
,
D
,
S
) policy, then (1)
at the best base stock levelS
*,
(16)
D
V
(
R
*
,
D
*
,
S
*
)
(
S
*
+
1
)
≤
b
<
D
V
(
R
*
,
D
*
,
S
*
)
(
S
*
)
;
(2)
at the best outsourcing thresholdD
*,
(17)
D
V
(
R
*
,
D
*
,
S
*
)
(
D
*
+
1
)
≤
c
<
D
V
(
R
*
,
D
*
,
S
*
)
(
D
*
)
;
(3)
at the best price switch thresholdR
*,
(18)
D
V
(
R
*
,
D
*
,
S
*
)
(
R
*
+
1
)
≤
λ
2
p
2
-
λ
1
p
1
λ
2
-
λ
1
<
D
V
(
R
*
,
D
*
,
S
*
)
(
R
*
)
.Furthermore, we have that V
(
R
*
,
D
*
,
S
*
)
(
x
) is concave in x.Finally, we obtain the main result of the paper based on the characteristics of Theorem4 as stated above.Theorem 5.
If the(
R
*
,
D
*
,
S
*
) policy is the best (
R
,
D
,
S
) policy, then its profit function V
(
R
*
,
D
*
,
S
*
)
(
x
) satisfies the HJB equation (6). Hence, the (
R
*
,
D
*
,
S
*
) policy is optimal amongst all nonanticipative policies of U.The optimal(
R
*
,
D
*
,
S
*
) policy has some important properties which we list below.Theorem 6.
In the best(
R
*
,
D
*
,
S
*
) policy, S
*
≥
0; and D
*
>
R
* if (
λ
2
p
2
-
λ
1
p
1
)
/
(
λ
2
-
λ
1
)
>
c; otherwise R
*
≥
D
*.Theorem6 tells us two results as the following. (i) Under the best thresholds (
R
*
,
D
*
,
S
*
) policy, the maximum inventory level S
* is nonnegative; and it is optimal to idle both the in-house and outsourced facilities when the stock level is above the maximum inventory level S
*. (ii) When production is on at both in-house and outsourced facilities and the inventory is increasingly building up, if the marginal profit gain when switching price from high to low is greater than the outsourced production cost c, it is more profitable to first stop outsourced production than to first stop in-house production and then decrease price. The converse is true if the marginal profit gain when switching price from high to low is smaller than the outsourced production cost c.Theorem 7.
As for fixed sourcing production ratea, suppose that the variable cost c of outsourced production is negotiable. Let (
R
*
(
c
)
,
D
*
(
c
)
,
S
*
(
c
)
) policy be the optimal threshold policy associated with a cost c. Then, (1)
the optimal thresholdsR
*
(
c
), S
*
(
c
) are piecewise constant, increasing functions of c, but D
*
(
c
) is a piecewise constant, decreasing function of c.
(2)
as for the optimal profit function,V
(
R
*
(
c
)
,
D
*
(
c
)
,
S
*
(
c
)
)
(
x
;
c
) is decreasing in c.Theorem7 concludes the following results. (i) A lower variable cost from outsourced production will lead to lower product selling price but higher safety stock level and more outsourced production. And (ii) a lower variable outsourced production cost will lead to higher optimal long-term discounted profit.


## 4. Extension to Multiple Price Choices

In this section, we briefly discuss the extension of the results in the previous section to multiple price choices. Namely, we have nowK
≥
2 possible prices to choose from for the selling of the product: p
1
>
p
2
>
⋯
>
p
K
>
c, with corresponding K demand arrival rates: λ
1
<
λ
2
<
⋯
<
λ
K.We further assume that the profit rates are also increasingly ordered; that is,(19)
λ
1
(
p
1
-
c
)
<
λ
2
(
p
2
-
c
)
<
⋯
<
λ
K
(
p
K
-
c
)
.It can be shown that if the profit rates do not follow this order, adominating subset of the price levels can be chosen to make the other prices unattractive for selection. Readers are referred to Chen, Feng, and Ou [20] for the analysis on dominating prices.TheK price choice model can be optimized by a policy which maximizes the right-hand side of the following Hamilton-Jacobi-Bellman equation:
(20)
0
=
-
h
(
x
)
+
max
⁡
α
=
0
,
μ
α
[
V
(
x
+
1
)
-
V
(
x
)
-
c
]
+
max
⁡
β
=
0
,
a
μ
[
V
(
x
+
1
)
-
V
(
x
)
-
c
]
+
max
⁡
i
=
1
,
…
,
K
λ
i
[
V
(
x
-
1
)
+
p
i
-
V
(
x
)
]
-
γ
V
(
x
)
,
where V
(
x
) is the profit function of the policy.The natural extension of the(
R
,
D
,
S
) policy as defined at the end of Section 2 is a K-level control policy characterized by K
+
1 parameters.Consider(
R
2
,
…
,
R
K
,
D
,
S
) with R
2
<
⋯
R
K
<
S and D
<
S. S is the base stock level on and above which there is no production at the in-house facility and also no production outsourcing. And when it is below S and above or equal to D, production is on at the in-house facility but no outsourcing; when it is below D, production is on at the in-house and outsourced to the external facility. The other K
-
1 parameters are price switch thresholds: when the inventory is in the range (
R
i
,
R
i
+
1
], i
=
1
,
…
,
K (assuming R
1
=
-
∞ and R
K
+
1
=
+
∞), the product is sold at price p
i.For a given(
R
2
,
…
,
R
K
,
D
,
S
) policy, the profit function can be calculated recursively as
(21)
0
=
-
h
(
x
)
+
λ
K
[
p
K
-
D
V
(
x
)
]
-
γ
V
(
x
)
,
x
≥
S
;
0
=
-
h
(
x
)
+
μ
[
D
V
(
x
)
-
b
]
0
=
+
λ
K
[
p
K
-
D
V
(
x
)
]
-
γ
V
(
x
)
,
D
≤
x
<
S
;
0
=
-
h
(
x
)
+
μ
[
D
V
(
x
)
-
b
]
+
a
[
D
V
(
x
)
-
c
]
0
=
+
λ
i
[
p
i
-
D
V
(
x
)
]
-
γ
V
(
x
)
,
R
i
<
x
≤
R
i
+
1
.Qualitatively, theK
+
1-level (
R
2
,
…
,
R
K
,
D
,
S
) policies possess the following properties similar to those for the two-price case.Proposition 8.
LetV
(
R
2
,
…
,
R
K
,
D
,
S
)
(
x
) be the long run total discounted profit of the K-level (
R
2
,
…
,
R
K
,
D
,
S
) policy when the initial inventory is x; then (1)
the
(
R
2
,
…
,
R
K
,
D
,
S
+
1
)
policy is better than the
(
R
2
,
…
,
R
K
,
D
,
S
)
policy if and only if
(22)
D
V
(
R
2
,
…
,
R
K
,
D
,
S
)
(
S
+
1
)
>
b
o
r
D
V
(
R
2
,
…
,
R
K
,
D
,
S
+
1
)
(
S
+
1
)
>
b
;
(2)
the
(
R
2
,
…
,
R
K
,
D
+
1
,
S
)
policy is better than the
(
R
2
,
…
,
R
K
,
D
,
S
)
policy if and only if
(23)
D
V
(
R
2
,
…
,
R
K
,
D
,
S
)
(
D
+
1
)
>
c
o
r
D
V
(
R
2
,
…
,
R
K
,
D
+
1
,
S
)
(
D
+
1
)
>
c
;
(3)
the
(
R
2
,
…
,
R
K
,
D
,
S
)
policy cannot be better than the
(
R
2
,
…
,
R
K
,
D
,
S
+
1
)
policy if
S
<
0
;
(4)
the
(
R
2
,
…
,
R
i
+
1
,
…
,
R
K
,
D
,
S
)
policy is better than the
(
R
2
,
…
,
R
i
,
…
,
R
K
,
D
,
S
)
policy if and only if
(24)
D
V
(
R
2
,
…
,
R
i
,
…
,
R
K
,
D
,
S
)
(
R
i
+
1
)
>
λ
i
p
i
-
λ
i
-
1
p
i
-
1
λ
i
-
λ
i
-
1
or
(25)
D
V
(
R
2
,
…
,
R
i
+
1
,
…
,
R
K
,
D
,
S
)
(
R
i
+
1
)
>
λ
i
p
i
-
λ
i
-
1
p
i
-
1
λ
i
-
λ
i
-
1
.We also have the following characterization of the bestK-level policy.Theorem 9.
The bestK-level (
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
) policy is characterized by the following relations: (1)
the profit functionV
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
x
) is concave in integer value of x;
(2)
the best base stock levelS
* satisfies S
*
≥
0 and
(26)
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
S
*
+
1
)
≤
b
<
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
S
*
)
;
(3)
at the optimal thresholdD
*,
(27)
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
D
*
+
1
)
≤
c
<
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
D
*
)
;
(4)
at the best price switch thresholdsR
i
*, i
=
2
,
…
,
K
-
1,
(28)
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
R
i
*
+
1
)
≤
λ
i
p
i
-
λ
i
-
1
p
i
-
1
λ
i
-
λ
i
-
1
<
D
V
(
R
2
*
,
…
,
R
K
*
,
D
*
,
S
*
)
(
R
i
*
)
.The following Theorem10 characterizes the impact of souring cost c on the optimal discounted profit and its optimal thresholds.Theorem 10.
As for fixed sourcing production ratea, suppose that the variable cost c of outsourced production is negotiable. Let (
R
2
*
(
c
)
,
…
,
R
K
*
(
c
)
,
D
*
(
c
)
,
S
*
(
c
)
) policy be the optimal threshold policy associated with a cost c. Then, (1)
the optimal thresholdsR
2
*
(
c
)
,
…
,
R
K
*
(
c
)
,
S
*
(
c
) are piecewise constant, increasing functions of c, but D
*
(
c
) is a piecewise constant, decreasing function of c.
(2)
the optimal profit functionV
(
R
2
*
(
c
)
,
…
,
R
K
*
(
c
)
,
D
*
(
c
)
,
S
*
(
c
)
)
(
x
;
c
) is decreasing in c.


## 5. Concluding Remarks

We have analyzed the optimal control of a single-product make-to-stock system that has the option to increase production capacity by outsourcing to external contract manufacturers and the option to vary product selling prices. Idealizing the system by a simpleM
/
M
/
1 make-to-stock queue model with discretely adjustable production and demand rates, we obtain a complete characterization of the optimal threshold control policy and prove beneficial impact of low cost outsourced production to the system. We wish to point out that the results can be easily extended multiple outsourcing alternatives and multiple price choices. It should also be possible to extend to the case of multiple demand classes that compete for the same product. More challenging extensions will be on incorporating fixed cost of outsourcing into the model as well as having multiple product classes in the model.



---

*Source: 301309-2014-07-06.xml*