1. health and fitness
2. disease
3. cholesterol

Optimal Selection of Customers from the Perspective of
Manufacturers in Continuous Replenishment Program
Payam Parsa
[email protected], (479) 422 6937
Industrial Engineering Department, University of Arkansas
Fayetteville, AR, U.S.A.
I am Payam Parsa, a Ph.D. student in the Industrial Engineering Department at the University of
Arkansas (UofA). I am also a research assistant (RA) in the Center for Excellence in Logistics
and Distribution (CELDi) at UofA. I received my Master’s degree in Industrial Engineering from
Southern Illinois University in 2012.
Optimal Selection of Customers from the Perspective of
Manufacturers in Continuous Replenishment Program
ABSTRACT
A continuous replenishment program (CRP) is a supply chain initiative in which the
manufacturer manages the replenishment process using shared demand information provided
by the customer. This paper presents an optimization model for partner selection process from
the perspective of the manufacturer, who faces a set of customers. Several factors such as
volume, customer location, requested product mix and desired service level are considered as
inputs for this selection process.
1
INRODUCTION
A continuous replenishment program (CRP) is a supply chain initiative in which the
manufacturer manages the replenishment process using the shared demand information
provided by the customer. A CRP relationship often comes with mutual benefit propositions in
which, both manufacturer and customer share the costs and the benefits of the collaboration.
The benefits include but are not limited to higher service level, lower transportation, holding,
and ordering cost. The cost efficiency of CRP encourages both manufacturers and customers to
move towards not only adoption, but selecting a good partner, which is very critical and has a
significant effect on the success of the entire program. The focus of this study is on the partner
selection process from the perspective of manufacturer who faces a set of customers. Several
factors such as volume, customer location, requested product mix and desired service level are
considered as the inputs of this selection process. The main criterion of partner selection is
total cost savings in the entire supply chain. In real world situations, both the manufacturer and
its customers have multiple distribution centers (DC) across the country (or possibly world). A
channel is referred to a pair between the manufacture’s DC and a customer’s DC. For some
channels, it is more efficient if the customer joins the CRP relationship with more consolidated
shipments, less manual handling, etc. However, for some channels of the same customer,
joining the CRP relationship may not benefit. For example, for the channels that have small
volume but more frequent demands, there may not be much transportation savings. Normally,
once a customer enters the CRP relationship, all channels of the customer will be included in
the CRP planning therefore; the aggregate impact of each customer’s channels is considered as
the selection criteria. Figure 1 illustrates the concept of channel and customers.
2
Customer DC
Manufacturer DC
Manufacturer DC
Customer DC
FIGURE 1: Supply network and channels representation
3
Customer DC
LITERATURE REVIEW
The most relevant topic in the literature to this study is supplier selection in which different
approaches have been proposed to tackle the problem. Selecting a partner is a comparatively
complex decision making process in which maintaining a long term partnership with good
suppliers has been sought by decision makers. The decision making approaches that have been
used for supplier selection problem could be each considered somewhere on the range of
qualitative methods to quantitative methods. Analytic hierarchy process (Chan and Kumar,
2007) (Akarte, et al, 2001), fuzzy set theory (Chen, et al, 2006) (Florez-Lopez, 2007), case-based
reasoning (Choy and Lee, 2002), mathematical modeling (Ghodsypour and O’brien, 2001)
(Kasilingam and Lee, 1996), multi attribute rating technique (Kwong, et al, 2002) are among the
most common approaches used in the literature. As you can see, various approaches are used
for solving the supplier selection problem but mathematical modeling has been the most
popular approach in the literature since most of the time a variation of it is used in constructing
the hybrid approaches. (Ho, et al, 2010)
Different forms of mathematical modeling have been proposed in the literature for supplier
selection problem. (Talluri and Narasimhan, 2003) modeled this problem by evaluating
alternative suppliers using their performance variability measures. Two linear programs are
used to maximize and minimize the performance of suppliers against the target measures.
Measures are set by buyers and performance of each supplier is evaluated by measuring the
maximum and minimum efficiencies metrics. (Ng, 2008) developed a weighted linear program
in which the decision maker determines the weights by their relative importance. The model is
4
maximization of the supplier score. (Narasimhan, et al, 2006) also proposed a multi-objective
model using the concept of criteria weighting. The model determines the optimal suppliers and
optimal order quantity while five different criteria are used to evaluate the performance of
each supplier. (Hong, et al, 2005) modeled the supplier selection problem by a mixed integer
linear model. The model is maximizing the revenue while determines the optimal number of
suppliers and the optimal order quantity. (Ghodsypour and O’brien, 2001) presented a mixed
integer nonlinear model to determine the optimal allocation of products to suppliers while the
annual purchasing cost is minimized. (Wadhwa and Ravindran, 2007) constructed a supplier
selection model using a multi-objective programming. Three objective functions are considered
for the model, minimization of price, lead time, and rejects. They used solution approaches
such as weighted objective method and goal programming to compare the solutions. In a very
recent study (Yu, et al, 2013) optimal selection of retailers for a vendor is investigated through
mathematical modeling. In this study, market scale of each retailer is considered as an
influential factor in decision making. This study selects the retailers for a vendor and sets the
contractual parameters between them such that both sides are in the optimal conditions from
their own perspective. It should be noted that the first priority is to maximize the vendor’s
profit. Hybrid heuristic methods used to find sufficiently good solutions for the mathematical
model.
5
MATHEMATICAL MODEL
In this section we are going to develop a mathematical model to identify an optimal set of
customers to join the CRP relationship from the perspective of manufacturer. We will develop a
maximization model that seeks to maximize the total amount of savings that can be achieved
by taking non-CRP channels to the CRP program. A channel cannot be moved to CRP
relationship unless all the associated channels of that non-CRP customer move to CRP program.
We assumed a single item system with the demand that occurs according to a Poisson process.
The replenishment policy is assumed to be a (r, Q) system.
Let
𝑚
be the total number of customer DCs
Let
𝑛
be the total number of manufacturer DCs
Let
𝜆𝑖𝑗
be the demand rate on channel 𝑖 − 𝑗
Let
𝑐
be the unit cost of item
Let
𝑣
be the unit volume of item
Let
𝑄𝑖𝑗
be the order quantity of channel 𝑖 − 𝑗
Let
𝑟𝑖𝑗
be the reorder point of channel 𝑖 − 𝑗
Let
𝐾𝑛𝑐𝑖𝑗
be the ordering cost of channel 𝑖 − 𝑗 in non-CRP relationship
Let
𝐾𝑐𝑖𝑗
be the ordering cost of channel 𝑖 − 𝑗 in CRP relationship
Let
ℎ
Let
̅
𝐼𝑛𝑐𝑖𝑗
be the average inventory level of channel 𝑖 − 𝑗 in non-CRP relationship
Let
̅
𝐼𝑐𝑖𝑗
be the average inventory level of channel 𝑖 − 𝑗 in CRP relationship
be the unit holding cost
6
Let
b
be the unit backorder cost
Let
𝐵̅𝑛𝑐𝑖𝑗
be the average backorder level of channel 𝑖 − 𝑗 in non-CRP relationship
Let
𝐵̅𝑐𝑖𝑗
be the average backorder level of channel 𝑖 − 𝑗 in CRP relationship
Let
𝑑𝑖𝑗
be the distance on channel 𝑖 − 𝑗 (mile)
Let
𝑉𝑗
be the volume limit of a full truckload shipment out of manufacturer’s DC
j
Let
𝐹𝑇𝑖𝑗
be the FTL rate of channel 𝑖 − 𝑗 ($/mile)
Let
𝐸𝑆𝑖𝑗
be the transportation efficiency score of channel 𝑖 − 𝑗
Let
𝑇𝑛𝑐𝑖𝑗
be the non-CRP transportation cost adjuster = 𝑀𝑎𝑥 [1,
Let
𝑇𝑐𝑖𝑗
be the CRP transportation cost adjuster = 𝑀𝑖𝑛 [1,
Let
𝐷𝑖𝑗
be the fixed additional cost of managing channel 𝑖 − 𝑗 in CRP program
Let
𝐸𝑖𝑗
be the variable additional cost of managing channel 𝑖 − 𝑗 in CRP program
($/ unit)
Let
𝑃
be the capacity of CRP program in terms of demand (quantity)
𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑛𝑜𝑛−𝐶𝑅𝑃 𝐸𝑆
𝐸𝑆𝑖𝑗
𝐸𝑆𝑖𝑗
𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝐶𝑅𝑃 𝐸𝑆
]
]
Let
1 be the binary decision variable, 1 if channel 𝑖 − 𝑗 is selected for CRP and
𝑥𝑖𝑗 = {
0 zero otherwise
Let
1
𝑥𝑖 = {
0
be the binary decision variable, 1 if customer 𝑖 is selected for CRP and
zero otherwise
The first step is to construct an objective function that represents the possible savings of
managing a channel in CRP. When the relationship between a customer and a manufacturer
changes from non-CRP to CRP, every cost component associated with replenishing the channel
could possibly change except the purchasing cost. The total purchasing cost is depended on two
7
parameters, demand (𝜆) and unit cost (𝑐), and both of them will remain at the same level when
the channel moves to CRP program.
On the other hand, the other cost components including ordering cost (OC), inventory holding
cost (HC), backorder cost (BC) and transportation cost (TC) would change when the channel
moves from non-CRP to CRP. Therefore, the objective function of our mathematical model
should reflect the possible savings in these cost components. Below is the formulation of the
total savings (𝑇𝑆𝑖𝑗 ) function of channel 𝑖 − 𝑗 that we are going to use in the model. We will
briefly explain how each component is constructed.
𝑻𝒐𝒕𝒂𝒍 𝑺𝒂𝒗𝒊𝒏𝒈𝒔 (𝑻𝑺𝒊𝒋 ) =
𝑂𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝐶𝑜𝑠𝑡 𝑆𝑎𝑣𝑖𝑛𝑔𝑠 (𝑂𝑆𝑖𝑗 ) +
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝐶𝑜𝑠𝑡 𝑆𝑎𝑣𝑖𝑛𝑔𝑠 (𝐻𝑆𝑖𝑗 ) +
𝐵𝑎𝑐𝑘𝑜𝑟𝑑𝑒𝑟 𝐶𝑜𝑠𝑡 𝑆𝑎𝑣𝑖𝑛𝑔𝑠 (𝐵𝑆𝑖𝑗 ) +
𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝐶𝑜𝑠𝑡 𝑆𝑎𝑣𝑖𝑛𝑔𝑠 (𝑇𝑆𝑖𝑗 ) −
𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑚𝑎𝑛𝑎𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑖𝑛 𝐶𝑅𝑃 (𝐴𝐶𝑖𝑗 ) =
𝜆𝑖𝑗
(𝐾
− 𝐾𝑐𝑖𝑗 ) +
𝑄𝑖𝑗 𝑛𝑐𝑖𝑗
̅
̅ )+
ℎ(𝐼𝑛𝑐𝑖𝑗
− 𝐼𝑐𝑖𝑗
𝑏(𝐵̅𝑛𝑐𝑖𝑗 − 𝐵̅𝑐𝑖𝑗 ) +
𝜆𝑖𝑗 𝑄𝑖𝑗 × 𝑣
[
× 𝑑𝑖𝑗 × 𝐹𝑇𝑖𝑗 ] × [𝑇𝑛𝑐𝑖𝑗 − 𝑇𝑐𝑖𝑗 ] −
𝑄𝑖𝑗
𝑉𝑗
[𝐷𝑖𝑗 + 𝐸𝑖𝑗 × 𝜆𝑖𝑗 ]
8
In the first term, ordering cost component, the unit ordering cost (K) is making the cost
difference between non-CRP and CRP relationships. In the second and third terms, holding and
backordering cost components, average inventory level (𝐼 )̅ and average backorder level (𝐵̅) are
expected to change when a channel moves to CRP program thus, they are making the cost
difference. The average inventory and backorder levels in (r, Q) system can be computed using
the following formulas:
𝐵̅ (𝑟, 𝑄) =
1
× [𝐺 2 (𝑟) − 𝐺 2 (𝑟 + 𝑄)]
𝑄
𝐼 (̅ 𝑟, 𝑄) =
1
× (𝑄 + 𝐿) + 𝑟 − 𝜆𝐿 + 𝐵̅ (𝑟, 𝑄)
2
Where 𝐺 2 is the second order loss function for the demand during lead time (𝐿). The fourth
term is representing the transportation savings in which the cost adjuster parameters (𝑇𝑛𝑐𝑖𝑗 ,
𝑇𝑐𝑖𝑗 ) are making the difference between non-CRP and CRP relationships. These parameters are
computed based on the channel transportation efficiency score. In the fourth term, [
𝑄𝑖𝑗 ×𝑣
𝑉𝑗
×
𝑑𝑖𝑗 × 𝐹𝑇𝑖𝑗 ] represents the cost of transporting 𝑄 units, which is the order quantity of the
channel, with FTL rate. Multiplying the cost adjuster parameters (𝑇𝑛𝑐𝑖𝑗 , 𝑇𝑐𝑖𝑗 ) and the order
𝜆
frequency (𝑄𝑖𝑗 ) of the channel makes the total transportation cost component of the channel in
𝑖𝑗
either CRP or non-CRP relationships. The last term of the total savings function represents the
additional cost of managing a channel in CRP program instead of in non-CRP. As we know,
initiating and also maintaining a supply chain collaboration program such as CRP incurs cost to
the system. Part of the cost is represented as a fixed cost (𝐷) (e.g., technological needs), while
9
there is also a variable cost component (e.g., higher skilled employees) which is depended on
the demand size (𝜆𝑖𝑗 ).
The critical constraint that is mentioned in the problem is once a customer enters into the CRP
relationship with the manufacturer; all the channels of the customer should be included in the
CRP program. In order to include this to our model we need to add the two following
constraints in which 𝑥𝑖 is a binary decision variable for selecting customer 𝑖:
𝑛
𝑛𝑥𝑖 ≤ ∑ 𝑥𝑖𝑗
𝑗=1
𝑥𝑖𝑗 ≤ 𝑥𝑖
The first constraint ensures if one channel is selected, then all the channels of that customer
needs to be selected which means partially selection of channels of a customer is not feasible. It
also ensures that if none of the channels of a customer is selected then the customer is not
allowed to be selected. The second constraint guarantees the selection of the customer if one
of its channels is selected.
By adding these two constraints, we reduced our channel evaluation model to a customer
evaluation model with the binary decision variable 𝑥𝑖 . In order to construct such mathematical
model we need to compute the possible savings that each customer can contribute to the
system.
Let 𝑇𝑆𝑖 represents the total savings that could be obtained by moving customer 𝑖 from non-CRP
relationship to CRP:
10
𝑛
𝑇𝑆𝑖 = ∑ 𝑇𝑆𝑖𝑗 =
𝑗=1
𝑛
∑
𝑗=1
𝜆𝑖𝑗
(𝐾
− 𝐾𝑐𝑖𝑗 ) +
𝑄𝑖𝑗 𝑛𝑐𝑖𝑗
𝑛
̅
̅ )+
∑ ℎ(𝐼𝑛𝑐𝑖𝑗
− 𝐼𝑐𝑖𝑗
𝑗=1
𝑛
∑ 𝑏(𝐵̅𝑛𝑐𝑖𝑗 − 𝐵̅𝑐𝑖𝑗 ) +
𝑗=1
𝑛
∑
𝑗=1
𝜆𝑖𝑗 𝑄𝑖𝑗 × 𝑣
[
× 𝑑𝑖𝑗 × 𝐹𝑇𝑖𝑗 ] × [𝑇𝑛𝑐𝑖𝑗 − 𝑇𝑐𝑖𝑗 ] −
𝑄𝑖𝑗
𝑉𝑗
𝑛
[∑ 𝐷𝑖𝑗 + 𝐸𝑖𝑗 𝜆𝑖𝑗 ]
𝑗=1
As you might have noticed, the value of 𝑇𝑆𝑖 could be computed for any customer 𝑖 using the
historical demand data. Therefore, the mathematical model for selecting the optimal set of
customers for CRP relationship could be formulized as the following:
𝑚
𝑀𝑎𝑥 ∑ 𝑇𝑆𝑖 × 𝑥𝑖
𝑖=
𝑚
𝑠. 𝑡.
∑ 𝑥𝑖 × 𝜆𝑖 ≤ 𝑃
𝑖=1
By looking at this formulation, you would realize that the problem is formulated as a classic
Knapsack problem, in which the customers are the “items”, 𝑇𝑆𝑖 and 𝜆𝑖 represents the value and
weight of each item and 𝑃 represents the total allowable weight.
11
12
NUMERICAL STUDY
The numerical example that is explained in this section is based on a real study that we
performed for a manufacturer in healthcare industry. They asked us to analyze the pairing of
top 10 customer locations with their top distribution centers and show how much savings could
be achieved from moving those channels to CRP program (30 channels in total that are
associated with 10 customer locations). The cost savings are computed by using the exact same
function that is discussed in the previous section (𝑇𝑆𝑖𝑗 ) excluding the last term (𝐷𝑖𝑗 + 𝐸𝑖𝑗 × 𝜆𝑖𝑗 )
which is representing the additional cost of CRP program compared to non-CRP (This savings
value is called “Gross Savings” in Table 1). Regarding the last term, we assumed two different
values for 𝐷𝑖𝑗 and 𝐸𝑖𝑗 and deduct this term from the gross savings (The result is called “Net
Savings” in Table 1). The gross savings values are calculated using a verified tool that
approximates the savings of each channel using different input information.
Table 1 shows the required information to compute the net savings values for each channel. I
assumed the fixed (𝐷𝑖𝑗 ) and variable cost (𝐸𝑖𝑗 ) of managing a channel in CRP to be $500 per
month and $0.05 per item per month respectively. The net savings values (𝑇𝑆𝑖𝑗 ) of each
channel is computed in the last column.
13
TABLE 1: Channel savings information
Gross Savings
Demand (Qty)
(6 months)
(6 months)
1
$111,856
91905
$30,571.50
$81,284.48
B
1
$107,344
38052
$14,415.60
$92,928.80
C
1
$120,680
69646
$23,893.80
$96,786.66
D
1
$45,333
51597
$18,479.10
$26,853.70
E
1
$140,726
44645
$16,393.50
$124,332.96
F
1
$149,037
10957
$6,287.10
$142,749.61
G
1
$58,265
34102
$13,230.60
$45,033.99
H
1
$62,634
36564
$13,969.20
$48,665.10
J
1
$87,444
33635
$13,090.50
$74,353.84
K
1
$23,222
26729
$11,018.70
$12,203.34
A
2
$251,571
262189
$81,656.70
$169,914.44
B
2
$359,732
151373
$48,411.90
$311,319.82
C
2
$124,724
115082
$37,524.60
$87,199.56
D
2
$122,825
96479
$31,943.70
$90,881.12
E
2
$193,125
72681
$24,804.30
$168,321.17
F
2
$152,331
15570
$7,671.00
$144,660.05
G
2
$70,929
85536
$28,660.80
$42,267.75
H
2
$58,631
69949
$23,984.70
$34,646.48
J
2
$95,611
74027
$25,208.10
$70,403.22
K
2
$49,313
78875
$26,662.50
$22,650.60
A
3
$246,444
9983
$5,994.90
$240,449.29
B
3
$68,610
16661
$7,998.30
$60,611.90
C
3
$301,038
7725
$5,317.50
$295,720.16
D
3
$180,227
3405
$4,021.50
$176,205.25
E
3
$228,715
7115
$5,134.50
$223,580.15
F
3
$227,853
11166
$6,349.80
$221,503.61
G
3
$125,561
6821
$5,046.30
$120,514.59
H
3
$109,591
3867
$4,160.10
$105,430.53
J
3
$158,051
6489
$4,946.70
$153,104.75
K
3
$97,349
5030
$4,509.00
$92,839.95
Customer
Channel
A
14
CRP cost
Net Savings
TS(ij)
Table 2 shows the net savings values (TSi ) for each customer in addition to the binary variable
values (xi ) that are used in the Excel solver to select the optimal customer for CRP relationship.
As you remember there is a capacity constraint in our mathematical model (∑𝑚
𝑖=1 𝑥𝑖 × 𝜆𝑖 ≤ 𝑃).
The last column of Table 2 is used to compute the left hand side of this constraint while the
right hand side value (𝑃) is set to be 800,000 items.
Table 2: Customers savings information
Customer
Demand
TS(i)
A
364077
$491,648
B
206086
C
Binary Variable
x(i)
TS(i) * Binary Variable
Demand * Binary Variable
0
$0
0
$464,861
1
$464,861
206086
192453
$479,706
1
$479,706
192453
D
151481
$293,940
0
$0
0
E
124441
$516,234
1
$516,234
124441
F
37693
$508,913
1
$508,913
37693
G
126459
$207,816
0
$0
0
H
110380
$188,742
1
$188,742
110380
J
114151
$297,862
1
$297,862
114151
K
110634
$127,694
0
$0
0
Table 2 shows the optimal selection while the objective function value maximized and became
$2,456,318. The constraint of our model is also satisfied since ∑𝑚
𝑖=1 𝑥𝑖 × 𝜆𝑖 = 785,204 ≤ 800,000
At the end, I should mention that while in this example we assumed selecting a customer location with
all its channels, the developed model could also be used for selecting optimal set of companies with the
mandatory requirement of selecting all of their channels. In such case, 𝑇𝑆𝑖𝑗 represents the total
achievable savings of 𝑗𝑡ℎ channel of customer 𝑖.
15
DISCUSSION AND CONCLUSION
As discussed earlier, the motivation behind this project is the ambiguity of partner selection process in
supply chain collaboration programs. For example, one of the leaders in the healthcare industry
collaborated with CELDi to gain the ability of selecting a good partner for a CRP relationship based on
the achievable savings for both parties. This selection process is usually more meaningful from the
perspective of vendors when they have limited capacity of resources to satisfy all the channels. While
CRP relationship lowers the cost of supply chain, some customers might be more profitable or less risky
which make them the best candidates for this relationship. Capability of identifying these customers is
very desirable for vendors since they could maximize their profit margin. The optimization model that is
discussed in this paper could be a great decision making tool for the manufacturers with hundreds of
channels in their supply chain and limited capacity in CRP program.
16
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