- Email: [email protected]
Combined pricing and supply chain operations under price-dependent stochastic demand
Accepted Manuscript Combined pricing and supply chain operations under price-dependent stochastic demand Cheng-Chang Lin, Yi-Chen Wu PII: DOI: Reference:
S0307-904X(13)00595-7 http://dx.doi.org/10.1016/j.apm.2013.09.017 APM 9690
To appear in:
Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
29 August 2011 13 January 2013 22 September 2013
Please cite this article as: C-C. Lin, Y-C. Wu, Combined pricing and supply chain operations under price-dependent stochastic demand, Appl. Math. Modelling (2013), doi: http://dx.doi.org/10.1016/j.apm.2013.09.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Combined pricing and supply chain operations under price-dependent stochastic demand Cheng-Chang Lin1,a and Yi-Chen Wu2 Department of Transportation and Communication Management Science, National Cheng Kung University, 1 University Road, Tainan, Taiwan 701 ROC a
Abstract In this study, we determined product prices and designed an integrated supply chain operations plan that maximized a manufacturer’s expected profit. The computational results of this study revealed that as the variance of the demand distribution increases, a manufacturer will increase its inventory to levels that are greater than the anticipated demand to prevent the potential loss of sales and will simultaneously raise product prices to obtain a greater profit.
In
the cost minimization approach, the manufacturer may earn the highest possible profits, as determined by the profit optimization approach, only if this firm precisely forecasts the mean market demand for its products.
Greater inaccuracies in this forecast will produce lower levels
of expected profit. Keywords: Supply chain operational planning, pricing, price-dependent demand, demand uncertainty 1.
Introduction Supply chain design integrates supply-side, manufacturing and demand-side operations
networks to determine the most cost-efficient operations plan that meets the customer’s demands in a timely way.
Cooperation and collaboration that range from the suppliers of a firm’s
suppliers to the customers of a firm’s customers involve three levels of decisions. Strategic decisions address supplier selection by the supply sub-network, the production facility locations
1 2
Corresponding author, Phone: 886-6-2757575 ext.53240, Fax: 886-6-2753882, E-mail: [email protected]. E-mail: [email protected].
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of the manufacturing sub-network and the warehouses and the distribution center locations of the distribution sub-network. Tactical decisions relate to inventories of components and finished products, whereas operational decisions involve the fulfillment of customer demands. Disruptions in the supply chain refer to unintentional and undesirable situations that result in negative performance. agile supply chain.
Therefore, disruptions must be implicitly considered in the design of an
Demand-side uncertainty is one of the sources of supply chain disruptions
(Wagner and Bode, 2008).
Previously published studies have investigated how the design
and/or operations of a supply chain can minimize total expected operating costs under conditions of demand uncertainty (Shu et al., 2005; Tsiakis et al., 2001; Santoso et al., 2005; Schutz et al., 2009). Supply chain operations involve a two-stage decision process.
In the first-stage, the
components are procured, and the products are manufactured and become inventories that are staged in warehouses and/or distribution centers.
In the second-stage, demand is realized by
being fulfilled with shipments from distribution centers. Studies that have addressed the problem of designing supply chains to minimize cost under uncertain demand have raised two issues: the purpose of supply chain integration and the decision processes of supply chains. First, one of the purposes of supply chain cooperation is to operate in a maximally cost-effective way. The overarching goal of this cooperation is the pursuit of the greatest possible profits. The cost minimization approach implicitly assumes a demand that is perfectly inelastic with respect to prices. Under this assumption, profit optimization is equivalent to cost minimization. This implicit assumption suggests that manufacturers and/or brand-name enterprises are unable to use pricing to alter their revenues to increase their profits.
However, most products are price sensitive; therefore, in this study, we
assume that demand is price dependent. Second, product manufacturers and/or brand-name
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
enterprises often announce their product prices prior to receiving customer orders; for instance, these announcements have occurred for Apple’s iPhone and iPad products and Microsoft’s Windows operating systems. Therefore, for manufacturers that wish to maximize their profits, the establishment of product prices and operations plans involves mutually interactive decisions that must be concurrently finalized during the first-stage of the supply chain decision process. Customers will then react to product prices and determine their purchases.
Customer demands
are realized and fulfilled in the second-stage of the supply chain decision process. In this research, we assume that the uncertain demand for a product is price dependent; in particular, we conjecture that mean demand is a function of price.
In addition, we presume that
the realized demand is uncertain but that this demand will follow a discrete statistical distribution. Given these assumptions, our study determines the appropriate decisions that will maximize a manufacturer’s expected profits under demand uncertainty.
These decisions relate to not only
product prices and inventories but also operational levels of procurement, production and distribution. The contributions of this research are as follows.
First, previous research has
studied demand-side uncertainty in supply chain network design under the assumption that demand is independent of price; in other words, previously published studies have assumed the existence of perfectly inelastic demand. perfectly inelastic demand.
In this investigation, we relax the assumption of
Thus, the combination of pricing policy and supply chain network
design will determine appropriate product prices and an integrated supply-side, manufacturing and demand-side operations plan in a manner that maximizes a manufacturer’s total expected profit. Thus, the model of this study integrates the tactical issues of product pricing and inventories with the operational issue of demand fulfillment. Second, the mathematical model for the combined pricing and operational planning problem
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
involves a two-stage stochastic program.
In the deterministic first-stage master problem, the
manufacturer makes tactical decisions regarding prices, inventory levels and inventory locations. In addition, operational decisions regarding component procurement and product manufacturing are also finalized during this stage of the model. products are shipped to meet realized demand.
In the second-stage recourse subproblem, An L-shaped decomposition-based algorithm
provides one possible computational approach for addressing two-stage stochastic problems. However, if prices are the decision variables, then the optimality cuts that are determined in this second-stage of the model are not necessarily linear functions.
In fact, these conditions produce
a non-fixed recourse subproblem. We explore the functional properties of optimality cuts to ensure that the expanded first-stage master problem remains a concave program that possesses an optimal solution.
By considering elastic demand, we can examine the mutual impacts of prices
and operations plans. Furthermore, we can investigate the economic implications of profit optimization and cost minimization approaches. The organization of this research is as follows.
In Section 2, we review two streams of
relevant research. The first research stream that is discussed examines how supply chains may be designed to minimize total operating costs under conditions of uncertain demand. The second research stream that is reviewed addresses the dynamic pricing problem that governs inventory replenishment in a monopolistic and uncertain market.
In Section 3, we present the
supply chain network and operations that are examined in this study. The supply chain consists of supply-side, manufacturing and demand-side operations; each of these aspects of the supply chain will be discussed.
We then present the mathematical model for combined pricing and
supply chain operations.
These operations may be represented as a two-stage nonlinear
stochastic programming problem.
In this paper, we present the deterministic equivalent
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
formulation of this problem.
In Section 4, we propose an L-shaped decomposition-based
algorithm. The second-stage recourse subproblem determines the recourse cost, which is the sum of a firm’s unrealized expected revenue and its expected cost of fulfillment. The dual objective function of the second-stage subproblem defines optimality cuts for the first-stage master problem. Due to the price-dependent nature of the demand function that is considered in this investigation, these cuts are nonlinear functions. optimality cuts may be determined.
In this section, we discuss how valid
Although the cuts are nonlinear functions, the embedded
master problem remains a concave objective function that is constrained by a convex set.
In
Section 5, we perform numerical testing. We compare the profit projections that are produced by profit maximization and cost minimization models. We also perform sensitivity analyses with respect to demand variations. The conclusions of this research are presented in Section 6. 2.
Literature review Demand uncertainty is one of the sources of supply chain disruptions.
have examined supply chain design under demand uncertainty.
Previous studies
In particular, two streams of
related research address this topic. The first of these two research streams examines how supply chains may be designed to address uncertain demand in a manner that minimizes total costs. Shu et al. (2005) studied a two-echelon supply chain distribution network design that included one supplier and multiple retailers. A two-stage network design problem minimizes fixed and operating costs by selecting certain retailers that function as distribution centers and assigning other retailers to these designated distribution centers. achieves the benefits of risk pooling.
Cooperation within the supply chain
Tsiakis et al. (2001) integrated production with the
distribution sub-network to study multi-echelon production and distribution networks under uncertain demand. These researchers examined this issue by constructing a two-stage stochastic
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
programming problem to determine how warehouses and distribution centers could be located in a manner that minimizes fixed and operating costs. The first-stage of this problem entails establishing facility locations, whereas the second-stage of this problem involves the creation of a supply chain operations plan.
Santoso et al. (2005) analyzed an integrated multiple-echelon
supply chain network design under demand uncertainty. The model that was designed by these researchers determines how processing facilities and finishing machines could be located such that total fixed and operating costs are minimized. These researchers solved the resulting two-stage stochastic problem on a realistic scale by integrating a sampling strategy with an accelerated Benders decomposition approach.
By contrast, Schutz et al. (2009) proposed a
sample average approximation and dual decomposition procedure to solve the supply chain design problem under demand uncertainty. demand is price inelastic.
All of the aforementioned studies assume that
This assumption implies that manufacturers/retailers are unable to use
pricing to change their revenues. The second stream of research that relates to the current study examines the dynamic pricing problem that governs the replenishment of inventory under monopolistic and uncertain market conditions. This problem involves how prices may be established over time in a manner that maximizes a firm’s total expected profit.
Netessine (2006) studied a pricing problem involving
a limited number of price changes in a dynamic but deterministic environment. environment, demand is dependent on the current price and time.
In this
Netessine’s research analyzed
inventory, capacity and pricing decisions for a product that is sold by a monopolist over the course of a short season.
This investigation revealed the impact of capacity constraints on
optimal prices and the timing of price changes. In a stochastic environment, Chen and Lee (2004) studied a multi-product, multi-stage and
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
multi-period scheduling model in a multi-echelon supply chain network involving uncertain product demands and product prices. These researchers used fuzzy sets to describe sellers’ and buyers’ incompatible preferences with respect to product prices. The product demands with known probabilities were assumed to be independent of product prices.
Under these conditions,
the supply chain scheduling model becomes a mixed-integer nonlinear programming problem involving several conflicting objectives, such as maximizing profits and maintaining high service levels.
Chen and Lee address this problem by presenting a two-phase fuzzy decision-making
method to determine a compromise solution in a multi-echelon supply chain network under uncertain conditions.
Wafa et al. (2008) developed a multi-period supply chain network design
model for a petroleum organization in an oil-producing country under uncertain market conditions.
These researchers proposed a stochastic formulation of this situation that was based
on the two-stage problem with a finite number of demand and price realizations. They concluded that the impact of demand and price uncertainties may be tolerable given an appropriate balance between crude exports and processing capacities. In the current study, we address the pricing problem that involves the replenishment of inventory in a monopolistic and uncertain market.
Using the assumption that stochastic
demands are price dependent, we examine the mutually interactive relationship between pricing and supply chain operations. 3.
The combined pricing and operations mathematical model In this section, we describe the supply chain planning network and define the scope of this
network.
We then present our mathematical model for combined pricing and operational
planning under demand uncertainty. 3.1 Supply chain network and operations
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The objective of pricing and supply chain operational planning under demand uncertainty is to determine the product prices in combination with an integrated supply, manufacturing and demand-side operations plan in a manner that maximizes total expected profit. network consists of supply, manufacturing and distribution sub-networks. sub-network is composed of contracted component providers.
A supply chain
A supply
In this study, the manufacturing
sub-network is assumed to consist of a single production plant that outputs finished products. To meet uncertain customer demands, the distribution sub-network is assumed to include both a central warehouse and a field warehouse. A four-echelon supply chain network for dual products is illustrated in Figure 1.
In this planning network, there are two finished products.
Both of these products require five components and are stocked at the central and/or field warehouses. The assumptions of this study are discussed below. 1.
We assume that the mean demand function is linear.
Therefore, the inverse mean demand
must exist and must also be a linear function. As a result, the revenue function is a concave function. 2.
The supply chain network consists of component providers; manufacturing plants; and centralized and field warehouses. To study the functional properties of optimality cuts for the L-shaped decomposition-based algorithm, we assume that the supply chain consists of a single manufacturing plant, a single centralized warehouse and a single field warehouse. Therefore, we assume the existence of monopolistic market conditions.
3.
We assume that the examined manufacturer sources all of its components and that this firm produces and supplies dual products.
The manufacturer is presumed to execute a
make-for-stock (MTS) manufacturing process, which is a mass production process that
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
achieves production at economies of scale.
Thus, we assume that the total production cost
function is a concave function and that the average cost is a convex function. 4.
Upon production, products are assumed to be shipped to the central and/or field warehouses for storage; no inventory is retained at the manufacturing plant.
The products become
inventory prior to demand realization and are directly shipped to meet customer orders. We assume that the dead inventory cost at the field warehouse is no less than the dead inventory cost at the central warehouse.
In addition, the unit shipment cost of delivering products
from the field warehouse to customers is presumed to be no higher than the unit shipment cost of delivering products from the central warehouse to these customers. 3.2 Mathematical model The combination of pricing and operational planning tactically determines the prices and inventory levels at any echelon of the supply chain network. Operationally, the manufacturer seeks to purchase, manufacture and deliver finished products to satisfy uncertain demand in a manner that maximizes expected profit.
These processes can be modeled in two stages.
In the
first-stage of these processes, the manufacturer determines product prices and the operations plan with respect to procurement, production and inventory management.
In this stage, components
are procured and products are created at the manufacturing plant. These products are shipped and positioned at the central and/or field warehouses.
In the second-stage of supply chain
processes, products are shipped to customers once customer orders have been received. The model of this study seeks to maximize the total expected profit, which is the sum of deterministic procurement, manufacturing and inventory costs; and expected revenue, product distribution expenses and obsolescence costs.
A two-stage stochastic programming problem was formulated
in a deterministic equivalent formulation.
We compare this profit optimization approach with
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the cost minimization approach.
For readability, the cost minimization model for uncertain
demands is provided in Appendix A of this paper. A.
Parameters and decision variables
1. Sets and parameters: h
denotes the consumption market;
g
denotes a product;
G
= {..., g ,...} ;
m
denotes the single production plant;
s
denotes a component;
S
= {..., s,...} ;
cs
is the unit purchase cost of component s S ;
c sm
is the freight rate to ship a unit of component s S to the production plant m;
c m, g Q m, g
is the average production cost of product g G at the production plant, which is a function of Q, the plant’s production level;
c mr , g / c rf , g
are the freight rates to ship a unit of product g G from the production plant to the central warehouse/from the central warehouse to the field warehouse;
c rh, g / c fh, g
are the freight rates to ship a unit of product g G from the central/field warehouse to the consumption market;
I r,g / I f ,g
are the unit inventory costs of product g G in the central (field) warehouse;
O r,g / O f ,g
are the obsolescence costs for a unit of unused product g G in the central/field warehouse;
Um
is the production capacity;
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s, g
is the number of components s S that are used for a product g G ;
2. Decision variables: d h, g
is the loss of sale of product g G in the consumption market h;
q h, g
is the anticipated mean demand (with an associated price p h, g (q h, g ) ) of product g G in the consumption market h;
Q m, g
is the production quantity of product g G at production plant m;
u r ,g / u f ,g
are the number of products g G stored in the central warehouse r/the field warehouse f;
v r,g / v f ,g
is the number of unused products g G in the central warehouse r/the field warehouse f; is the number of components s S shipped from the supplier to the production
x sm
plant m; y mr , g / y rf , g
are the number of products g G shipped from the production plant m to the central warehouse r/from the central warehouse r to the field warehouse f.
z rh, g / z fh, g
is the number of products g G shipped from the central warehouse r/the field warehouse f to the consumption market h;
B. A second-stage recourse subproblem For a given procurement with the production and inventory operations plan of {qˆ h, g , uˆ r , g , uˆ f , g } , the second-stage recourse subproblem under demand scenario may be expressed as follows:
Max (qˆ ( ), uˆ ) p h, g (qˆ h, g )d h, g c rh, g z rh, g c fh, g z fh, g O r , g v r , g O f , g v f , g g
11
(1)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Subject to: z rh, g z fh, g d h, g qˆ h, g ( )
g G
(2)
z rh, g v r , g uˆ r , g
g G
(3)
z fh, g v f , g uˆ f , g
g G
(4)
z
rh, g
, z fh, g , d h, g , v r , g , v f , g
The objective function (1) represents the total fulfillment cost, which is the sum of unrealized revenue from insufficient inventory; distribution expenses; and product obsolescence/dead inventory costs from overstocked inventory. Constraint (2) states that the sum of the total shipments to the consumption market and losses of sales (if any) must be equal to the actual demand of scenario . Constraint (3) represents product flow conservation at the central warehouse and states that the sum of the fulfilled demand and unused inventory must be equal to the inventory level that existed in the first-stage. Similarly, the conservation of product flow at the field warehouse is stated in constraint (4). Note that in this formulation, there are no loss of sale terms in the objective function.
By contrast, in the cost minimization formulation,
losses of sales serve as a proxy for potential profits.
In cost minimization approaches, in the
absence of lost sales, manufacturers would cease production to achieve an optimal cost of zero. However, in the profit optimization formulation, the recourse function represents the actual profit upon demand realization. The profit optimization approach involves a linear program, thus, the associated dual subproblem for this approach may be expressed as follows:
(qˆ ( ), uˆ ) Min h, g qˆ h, g ( ) r , g uˆ r , g f , g uˆ f , g
(5)
g
Subject to
h, g p h, g qˆ h, g
g G
12
(6)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h, g r , g c rh, g
g G
(7)
h, g f , g c fh, g
g G
(8)
r , g O r , g
g G
(9)
f , g O f , g
g G
(10)
{ h, g , r , g , f , g }
In the above expressions, { h, g , r , g , f , g } are dual variables that are associated with constraints (2), (3) and (4), respectively. Thus, the expected fulfillment cost of the second-stage recourse subproblem is the sum of the costs of each demand scenario multiplied by the corresponding probability for the scenario in question, (qˆ ( ), uˆ ) .
C.
The combined pricing and operational planning model Given the expected second-stage fulfillment cost that is discussed above, the combined
pricing and supply chain operational planning model in its deterministic equivalent formulation may be expressed as follows: Max
q p q h, g
h, g
g
I
r ,g
g
h, g
( ) (c s c sm ) x sm s
c Q Q m, g
m, g
g
u r , g c mf , g y mf , g I f , g u f , g (q( ), u ) g
g
m, g
c mr , g y mr , g g
(11)
Subject to: x sm s , g Q m, g
s S
(12)
Q m, g U m, g
g G
(13)
q h, g Q m, g
g G
(14)
y mr , g y mf , g Q m, g
g G
(15)
g
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
y mr , g u r , g
g G
(16)
y mf , g u f , g
g G
(17)
{q h, g , Q m, g , x sm , y mr , g , y mf , g , u r , g , u f , g }
The objective of function (11) is to maximize expected profit. The terms of this function are the expected revenue at the anticipated mean demand; the deterministic component procurement, product manufacturing and inventory costs at the central and field warehouses; and the expected second-stage fulfillment cost. Constraint (12) is the procurement constraint, which requires a manufacturer to purchase sufficient components for production.
Constraint (13) is the
plant production capacity constraint, which states that the total units of produced products cannot exceed the plant’s production capacity. Constraint (14) is the production/inventory level constraint, which requires the production of sufficient product to meet the anticipated mean demand.
In other words, this constraint specifies that upon the determination of product prices
(through the use of the inverse demand function), the manufacturer will produce and stock sufficient product to at least satisfy the anticipated mean demand for the consumption market. Constraint (15) is the product flow conservation constraint at the plant, which states that all of the produced products must be shipped to central and/or field warehouses to become inventory. Constraints (16) and (17) are flow conservation constraints at the central and field warehouses, respectively, that require received products to become inventory for consumption.
Finally, the
decision variables of anticipated mean demand, production, procurement level, production shipments and inventory level must all have non-negative values. 4.
The L-shaped solution algorithm The combination of pricing and supply chain operations planning with price-dependent
demand uncertainty may be represented as a two-stage stochastic program.
14
The L-shaped
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
approach, which is also known as the decomposition approach, separates the two-stage stochastic problem into a first-stage deterministic master problem and a second-stage stochastic recourse subproblem (Birge and Louveaux, 1997; Kall and Wallace, 1994).
In this case, to implement
this approach, the combined pricing and operational planning model expressed in (11)-(17) is decomposed into the following first-stage deterministic master problem and the second-stage stochastic recourse subproblem that is described by (1)-(4). Max
q p q h, g
h, g
g
I
r,g
g
h, g
( ) (c s c sm ) x sm s
c Q Q m, g
g
m, g
m, g
c mr , g y mr , g g
u r , g c mf , g y mf , g I f , g u f , g g
(18)
g
Subject to: (12)-(17)
h, g q h, g ( ) r , g u r , g f , g u f , g
(19)
g
{q h, g , Q m, g , x sm , y mr , g , y mf , g , u r , g , u f , g } ;{ }
Constraint (19), which is the dual objective function of the second-stage recourse subproblem (5), defines the optimality cuts.
In particular, this constraint replaces the
second-stage expected fulfillment cost function (q( ), u ) with a set of optimality cuts
that represent the upper bounds of this function. The computational process solves the second-stage recourse subproblem that is expressed in (1)-(4).
An optimality cut for (5) is
determined and subsequently embedded in the first-stage master problem as the additional constraint (19). This process forms an expanded first-stage master problem (18) that is subject to constraints (12)-(17) and (19).
In a fixed recourse subproblem with parameters
15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{ h, g , r , g , f , g , g G} , the optimality cut is linear. Therefore, the expected fulfillment cost
function can be replaced with a piecewise linear function. With a concave revenue function, a convex cost function and linear constraints, the expanded master problem remains a concave program.
Unfortunately, under these conditions, using prices and anticipated mean demands as
decision variables, the second-stage of the model becomes a non-fixed recourse subproblem. Therefore, we analyze the functional form of optimality cuts and the type of mathematical model that exists for the expanded master problem. 4.1 The valid functional forms for optimality cuts In section 3.1, we assume that the dead inventory cost at the field warehouse is no less than the dead inventory cost at the central warehouse.
In addition, the unit shipment cost from the
field warehouse to customers is assumed to be no higher than the unit shipment cost from the central warehouse to the same customers.
These assumptions lead to the following lemma.
Lemma 1. The manufacturer uses the inventory at the central warehouse to fulfill demand only if no available inventory exists at the field warehouse. In other words, z rh, g 0 v f , g 0. Proof. The manufacturer will fulfill demand with inventory from the field warehouse if possible. Assume the contrary. (O r , g c rh, g ) (O f , g c fh, g ) .
Because O r , g O f , g and c rh, g c fh, g , it follows that
If v f , g 0 , we may increase z fh, g by one unit and decrease
z rh, g by one unit, and the firm’s overall profit will be increased by (c fh, g O f , g ) (c rh, g O r , g ) 0 . The contradiction is now established.
□
We now analyze the functional form of optimality cuts. Lemma 2. The optimality cuts are quadratic functions. Proof. The optimality cut is a combination/summation of the following parts, which are
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
derived and demonstrated in Appendix B:
O f , g c fh, g qˆ h, g ( ) O r , g uˆ r , g O f , g uˆ f , g ; for d h, g 0, v r , g uˆ r , g , v f , g 0 (a) f , g fh , g qˆ h, g ( ) O f , g uˆ f , g ; for d h, g 0, v r , g uˆ r , g 0, v f , g 0 O c
O r , g c rh, g qˆ h, g ( ) O r , g uˆ r , g O r , g c rh, g c fh, g uˆ f , g ; (b) for d h, g 0, uˆ r , g v r , g 0, uˆ f , g v f , g 0 O r , g c rh, g qˆ h , g ( ) O r , g uˆ r , g ; for d h , g 0, 0 v r , g uˆ r , g , uˆ f , g 0 (c)
p qˆ uˆ h, g
h, g
r,g
uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , if d h, g 0 and v r , g v f , g 0 .
The functional forms of (a) and (b) above are linear functions for anticipated mean demands and inventories, respectively. The functional form that is expressed in (c) is the sum of unrealized revenues (in cases of insufficient supply) and the costs of transporting products from the central and field warehouses to the consumption market.
The unrealized revenue is a
function of the product prices and the associated anticipated mean demand of qˆ h , g that has been determined in the master problem. mean demand is altered.
Therefore, the unrealized revenue changes as the anticipated
If we simply treat h, g as a parameter and derive an optimality cut
of h, g uˆ r , g uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , then this cut may be embedded as an additional constraint for the master problem.
In the subsequent computation of the expanded
master problem, a set of new anticipated mean demands and the product inventories at the central
and field warehouses are determined q h, g , u r , g , u f , g , g G , and assumed to
be p h, g q h, g p h, g qˆ h, g .
Given a new set of product prices p h, g q h, g , the upper bound on the
expected fulfillment cost that is determined by this optimality cut is
p h, g qˆ h, g u r , g u f , g (q h, g ()) c rh, g u r , g c fh, g u f , g because h, g p h, g qˆ h, g .
In fact, the
correct upper bound is p h, g q h, g u r , g u f , g (q h, g ()) c rh, g u r , g c fh, g u f , g with respect to the
set of new product prices p h, g q h, g . The discrepancy between these two expressions implies
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
that h, g u r , g u f , g (q h, g ( )) c rh, g u r , g c fh, g u f , g , for h, g p h, g qˆ h, g , either over- or under-estimates the actual upper bound of the expected fulfillment cost in the expanded master problem. Thus, h, g cannot be treated as a parameter.
Instead, an appropriate functional form
of (c) is p h, g qˆ h, g uˆ r , g uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , a quadratic function.
□
Whenever an optimality cut (qˆ ( ), uˆ ) is derived for a given qˆ h, g , uˆ r , g , uˆ f , g , g G ,
this cut is embedded in the master problem as an additional constraint. We must show that this optimality cut remains valid for all other prices.
Otherwise, the inclusion of this new constraint
will alter the feasible set of possibilities for other prices. Lemma 3.
qˆ
h, g
An optimality cut determined by a set of product prices and inventories
, uˆ r , g , uˆ f , g , g G is a valid optimality cut for all other product
prices q h, g , u r , g , u f , g , g G if the corresponding discrete demand probabilities remain unchanged by shifts in product prices. Proof.
The first-stage master problem determines qˆ h, g , uˆ r , g , uˆ f , g , g G . The
associated product prices are p h, g qˆ h, g , g G . For given p h, g qˆ h, g , d h,g , v r ,g , v f ,g of the primal and { h, g , r , g , f , g } of the dual of the second-stage recourse subproblem may be determined. The dual objective function becomes the optimality cut that is the mathematical sum of (b.1) through (b.5) with the corresponding demand probabilities.
We discuss the
applicable cases as follows. (1)
uˆ f , g 0 and uˆ r , g 0 .
of (b.1), (b.2) and (b.5).
In this instance, the optimality cut
(qˆ ( ), uˆ ) consists
We partition the demand distribution of anticipated mean
demand qˆ h , g into three segments qˆ h, g (1 ) qˆ h, g ( 2 ) qˆ h, g ( 3 ) with associated probabilities
18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of w1 , w2 , w3 such that the conditions qˆ h, g (1 ) uˆ f , g , qˆ h, g ( 2 ) uˆ r , g uˆ f , g and qˆ h, g ( 3 ) uˆ r , g uˆ f , g are satisfied.
As a result, each of these demand segments corresponds to
the condition for either (b.1), (b.2) or (b.5).
For any anticipated mean demand q h, g qˆ h, g , under
the probability distribution w1 , w2 , w3 , the demand distribution is partitioned
into q h, g (1 ) q h, g ( 2 ) q h, g ( 3 ) . We may define u r , g 0, u f , g 0 to satisfy the condition q h, g (1 ) u f , g q h, g ( 2 ) u r , g u f , g q h, g ( 3 ) . These conditions apply for (b.1), (b.2) and (b.5), implying that (q ( ), u ) (qˆ ( ), uˆ ) is a valid optimality cut for
q h, g .
(2)
uˆ f , g 0 but uˆ r , g 0 .
The optimality cut
(qˆ ( ), uˆ ) consists of (b.2) and
(b.5). We partition the demand distribution of anticipated mean demand qˆ h , g into two segments
qˆ
h, g
(1 ) qˆ h, g ( 2 ) with probabilities of w1 , w2 , respectively. These segments
satisfying qˆ h, g (1 ) uˆ f , g and qˆ h, g ( 2 ) uˆ f , g . Thus, for any anticipated mean
demand q h, g qˆ h, g , any u r , g 0, u f , g 0 that satisfies q h, g (1 ) u f , g q h, g ( 2 ) will result in
(q ( ), u ) (qˆ ( ), uˆ ) . (3)
The optimality cut (qˆ ( ), uˆ ) consists of (b.4) and (b.5).
uˆ f , g 0 .
We
partition the demand distribution of the anticipated mean demand qˆ h , g into two segments
qˆ
h, g
(1 ) qˆ h, g ( 2 ) with w1 , w2 satisfying qˆ h, g (1 ) uˆ r , g and with qˆ h, g ( 2 ) uˆ r , g .
Thus, for any anticipated mean demand q h, g qˆ h, g , any u r , g 0, u f , g 0 that satisfies
19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
q h, g (1 ) u r , g q h, g ( 2 ) will result in (q ( ), u ) (qˆ ( ), uˆ ) .
□
Finally, we demonstrate that optimality cuts are concave functions. Lemma 4. Optimality cuts are concave functions.
Proof. We demonstrate that p h, g q h, g u r , g u f , g (q h, g ( )) c rh, g u r , g c fh, g u f , g is a concave function.
Because we assume that the demand function is linear, the inverse demand
function is also linear and may be expressed in the form of p a bq . This function therefore possesses the following Hessian matrix:
2b b b b 0 0 . Because u r , g u r , g Q m q h, g , b 0 0
q
h, g
u r,g
u f ,g
h, g 2b b b q b 0 u r , g 2bq h , g q h , g u r , g u r , g 0 . Thus, this matrix is 0 f ,g 0 u b 0
semi-definite. □ From the above lemmas, we may derive the following theorem regarding the embedded first-stage master problem. Theorem 5. The expanded first-stage master problem is a concave program. Proof. By Lemma 4, all optimality cuts are concave functions. Thus,
h, g q h, g ( ) r , g u r , g f , g u f , g defines a convex set for q h, g , u r , g , u f , g , g G. g
g
g
Constraints (12)-(17) are linear; therefore, the feasible set remains convex.
Given a concave
objective function (18), the expanded master problem must remain a concave program. 4.2 The L-shaped decomposition algorithm
20
□
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The L-shaped algorithm includes the following detailed steps. Step 0 (Initialization). We initialize the number of iterations for the master problem T=0 and the set of demand fulfillment optimality cuts . Step 1 (First-stage master problem).
We increment the number of iterations such that
T=T+1. At the Tth master problem, we solve (18) under constraints (12)-(17) and K, which determines the anticipated mean demand (with an associated price), the product production, the component procurement, and the product inventories at the central and field warehouses {(q h, g )T , (Q m, g )T , ( x sm )T , ( y mr , g )T , ( y mf , g )T , (u r , g )T , (u f , g )T } .
This solution also
determines ( )T . Step 2 (Second-stage recourse subproblem). At the Tth master problem, for the determined anticipated mean demand, manufacturing and inventory plan {(q h, g )T , (u r , g )T , (u f , g )T } , we solve (1)-(4), which determines the distribution
plan ( z rh, g )T , ( z fh, g )T , (v r , g )T , (v f , g )T , (d h, g )T with duals ( h, g )T , ( r , g )T , ( f , g )T . If the second-stage optimal objective value is bounded by ( )T , the solution {(q h, g )T , (Q m, g )T , ( x sm )T , ( y mr , g )T , ( y mf , g )T , (u r , g )T , (u f , g )T } is optimal, and the program
terminates. Otherwise, we generate
( h, g )T q h, g ( ) ( r , g )T (u r , g )T ( f , g )T (u f , g )T ,
T
g
(20)
a demand fulfillment optimality cut for the second-stage recourse subproblem, which becomes an additional constraint on the expanded master problem K such that { (20)} . return to step 1. 5.
Numerical examples
21
We then
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The supply chain operations planning network for the numerical example of this study is displayed in Figure 1.
There are three common components (labeled 1–3) and two
differentiated components (labeled 4–5) for two distinct products A and B.
Product A uses
components 1-4, whereas product B utilizes components 1-3 and component 5.
The
manufacturer sources components 1-5; produces A and B; and subsequently stocks A and B at a central and/or field warehouse.
S0307-904X(13)00595-7 http://dx.doi.org/10.1016/j.apm.2013.09.017 APM 9690
To appear in:
Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
29 August 2011 13 January 2013 22 September 2013
Please cite this article as: C-C. Lin, Y-C. Wu, Combined pricing and supply chain operations under price-dependent stochastic demand, Appl. Math. Modelling (2013), doi: http://dx.doi.org/10.1016/j.apm.2013.09.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Combined pricing and supply chain operations under price-dependent stochastic demand Cheng-Chang Lin1,a and Yi-Chen Wu2 Department of Transportation and Communication Management Science, National Cheng Kung University, 1 University Road, Tainan, Taiwan 701 ROC a
Abstract In this study, we determined product prices and designed an integrated supply chain operations plan that maximized a manufacturer’s expected profit. The computational results of this study revealed that as the variance of the demand distribution increases, a manufacturer will increase its inventory to levels that are greater than the anticipated demand to prevent the potential loss of sales and will simultaneously raise product prices to obtain a greater profit.
In
the cost minimization approach, the manufacturer may earn the highest possible profits, as determined by the profit optimization approach, only if this firm precisely forecasts the mean market demand for its products.
Greater inaccuracies in this forecast will produce lower levels
of expected profit. Keywords: Supply chain operational planning, pricing, price-dependent demand, demand uncertainty 1.
Introduction Supply chain design integrates supply-side, manufacturing and demand-side operations
networks to determine the most cost-efficient operations plan that meets the customer’s demands in a timely way.
Cooperation and collaboration that range from the suppliers of a firm’s
suppliers to the customers of a firm’s customers involve three levels of decisions. Strategic decisions address supplier selection by the supply sub-network, the production facility locations
1 2
Corresponding author, Phone: 886-6-2757575 ext.53240, Fax: 886-6-2753882, E-mail: [email protected]. E-mail: [email protected].
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of the manufacturing sub-network and the warehouses and the distribution center locations of the distribution sub-network. Tactical decisions relate to inventories of components and finished products, whereas operational decisions involve the fulfillment of customer demands. Disruptions in the supply chain refer to unintentional and undesirable situations that result in negative performance. agile supply chain.
Therefore, disruptions must be implicitly considered in the design of an
Demand-side uncertainty is one of the sources of supply chain disruptions
(Wagner and Bode, 2008).
Previously published studies have investigated how the design
and/or operations of a supply chain can minimize total expected operating costs under conditions of demand uncertainty (Shu et al., 2005; Tsiakis et al., 2001; Santoso et al., 2005; Schutz et al., 2009). Supply chain operations involve a two-stage decision process.
In the first-stage, the
components are procured, and the products are manufactured and become inventories that are staged in warehouses and/or distribution centers.
In the second-stage, demand is realized by
being fulfilled with shipments from distribution centers. Studies that have addressed the problem of designing supply chains to minimize cost under uncertain demand have raised two issues: the purpose of supply chain integration and the decision processes of supply chains. First, one of the purposes of supply chain cooperation is to operate in a maximally cost-effective way. The overarching goal of this cooperation is the pursuit of the greatest possible profits. The cost minimization approach implicitly assumes a demand that is perfectly inelastic with respect to prices. Under this assumption, profit optimization is equivalent to cost minimization. This implicit assumption suggests that manufacturers and/or brand-name enterprises are unable to use pricing to alter their revenues to increase their profits.
However, most products are price sensitive; therefore, in this study, we
assume that demand is price dependent. Second, product manufacturers and/or brand-name
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
enterprises often announce their product prices prior to receiving customer orders; for instance, these announcements have occurred for Apple’s iPhone and iPad products and Microsoft’s Windows operating systems. Therefore, for manufacturers that wish to maximize their profits, the establishment of product prices and operations plans involves mutually interactive decisions that must be concurrently finalized during the first-stage of the supply chain decision process. Customers will then react to product prices and determine their purchases.
Customer demands
are realized and fulfilled in the second-stage of the supply chain decision process. In this research, we assume that the uncertain demand for a product is price dependent; in particular, we conjecture that mean demand is a function of price.
In addition, we presume that
the realized demand is uncertain but that this demand will follow a discrete statistical distribution. Given these assumptions, our study determines the appropriate decisions that will maximize a manufacturer’s expected profits under demand uncertainty.
These decisions relate to not only
product prices and inventories but also operational levels of procurement, production and distribution. The contributions of this research are as follows.
First, previous research has
studied demand-side uncertainty in supply chain network design under the assumption that demand is independent of price; in other words, previously published studies have assumed the existence of perfectly inelastic demand. perfectly inelastic demand.
In this investigation, we relax the assumption of
Thus, the combination of pricing policy and supply chain network
design will determine appropriate product prices and an integrated supply-side, manufacturing and demand-side operations plan in a manner that maximizes a manufacturer’s total expected profit. Thus, the model of this study integrates the tactical issues of product pricing and inventories with the operational issue of demand fulfillment. Second, the mathematical model for the combined pricing and operational planning problem
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
involves a two-stage stochastic program.
In the deterministic first-stage master problem, the
manufacturer makes tactical decisions regarding prices, inventory levels and inventory locations. In addition, operational decisions regarding component procurement and product manufacturing are also finalized during this stage of the model. products are shipped to meet realized demand.
In the second-stage recourse subproblem, An L-shaped decomposition-based algorithm
provides one possible computational approach for addressing two-stage stochastic problems. However, if prices are the decision variables, then the optimality cuts that are determined in this second-stage of the model are not necessarily linear functions.
In fact, these conditions produce
a non-fixed recourse subproblem. We explore the functional properties of optimality cuts to ensure that the expanded first-stage master problem remains a concave program that possesses an optimal solution.
By considering elastic demand, we can examine the mutual impacts of prices
and operations plans. Furthermore, we can investigate the economic implications of profit optimization and cost minimization approaches. The organization of this research is as follows.
In Section 2, we review two streams of
relevant research. The first research stream that is discussed examines how supply chains may be designed to minimize total operating costs under conditions of uncertain demand. The second research stream that is reviewed addresses the dynamic pricing problem that governs inventory replenishment in a monopolistic and uncertain market.
In Section 3, we present the
supply chain network and operations that are examined in this study. The supply chain consists of supply-side, manufacturing and demand-side operations; each of these aspects of the supply chain will be discussed.
We then present the mathematical model for combined pricing and
supply chain operations.
These operations may be represented as a two-stage nonlinear
stochastic programming problem.
In this paper, we present the deterministic equivalent
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
formulation of this problem.
In Section 4, we propose an L-shaped decomposition-based
algorithm. The second-stage recourse subproblem determines the recourse cost, which is the sum of a firm’s unrealized expected revenue and its expected cost of fulfillment. The dual objective function of the second-stage subproblem defines optimality cuts for the first-stage master problem. Due to the price-dependent nature of the demand function that is considered in this investigation, these cuts are nonlinear functions. optimality cuts may be determined.
In this section, we discuss how valid
Although the cuts are nonlinear functions, the embedded
master problem remains a concave objective function that is constrained by a convex set.
In
Section 5, we perform numerical testing. We compare the profit projections that are produced by profit maximization and cost minimization models. We also perform sensitivity analyses with respect to demand variations. The conclusions of this research are presented in Section 6. 2.
Literature review Demand uncertainty is one of the sources of supply chain disruptions.
have examined supply chain design under demand uncertainty.
Previous studies
In particular, two streams of
related research address this topic. The first of these two research streams examines how supply chains may be designed to address uncertain demand in a manner that minimizes total costs. Shu et al. (2005) studied a two-echelon supply chain distribution network design that included one supplier and multiple retailers. A two-stage network design problem minimizes fixed and operating costs by selecting certain retailers that function as distribution centers and assigning other retailers to these designated distribution centers. achieves the benefits of risk pooling.
Cooperation within the supply chain
Tsiakis et al. (2001) integrated production with the
distribution sub-network to study multi-echelon production and distribution networks under uncertain demand. These researchers examined this issue by constructing a two-stage stochastic
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
programming problem to determine how warehouses and distribution centers could be located in a manner that minimizes fixed and operating costs. The first-stage of this problem entails establishing facility locations, whereas the second-stage of this problem involves the creation of a supply chain operations plan.
Santoso et al. (2005) analyzed an integrated multiple-echelon
supply chain network design under demand uncertainty. The model that was designed by these researchers determines how processing facilities and finishing machines could be located such that total fixed and operating costs are minimized. These researchers solved the resulting two-stage stochastic problem on a realistic scale by integrating a sampling strategy with an accelerated Benders decomposition approach.
By contrast, Schutz et al. (2009) proposed a
sample average approximation and dual decomposition procedure to solve the supply chain design problem under demand uncertainty. demand is price inelastic.
All of the aforementioned studies assume that
This assumption implies that manufacturers/retailers are unable to use
pricing to change their revenues. The second stream of research that relates to the current study examines the dynamic pricing problem that governs the replenishment of inventory under monopolistic and uncertain market conditions. This problem involves how prices may be established over time in a manner that maximizes a firm’s total expected profit.
Netessine (2006) studied a pricing problem involving
a limited number of price changes in a dynamic but deterministic environment. environment, demand is dependent on the current price and time.
In this
Netessine’s research analyzed
inventory, capacity and pricing decisions for a product that is sold by a monopolist over the course of a short season.
This investigation revealed the impact of capacity constraints on
optimal prices and the timing of price changes. In a stochastic environment, Chen and Lee (2004) studied a multi-product, multi-stage and
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
multi-period scheduling model in a multi-echelon supply chain network involving uncertain product demands and product prices. These researchers used fuzzy sets to describe sellers’ and buyers’ incompatible preferences with respect to product prices. The product demands with known probabilities were assumed to be independent of product prices.
Under these conditions,
the supply chain scheduling model becomes a mixed-integer nonlinear programming problem involving several conflicting objectives, such as maximizing profits and maintaining high service levels.
Chen and Lee address this problem by presenting a two-phase fuzzy decision-making
method to determine a compromise solution in a multi-echelon supply chain network under uncertain conditions.
Wafa et al. (2008) developed a multi-period supply chain network design
model for a petroleum organization in an oil-producing country under uncertain market conditions.
These researchers proposed a stochastic formulation of this situation that was based
on the two-stage problem with a finite number of demand and price realizations. They concluded that the impact of demand and price uncertainties may be tolerable given an appropriate balance between crude exports and processing capacities. In the current study, we address the pricing problem that involves the replenishment of inventory in a monopolistic and uncertain market.
Using the assumption that stochastic
demands are price dependent, we examine the mutually interactive relationship between pricing and supply chain operations. 3.
The combined pricing and operations mathematical model In this section, we describe the supply chain planning network and define the scope of this
network.
We then present our mathematical model for combined pricing and operational
planning under demand uncertainty. 3.1 Supply chain network and operations
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The objective of pricing and supply chain operational planning under demand uncertainty is to determine the product prices in combination with an integrated supply, manufacturing and demand-side operations plan in a manner that maximizes total expected profit. network consists of supply, manufacturing and distribution sub-networks. sub-network is composed of contracted component providers.
A supply chain
A supply
In this study, the manufacturing
sub-network is assumed to consist of a single production plant that outputs finished products. To meet uncertain customer demands, the distribution sub-network is assumed to include both a central warehouse and a field warehouse. A four-echelon supply chain network for dual products is illustrated in Figure 1.
In this planning network, there are two finished products.
Both of these products require five components and are stocked at the central and/or field warehouses. The assumptions of this study are discussed below. 1.
We assume that the mean demand function is linear.
Therefore, the inverse mean demand
must exist and must also be a linear function. As a result, the revenue function is a concave function. 2.
The supply chain network consists of component providers; manufacturing plants; and centralized and field warehouses. To study the functional properties of optimality cuts for the L-shaped decomposition-based algorithm, we assume that the supply chain consists of a single manufacturing plant, a single centralized warehouse and a single field warehouse. Therefore, we assume the existence of monopolistic market conditions.
3.
We assume that the examined manufacturer sources all of its components and that this firm produces and supplies dual products.
The manufacturer is presumed to execute a
make-for-stock (MTS) manufacturing process, which is a mass production process that
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
achieves production at economies of scale.
Thus, we assume that the total production cost
function is a concave function and that the average cost is a convex function. 4.
Upon production, products are assumed to be shipped to the central and/or field warehouses for storage; no inventory is retained at the manufacturing plant.
The products become
inventory prior to demand realization and are directly shipped to meet customer orders. We assume that the dead inventory cost at the field warehouse is no less than the dead inventory cost at the central warehouse.
In addition, the unit shipment cost of delivering products
from the field warehouse to customers is presumed to be no higher than the unit shipment cost of delivering products from the central warehouse to these customers. 3.2 Mathematical model The combination of pricing and operational planning tactically determines the prices and inventory levels at any echelon of the supply chain network. Operationally, the manufacturer seeks to purchase, manufacture and deliver finished products to satisfy uncertain demand in a manner that maximizes expected profit.
These processes can be modeled in two stages.
In the
first-stage of these processes, the manufacturer determines product prices and the operations plan with respect to procurement, production and inventory management.
In this stage, components
are procured and products are created at the manufacturing plant. These products are shipped and positioned at the central and/or field warehouses.
In the second-stage of supply chain
processes, products are shipped to customers once customer orders have been received. The model of this study seeks to maximize the total expected profit, which is the sum of deterministic procurement, manufacturing and inventory costs; and expected revenue, product distribution expenses and obsolescence costs.
A two-stage stochastic programming problem was formulated
in a deterministic equivalent formulation.
We compare this profit optimization approach with
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the cost minimization approach.
For readability, the cost minimization model for uncertain
demands is provided in Appendix A of this paper. A.
Parameters and decision variables
1. Sets and parameters: h
denotes the consumption market;
g
denotes a product;
G
= {..., g ,...} ;
m
denotes the single production plant;
s
denotes a component;
S
= {..., s,...} ;
cs
is the unit purchase cost of component s S ;
c sm
is the freight rate to ship a unit of component s S to the production plant m;
c m, g Q m, g
is the average production cost of product g G at the production plant, which is a function of Q, the plant’s production level;
c mr , g / c rf , g
are the freight rates to ship a unit of product g G from the production plant to the central warehouse/from the central warehouse to the field warehouse;
c rh, g / c fh, g
are the freight rates to ship a unit of product g G from the central/field warehouse to the consumption market;
I r,g / I f ,g
are the unit inventory costs of product g G in the central (field) warehouse;
O r,g / O f ,g
are the obsolescence costs for a unit of unused product g G in the central/field warehouse;
Um
is the production capacity;
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s, g
is the number of components s S that are used for a product g G ;
2. Decision variables: d h, g
is the loss of sale of product g G in the consumption market h;
q h, g
is the anticipated mean demand (with an associated price p h, g (q h, g ) ) of product g G in the consumption market h;
Q m, g
is the production quantity of product g G at production plant m;
u r ,g / u f ,g
are the number of products g G stored in the central warehouse r/the field warehouse f;
v r,g / v f ,g
is the number of unused products g G in the central warehouse r/the field warehouse f; is the number of components s S shipped from the supplier to the production
x sm
plant m; y mr , g / y rf , g
are the number of products g G shipped from the production plant m to the central warehouse r/from the central warehouse r to the field warehouse f.
z rh, g / z fh, g
is the number of products g G shipped from the central warehouse r/the field warehouse f to the consumption market h;
B. A second-stage recourse subproblem For a given procurement with the production and inventory operations plan of {qˆ h, g , uˆ r , g , uˆ f , g } , the second-stage recourse subproblem under demand scenario may be expressed as follows:
Max (qˆ ( ), uˆ ) p h, g (qˆ h, g )d h, g c rh, g z rh, g c fh, g z fh, g O r , g v r , g O f , g v f , g g
11
(1)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Subject to: z rh, g z fh, g d h, g qˆ h, g ( )
g G
(2)
z rh, g v r , g uˆ r , g
g G
(3)
z fh, g v f , g uˆ f , g
g G
(4)
z
rh, g
, z fh, g , d h, g , v r , g , v f , g
The objective function (1) represents the total fulfillment cost, which is the sum of unrealized revenue from insufficient inventory; distribution expenses; and product obsolescence/dead inventory costs from overstocked inventory. Constraint (2) states that the sum of the total shipments to the consumption market and losses of sales (if any) must be equal to the actual demand of scenario . Constraint (3) represents product flow conservation at the central warehouse and states that the sum of the fulfilled demand and unused inventory must be equal to the inventory level that existed in the first-stage. Similarly, the conservation of product flow at the field warehouse is stated in constraint (4). Note that in this formulation, there are no loss of sale terms in the objective function.
By contrast, in the cost minimization formulation,
losses of sales serve as a proxy for potential profits.
In cost minimization approaches, in the
absence of lost sales, manufacturers would cease production to achieve an optimal cost of zero. However, in the profit optimization formulation, the recourse function represents the actual profit upon demand realization. The profit optimization approach involves a linear program, thus, the associated dual subproblem for this approach may be expressed as follows:
(qˆ ( ), uˆ ) Min h, g qˆ h, g ( ) r , g uˆ r , g f , g uˆ f , g
(5)
g
Subject to
h, g p h, g qˆ h, g
g G
12
(6)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h, g r , g c rh, g
g G
(7)
h, g f , g c fh, g
g G
(8)
r , g O r , g
g G
(9)
f , g O f , g
g G
(10)
{ h, g , r , g , f , g }
In the above expressions, { h, g , r , g , f , g } are dual variables that are associated with constraints (2), (3) and (4), respectively. Thus, the expected fulfillment cost of the second-stage recourse subproblem is the sum of the costs of each demand scenario multiplied by the corresponding probability for the scenario in question, (qˆ ( ), uˆ ) .
C.
The combined pricing and operational planning model Given the expected second-stage fulfillment cost that is discussed above, the combined
pricing and supply chain operational planning model in its deterministic equivalent formulation may be expressed as follows: Max
q p q h, g
h, g
g
I
r ,g
g
h, g
( ) (c s c sm ) x sm s
c Q Q m, g
m, g
g
u r , g c mf , g y mf , g I f , g u f , g (q( ), u ) g
g
m, g
c mr , g y mr , g g
(11)
Subject to: x sm s , g Q m, g
s S
(12)
Q m, g U m, g
g G
(13)
q h, g Q m, g
g G
(14)
y mr , g y mf , g Q m, g
g G
(15)
g
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
y mr , g u r , g
g G
(16)
y mf , g u f , g
g G
(17)
{q h, g , Q m, g , x sm , y mr , g , y mf , g , u r , g , u f , g }
The objective of function (11) is to maximize expected profit. The terms of this function are the expected revenue at the anticipated mean demand; the deterministic component procurement, product manufacturing and inventory costs at the central and field warehouses; and the expected second-stage fulfillment cost. Constraint (12) is the procurement constraint, which requires a manufacturer to purchase sufficient components for production.
Constraint (13) is the
plant production capacity constraint, which states that the total units of produced products cannot exceed the plant’s production capacity. Constraint (14) is the production/inventory level constraint, which requires the production of sufficient product to meet the anticipated mean demand.
In other words, this constraint specifies that upon the determination of product prices
(through the use of the inverse demand function), the manufacturer will produce and stock sufficient product to at least satisfy the anticipated mean demand for the consumption market. Constraint (15) is the product flow conservation constraint at the plant, which states that all of the produced products must be shipped to central and/or field warehouses to become inventory. Constraints (16) and (17) are flow conservation constraints at the central and field warehouses, respectively, that require received products to become inventory for consumption.
Finally, the
decision variables of anticipated mean demand, production, procurement level, production shipments and inventory level must all have non-negative values. 4.
The L-shaped solution algorithm The combination of pricing and supply chain operations planning with price-dependent
demand uncertainty may be represented as a two-stage stochastic program.
14
The L-shaped
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
approach, which is also known as the decomposition approach, separates the two-stage stochastic problem into a first-stage deterministic master problem and a second-stage stochastic recourse subproblem (Birge and Louveaux, 1997; Kall and Wallace, 1994).
In this case, to implement
this approach, the combined pricing and operational planning model expressed in (11)-(17) is decomposed into the following first-stage deterministic master problem and the second-stage stochastic recourse subproblem that is described by (1)-(4). Max
q p q h, g
h, g
g
I
r,g
g
h, g
( ) (c s c sm ) x sm s
c Q Q m, g
g
m, g
m, g
c mr , g y mr , g g
u r , g c mf , g y mf , g I f , g u f , g g
(18)
g
Subject to: (12)-(17)
h, g q h, g ( ) r , g u r , g f , g u f , g
(19)
g
{q h, g , Q m, g , x sm , y mr , g , y mf , g , u r , g , u f , g } ;{ }
Constraint (19), which is the dual objective function of the second-stage recourse subproblem (5), defines the optimality cuts.
In particular, this constraint replaces the
second-stage expected fulfillment cost function (q( ), u ) with a set of optimality cuts
that represent the upper bounds of this function. The computational process solves the second-stage recourse subproblem that is expressed in (1)-(4).
An optimality cut for (5) is
determined and subsequently embedded in the first-stage master problem as the additional constraint (19). This process forms an expanded first-stage master problem (18) that is subject to constraints (12)-(17) and (19).
In a fixed recourse subproblem with parameters
15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{ h, g , r , g , f , g , g G} , the optimality cut is linear. Therefore, the expected fulfillment cost
function can be replaced with a piecewise linear function. With a concave revenue function, a convex cost function and linear constraints, the expanded master problem remains a concave program.
Unfortunately, under these conditions, using prices and anticipated mean demands as
decision variables, the second-stage of the model becomes a non-fixed recourse subproblem. Therefore, we analyze the functional form of optimality cuts and the type of mathematical model that exists for the expanded master problem. 4.1 The valid functional forms for optimality cuts In section 3.1, we assume that the dead inventory cost at the field warehouse is no less than the dead inventory cost at the central warehouse.
In addition, the unit shipment cost from the
field warehouse to customers is assumed to be no higher than the unit shipment cost from the central warehouse to the same customers.
These assumptions lead to the following lemma.
Lemma 1. The manufacturer uses the inventory at the central warehouse to fulfill demand only if no available inventory exists at the field warehouse. In other words, z rh, g 0 v f , g 0. Proof. The manufacturer will fulfill demand with inventory from the field warehouse if possible. Assume the contrary. (O r , g c rh, g ) (O f , g c fh, g ) .
Because O r , g O f , g and c rh, g c fh, g , it follows that
If v f , g 0 , we may increase z fh, g by one unit and decrease
z rh, g by one unit, and the firm’s overall profit will be increased by (c fh, g O f , g ) (c rh, g O r , g ) 0 . The contradiction is now established.
□
We now analyze the functional form of optimality cuts. Lemma 2. The optimality cuts are quadratic functions. Proof. The optimality cut is a combination/summation of the following parts, which are
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
derived and demonstrated in Appendix B:
O f , g c fh, g qˆ h, g ( ) O r , g uˆ r , g O f , g uˆ f , g ; for d h, g 0, v r , g uˆ r , g , v f , g 0 (a) f , g fh , g qˆ h, g ( ) O f , g uˆ f , g ; for d h, g 0, v r , g uˆ r , g 0, v f , g 0 O c
O r , g c rh, g qˆ h, g ( ) O r , g uˆ r , g O r , g c rh, g c fh, g uˆ f , g ; (b) for d h, g 0, uˆ r , g v r , g 0, uˆ f , g v f , g 0 O r , g c rh, g qˆ h , g ( ) O r , g uˆ r , g ; for d h , g 0, 0 v r , g uˆ r , g , uˆ f , g 0 (c)
p qˆ uˆ h, g
h, g
r,g
uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , if d h, g 0 and v r , g v f , g 0 .
The functional forms of (a) and (b) above are linear functions for anticipated mean demands and inventories, respectively. The functional form that is expressed in (c) is the sum of unrealized revenues (in cases of insufficient supply) and the costs of transporting products from the central and field warehouses to the consumption market.
The unrealized revenue is a
function of the product prices and the associated anticipated mean demand of qˆ h , g that has been determined in the master problem. mean demand is altered.
Therefore, the unrealized revenue changes as the anticipated
If we simply treat h, g as a parameter and derive an optimality cut
of h, g uˆ r , g uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , then this cut may be embedded as an additional constraint for the master problem.
In the subsequent computation of the expanded
master problem, a set of new anticipated mean demands and the product inventories at the central
and field warehouses are determined q h, g , u r , g , u f , g , g G , and assumed to
be p h, g q h, g p h, g qˆ h, g .
Given a new set of product prices p h, g q h, g , the upper bound on the
expected fulfillment cost that is determined by this optimality cut is
p h, g qˆ h, g u r , g u f , g (q h, g ()) c rh, g u r , g c fh, g u f , g because h, g p h, g qˆ h, g .
In fact, the
correct upper bound is p h, g q h, g u r , g u f , g (q h, g ()) c rh, g u r , g c fh, g u f , g with respect to the
set of new product prices p h, g q h, g . The discrepancy between these two expressions implies
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
that h, g u r , g u f , g (q h, g ( )) c rh, g u r , g c fh, g u f , g , for h, g p h, g qˆ h, g , either over- or under-estimates the actual upper bound of the expected fulfillment cost in the expanded master problem. Thus, h, g cannot be treated as a parameter.
Instead, an appropriate functional form
of (c) is p h, g qˆ h, g uˆ r , g uˆ f , g (qˆ h, g ( )) c rh, g uˆ r , g c fh, g uˆ f , g , a quadratic function.
□
Whenever an optimality cut (qˆ ( ), uˆ ) is derived for a given qˆ h, g , uˆ r , g , uˆ f , g , g G ,
this cut is embedded in the master problem as an additional constraint. We must show that this optimality cut remains valid for all other prices.
Otherwise, the inclusion of this new constraint
will alter the feasible set of possibilities for other prices. Lemma 3.
qˆ
h, g
An optimality cut determined by a set of product prices and inventories
, uˆ r , g , uˆ f , g , g G is a valid optimality cut for all other product
prices q h, g , u r , g , u f , g , g G if the corresponding discrete demand probabilities remain unchanged by shifts in product prices. Proof.
The first-stage master problem determines qˆ h, g , uˆ r , g , uˆ f , g , g G . The
associated product prices are p h, g qˆ h, g , g G . For given p h, g qˆ h, g , d h,g , v r ,g , v f ,g of the primal and { h, g , r , g , f , g } of the dual of the second-stage recourse subproblem may be determined. The dual objective function becomes the optimality cut that is the mathematical sum of (b.1) through (b.5) with the corresponding demand probabilities.
We discuss the
applicable cases as follows. (1)
uˆ f , g 0 and uˆ r , g 0 .
of (b.1), (b.2) and (b.5).
In this instance, the optimality cut
(qˆ ( ), uˆ ) consists
We partition the demand distribution of anticipated mean
demand qˆ h , g into three segments qˆ h, g (1 ) qˆ h, g ( 2 ) qˆ h, g ( 3 ) with associated probabilities
18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of w1 , w2 , w3 such that the conditions qˆ h, g (1 ) uˆ f , g , qˆ h, g ( 2 ) uˆ r , g uˆ f , g and qˆ h, g ( 3 ) uˆ r , g uˆ f , g are satisfied.
As a result, each of these demand segments corresponds to
the condition for either (b.1), (b.2) or (b.5).
For any anticipated mean demand q h, g qˆ h, g , under
the probability distribution w1 , w2 , w3 , the demand distribution is partitioned
into q h, g (1 ) q h, g ( 2 ) q h, g ( 3 ) . We may define u r , g 0, u f , g 0 to satisfy the condition q h, g (1 ) u f , g q h, g ( 2 ) u r , g u f , g q h, g ( 3 ) . These conditions apply for (b.1), (b.2) and (b.5), implying that (q ( ), u ) (qˆ ( ), uˆ ) is a valid optimality cut for
q h, g .
(2)
uˆ f , g 0 but uˆ r , g 0 .
The optimality cut
(qˆ ( ), uˆ ) consists of (b.2) and
(b.5). We partition the demand distribution of anticipated mean demand qˆ h , g into two segments
qˆ
h, g
(1 ) qˆ h, g ( 2 ) with probabilities of w1 , w2 , respectively. These segments
satisfying qˆ h, g (1 ) uˆ f , g and qˆ h, g ( 2 ) uˆ f , g . Thus, for any anticipated mean
demand q h, g qˆ h, g , any u r , g 0, u f , g 0 that satisfies q h, g (1 ) u f , g q h, g ( 2 ) will result in
(q ( ), u ) (qˆ ( ), uˆ ) . (3)
The optimality cut (qˆ ( ), uˆ ) consists of (b.4) and (b.5).
uˆ f , g 0 .
We
partition the demand distribution of the anticipated mean demand qˆ h , g into two segments
qˆ
h, g
(1 ) qˆ h, g ( 2 ) with w1 , w2 satisfying qˆ h, g (1 ) uˆ r , g and with qˆ h, g ( 2 ) uˆ r , g .
Thus, for any anticipated mean demand q h, g qˆ h, g , any u r , g 0, u f , g 0 that satisfies
19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
q h, g (1 ) u r , g q h, g ( 2 ) will result in (q ( ), u ) (qˆ ( ), uˆ ) .
□
Finally, we demonstrate that optimality cuts are concave functions. Lemma 4. Optimality cuts are concave functions.
Proof. We demonstrate that p h, g q h, g u r , g u f , g (q h, g ( )) c rh, g u r , g c fh, g u f , g is a concave function.
Because we assume that the demand function is linear, the inverse demand
function is also linear and may be expressed in the form of p a bq . This function therefore possesses the following Hessian matrix:
2b b b b 0 0 . Because u r , g u r , g Q m q h, g , b 0 0
q
h, g
u r,g
u f ,g
h, g 2b b b q b 0 u r , g 2bq h , g q h , g u r , g u r , g 0 . Thus, this matrix is 0 f ,g 0 u b 0
semi-definite. □ From the above lemmas, we may derive the following theorem regarding the embedded first-stage master problem. Theorem 5. The expanded first-stage master problem is a concave program. Proof. By Lemma 4, all optimality cuts are concave functions. Thus,
h, g q h, g ( ) r , g u r , g f , g u f , g defines a convex set for q h, g , u r , g , u f , g , g G. g
g
g
Constraints (12)-(17) are linear; therefore, the feasible set remains convex.
Given a concave
objective function (18), the expanded master problem must remain a concave program. 4.2 The L-shaped decomposition algorithm
20
□
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The L-shaped algorithm includes the following detailed steps. Step 0 (Initialization). We initialize the number of iterations for the master problem T=0 and the set of demand fulfillment optimality cuts . Step 1 (First-stage master problem).
We increment the number of iterations such that
T=T+1. At the Tth master problem, we solve (18) under constraints (12)-(17) and K, which determines the anticipated mean demand (with an associated price), the product production, the component procurement, and the product inventories at the central and field warehouses {(q h, g )T , (Q m, g )T , ( x sm )T , ( y mr , g )T , ( y mf , g )T , (u r , g )T , (u f , g )T } .
This solution also
determines ( )T . Step 2 (Second-stage recourse subproblem). At the Tth master problem, for the determined anticipated mean demand, manufacturing and inventory plan {(q h, g )T , (u r , g )T , (u f , g )T } , we solve (1)-(4), which determines the distribution
plan ( z rh, g )T , ( z fh, g )T , (v r , g )T , (v f , g )T , (d h, g )T with duals ( h, g )T , ( r , g )T , ( f , g )T . If the second-stage optimal objective value is bounded by ( )T , the solution {(q h, g )T , (Q m, g )T , ( x sm )T , ( y mr , g )T , ( y mf , g )T , (u r , g )T , (u f , g )T } is optimal, and the program
terminates. Otherwise, we generate
( h, g )T q h, g ( ) ( r , g )T (u r , g )T ( f , g )T (u f , g )T ,
T
g
(20)
a demand fulfillment optimality cut for the second-stage recourse subproblem, which becomes an additional constraint on the expanded master problem K such that { (20)} . return to step 1. 5.
Numerical examples
21
We then
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The supply chain operations planning network for the numerical example of this study is displayed in Figure 1.
There are three common components (labeled 1–3) and two
differentiated components (labeled 4–5) for two distinct products A and B.
Product A uses
components 1-4, whereas product B utilizes components 1-3 and component 5.
The
manufacturer sources components 1-5; produces A and B; and subsequently stocks A and B at a central and/or field warehouse.
The cost parameters are organized in Table 1. The cost parameter rules are as follows. (1) Manufacturing.
We assume that there are economies of scale in production. The total
production cost function for each product is a monotonic increasing function that is given by 0.05q 2 500 . The corresponding average cost function is a strictly convex function. (2) Obsolescence/dead inventory cost. Finished goods are assumed to be obsolete products that will be sold at a relatively high loss through discounted retailers. (3) Service penalty. warehouse serves customers more promptly than the central warehouse.
The field
Thus, there is a penalty
for the manufacturer if demands are met by the central warehouse instead of the field warehouse. (4) Loss of sale. The penalty for a lost sale is the price of this sale, which is a parameter in the cost minimization model. The demand function is assumed to be a linear function for each product; in particular, p 300 1.5q . We now examine the effects of the pricing-related tactical decision. We assume that the demand function is linear and that the revenue function will therefore be strictly concave; furthermore, we presume that the total cost function is monotonically increasing.
We assume that there are economies of scale in production. The total
production cost function for each product is a monotonic increasing function that is given by 0.05q 2 500 . The corresponding average cost function is a strictly convex function. (2) Obsolescence/dead inventory cost. Finished goods are assumed to be obsolete products that will be sold at a relatively high loss through discounted retailers. (3) Service penalty. warehouse serves customers more promptly than the central warehouse.
The field
Thus, there is a penalty
for the manufacturer if demands are met by the central warehouse instead of the field warehouse. (4) Loss of sale. The penalty for a lost sale is the price of this sale, which is a parameter in the cost minimization model.