2013 International Conference on Management Science & Engineering (20th)
July 17-19, 2013
Harbin, P.R.China
A Lot Sizing Order Model and Solution Based on Win-win Negotiation
BAO Xiao-min，HE Xi-jun，JIANG Guo-rui
The Economics & Management School, Beijing University of Technology, P. R. China, 100124
Abstract: Recent years, in the field of supply chain
coordination, the collaborative planning mode based on
negotiation is getting more and more attention. For
solving the problem of order quantity between supplier
and demander in collaborative supply chain under the
condition of stochastic demand and asymmetric
information, this paper established lot sizing order
models: Retailer’s model and Manufacturer’s model
that are used to calculate their annual profit separately.
To get a satisfy solution that both sides accept, a win-win
algorithm is proposed. This algorithm applies the
equivalents curve theory to the negotiation theory under
the condition of time constraints. It is good to simulate
the negotiation between sealer and buyer in practice.
Also, the win-win algorithm breaks the restriction of
Master-slave status in Stackelberg mode and doesn’t
need the global information like the centralized
decision-making mode. The algorithm proposed in this
paper is shown more effective and practical by the
analysis of numerical example.
Keywords: supply chain, lot sizing order model,
win-win negotiation，Stackelberg, centralized decision
making
1 Introduction
With the development of economic globalization,
the environment that enterprises face is getting more and
more complex, and single company doesn’t have the
ability to meet the market demand [1]. All the partners
like producers, manufacturers, and the distributors have
to coordinate with each other for survival [2]. The
competition between enterprises is transforming into the
competition between supply chains, and research on
supply chain is getting more and more attention from the
companies in reality [3]. The key to improve the
competitiveness of the supply chain is cooperating with
each other in the supply chain. Meanwhile, negotiation is
very important way to the cooperation [4]. So, negotiation
has become the hot issue in the supply chain
collaboration.
In the supply chain, every enterprise node is not
only a supplier to the next node, but also a demander to
the precious node, the relationship between supplier and
Supported by the National Natural Science Foundation of
China (71071005)
978-1-4799-0474-7/13/$31.00 ©2013 IEEE
demander is always running through the entire supply
chain[5-7]. Therefore, the research on coordination
between supplier and demander is a basic and important
issue. Recently, there are three modes of supply chain
coordination: Stackelberg coordination mode [8],
centralized decision-making mode [9] and collaborative
planning mode based on negotiation [10]. Regarding the
Stackelberg mode, the leader grasps all the information
and announces the sales price firstly, and then the buyer
decides the optimal order quantity. P.C.Yang [11] studied
the problem of supplier’s optimal production quantity
and calculated the price discount that the buyer would
accept under the Stackelberg mode. Hojung Shin [12]
made a research on coordination of the supply chain
based on sharing inventory risk between the companies.
The result is that all the participants’ profits are
improved, but the overall interest of the whole supply
chain can’t achieve the highest. In response to the
shortcoming of the Stackelberg mode, centralized
decision-making mode is widely used, which makes the
overall optimal as the target for all enterprises. D. Jafari
[13]
got the range of price discount that all the partners can
accept under the centralized decision-making mode. Also,
Kamal Chaharsooghi [14] built the coordination model
that allowed a second order, and found that the final
interest depends on the capacity of negotiators, which is
actually the process of allocating the increase profit of
whole supply chain. Later, Wang nengmin [15] made the
comparison of those two modes, and found that the
Stackelberg mode couldn’t reach the best collaboration
due to lack of communication or one of the parties hold
the dominant position. At the same time, the hypothesis
of information sharing in centralized decision-making
mode must be recognized, which couldn’t meet the
reality enterprise circumstance.The supply chain
coordination under asymmetric information conditions
deserves further study.
In the negotiation-based collaborative planning
mode, every company in the supply chain is treated
equally. This mode breaks the restrictions in Stackelberg.
Meanwhile, the participants don’t need to know the
information of the others, and achieve the balance of
interests between members through several rounds of
interaction. There are also researches on that aspect. For
multi-player consultation, Nash had proposed a
bargaining mode of negotiations, and given the
well-known Nash bargaining solution. Harsanyi and
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Selten (1972) proposed the asymmetric Nash bargaining
mode on the basis of Nash’s work. Stanislaw Bylka [16]
solved the lot-sizing problem under the asymmetric Nash
bargaining mode. And then ， Ye fei [17] made a
comparison between Nash bargaining mode and
Stackelberg mode. He got the conclusion that Nash
bargaining mode is better than the other mode on the
supply chain’s global profit and the restrictions. Though
the Nash negotiation mode solved the problem of
information asymmetry in centralized mode, there is still
a problem that has to discuss is the difficult of measuring
the negotiating capacity. In 2007, Li ying [18] introduced
the bi-level programming theory with the application of
fuzzy satisfaction to solve this problem and used the
genetic algorithm to simulate the behavior of negotiation
in practice, and got the global interest close to the best.
But, every company has their own attitudes towards the
negotiation that has to be noticed.
This paper proposes a win-win negotiation
algorithm, by using the negotiation theory under time
constraints [19], learning the equivalents curve theory [20]
and considering the attitude of the negotiators [21]. In the
end, a numerical example is given to prove the
effectiveness and applicability of the algorithm this paper
proposed.
π r (Q ) = w * D − p * D − k r
D
−
Q
Where w is the purchase price of each unit of
product for the retailer; Q is retailer’s order quantity; k r
is the retailer’s setup cost; hr is the cost of holding for
each product per time of the retailer; B is the shortage
cost for each unit product; σ is the standard deviation
of customer’s demand in forecast period; k is the
constant coefficient of retailer’s safety stock which
follows a standard normal distribution; G (k ) is the
probability distribution function of the customer’s
demand in lead time.
∞
G (k ) = ∫ (u − k )
k
1
exp(−u 2 / 2)du
2π
The optimized amount of order size and the
maximum profit that the retailer expects can be
calculated through the following formulas:
EOQ =
2(k r + BσG (k )) D
hr
(2)
*
π r = ( w − p) D − hr kσ − 2(k r + BσG (k ))hr D
2 The lot sizing order model
2.1 Model hypothesis
(1)The commodity demand rate is a random
variable. When the market demand is stable, the demand
obeys normal distribution. Let D the mathematical
expectation of demand;
(2)The manufacturer's production capacity(R) is
much higher than the rate of demand, and the
manufacturer’s shortage cost is ignored;
(3)The production volume is an integer multiple ( n )
of the order quantity;
(4)Transportation cost is proportional to the number
of transport, and it is borne by the retailer;
(5)In steady state, the sales price of the retailer is
basically stable. Set a constant w as the sales price;
(6)The transaction price between supplier and
demander is p , the retailer’s expectation is p , and the
manufacturer’s expectation is p ( w > p > p > c) ;
(7)The price taker is retailer, who allows market
supply shortage, and uses ( s, Q ) as ordering policy.
When the inventory reduces to s , the buyer sends an
order batch of Q .
2.2 Model definition
Definition 1: Retailer’s model
Retailer’s revenue is sales income; its cost includes
the procurement cost, order processing (including
logistics) cost, inventory holding cost and shortage cost.
π r is the notation of retailer’s annual profit.
(1)
Q
D
+ kσ ) − BσG ( k )
h（
r
2
Q
(3)
Definition 2: Manufacturer’s model
Manufacturer’s revenue is also the sales income; its
cost includes production cost, preparation cost for each
order, order processing cost, and inventory holding cost.
π m is the notation of manufacturer’s annual profit.
D
D
− lm
nQ
Q
− [(n − 1) − (n − 2) D / R](Q / 2) * hm
π m (n, Q) = p * D − c * D − k m
Where
(4)
c is the cost of each unit product; k m is
the manufacturer’s setup cost;
lm
is the order
processing cost; hm is the cost of storage for each
product per time of the manufacturer; According to the
references [5],the annual average inventory of
manufacturer is [( n − 1) − (n − 2) D / R ](Q / 2) .
The optimized production volume of manufacturer
can be calculated as follows:
2k m D
(1 − D / R )Q 2 hm
(5)
2k m D
(1 − D / R ) EOQ 2 hm
is an integer ，
n=
If n =
then the manufacturer’s optimized production volume is
n;
Else if
n=
2k m D
is not an
(1 − D / R ) EOQ 2 hm
integer, then the optimized production volume of
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manufacturer is either [ n ] or [ n ] +1. To determine the
manufacturer’s choice, we have to calculate the fourth
formula with [n] and [n] +1 separately. Then we choose
the larger value and note n S .If the two values are equal.
We choose[n] as n S , because it can reduce the
manufacturer’s production pressure.
After the manufacturer’s optimized production
volume is fixed, there is also an expected order quantity
that the manufacturer has, and a maximum profit that the
manufacturer hopes to get can be calculated through the
following formulas:
EPQ =
2(k m / n + l m ) D
hm [(n S − 1) − (n S − 2) D / R]
π m* = ( p − c) D −
2(
km
+ l m ) * ((n S − 1) − (n S − 2) D / R) * hm
nS
(6)
(7)
2.3 Description of the coordination process
Because of the conflict between the manufacturer
and retailer, we give a description of coordination
process.
Firstly, the manufacturer announced the sales price
p . According to the market demand, the retailer
announced its economic order quantity ( EOQ ) and the
expected sales price p (Generally p ≠ p );
Secondly, the manufacturer determined its own
optimal production lot size n , and then offered its
economic production quantity ( EPQ ).
If (
p, EOQ )=( p, EPQ ), there is no need to
coordinate, then the two parties make a deal directly.
Else, the two sides went into the coordination process. In
this paper, Section 3 will introduce a win-win solution
algorithm to simulate the coordination process between
the manufacturer and retailer. Also, this algorithm can
make the partners reach their respective profit and
maximize the overall interests of the whole supply chain.
3 Win-win negotiation algorithm
3.1 Coordination agreement
The enterprises involved in this negotiation take
rounds to give proposals and counter-proposals. In each
round, the supplier and demander calculate their profit
according to the other’s proposals. If the value is greater
than or equal to the expected profit in the next round, the
transaction is done. Otherwise, they make a counter
proposal. In addition, both the supplier and buyer don’t
reduce the value of their own profit when they propose in
each round. Instead, they offer a set of proposals to the
other side to choose, which concludes M elements. Also,
it is a good reflection of desire for cooperation between
the enterprises. That is to say, in the negotiation process,
the participants make a horizontal search firstly (without
reducing one’s own profit).If an agreement is not reached,
then go to the next round. One or both two parties reduce
their expected profit, and go to the horizontal search
again. Make those searches repeatedly until reach a final
agreement. Otherwise, Exit the negotiation. The
consultation way proposed in this paper guarantees the
satisfactory of the enterprises and optimizes the overall
interest of the whole supply chain.
3.2 Win-win negotiation solution
In the initialization phase, the two parts are going to
offer their proposal from their own profit without
considering the other. There is one and only one
quotation to meet their interests. If the two parties’
proposals are the same, then they make a deal. If not,
they enter the negotiation phase. The two sides make a
vertical search (concession).This paper uses the classic
time-constraints concessions strategy. Learning from the
literature [19], we build the dynamic negotiation model.
Define three negotiation strategies as follows:
(1) Anxious transaction type: The concession
amplitude is going to become smaller as the negotiation
time goes to disappear. This pattern reflects that the
negotiator is eager to make a deal at the beginning, while
in the latter he adheres to the bottom line. At this time
ϕ > 1 ; ( ϕ reflects the time-preferences of the
negotiator)
(2) Stable transaction type: The concession
amplitude is a fixed constant. This pattern reflects that
the negotiator’s relatively stable characteristic. At this
time ϕ = 1 ;
(3) Patient transaction type: The concession
amplitude is going to become larger as the negotiation
time goes to disappear. This pattern reflects that the
negotiator is dawdling in the beginning, while the
concession amplitude increase as the negotiation goes to
the end. At this time ϕ < 1 ;
In this paper, the supplier and demander both
maximize the profits as the foundation of their
decision-making, and therefore, we will use the
following formula to make the concessions.
π (k ) = π min (k ) + (1 − Φ(k ))(π * − π min (k )) (8)
Where k is the k-th round of negotiation, π (k ) is
the profit the partner want to achieve, and Φ (k ) is the
utility that the partner planned, and π * is the expected
value that the partner want to attain at the beginning, and
the
acceptable
minimum
profit
π min (k) is
*
π m (0) = π m * , π mmin (0) = π m ; π r (0) = π r , π rmin (0) = π r .
π m , π r is the minimum interest that maintains the
normal operation of enterprises for the manufacturer and
retailer separately.
Here, the negotiation time is valuable resource for
both sides, and it is limited. The utility will decrease over
time, and the negotiation rounds reflect the time
consumption.
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0, k = 0

1
Φ(k ) = 
k ϕ
k p + (1 − k p )( ) , k ∈ [1, N ]
N

hp
la
so
p
or
p
ali
ti
nI
Supplier
Demander
First proposal
First proposal
(9)
Where k p is a constant, and 0 ≤ Φ (k ) ≤ 1 , and
S
E
Y
Match
NO
sea
hp
no
it
iat
og
e
N
Vertical search
Concession
Horizontal
search
Accepted
History
YES
Deal
Update
NO
N is the maximum negotiation rounds.
Through the mentioned concession based on time,
the partners will both have the expected profit they want
to achieve in the next round. And then one partner will
search horizontally to generate a proposal set for the
other. The specific horizontal search process is described
as follows:
(1) Retrieval the case base to find historical data
which can meet the concession profit. Here, we assume
that m groups of proposals meet the requirement;
(2) According to the scope of attributes which are
determined by the opponent, generate the other
attributes’ ranges. Choose (M-m) decision variables
randomly, and calculate the other attributes’ value. Thus,
there are M-m groups of new proposals.
(3) To sum up, the proponent gives set concluding
M proposals.
Above all, the win-win negotiation solution
procedure is described in detail as follows：
Step 1: The supplier and the demander both offer
their expected transaction value. If they are the same, go
to Step 6. Otherwise, go to Step 2;
Step 2: The negotiating parties both take the vertical
search, and reduce the expected profit value in current
round;
Step 3: One party as proponent take the horizontal
search and provide the alternative proposal collection for
the other;
Step 4: The proposed recipient calculates the
highest profit that he can get from the opponent’s
proposals. If the profit is higher than or equal to the
excepted one in the current round then go to Step 6.
Otherwise, go to Step 5;
Step 5: Detect the negotiation rounds to see whether
or not the maximum tolerated time has been reached. If
reached, go to Step 7. Otherwise, update the minimum
acceptable profit value and change the role as proponent
and go to Step 3;
Step 6: Make a deal;
Step 7: The transaction fails.
Through the description of the entire negotiation
process, we draw the win-win negotiation flow chart as
Fig 1.
k>K
YES
Fail
Fig.1 The flow chart of win-win negotiation
Retailer’s parameter: k r = 80 , hr = 1.6 , B = 0.5 ,
p = 7 , p = 0.1 , π r = 2500 ; The retailer selects the
anxious transaction type of negotiation strategy ϕ = 2 ;
the maximum tolerated negotiation rounds are N r = 30 ;
each round’s proposal number M r = 5 ;
Manufacturer’s parameter: c = 4 , k m = 200 ,
l m = 50 , hm = 0.2 , R = 3000 , p = 9 , π m = 3500 ;
Manufacturer selects the stable type of negotiation
strategy ϕ = 1 ; the maximum tolerated negotiation
rounds are N m = 50 ; each round’s proposal number
M m = 5.
The model parameters can be adjusted in
accordance with the actual situation of the enterprise, this
paper just gives an example.
4 Numerical example
4.2 Results analysis
According to the formula (2) and (3), the retailer’s
expected transaction value is ( p, EOQ) = (7,320) , and
4.1 Parameter information
In order to verify the effectiveness of the win-win
negotiation algorithm, it is assumed that the market
demand expectations D = 1000 , the variance of the
normal distribution σ = 80 , market sales price w = 12 ;
π r * = 4756.88 . Meanwhile, the optimal production
batch of the manufacturer is n = 5 and the expected
transaction
value
is
( p, EPQ) = (9,548) and
*
π m = 4989.61 . Simulated the consultation process
which is proposed in this paper by using Matlab2012a. In
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the 30th round, the partners make a final agreement, and
the value is(8.36,358).
In the process of applying the negotiation algorithm,
we set an array recording the actual maximum profit that
the retailer achieves and the expected profit and the
minimum acceptable profit changes in each round. These
changes are shown in Fig2.
5000
4500
Profit
4000
3500
4000
3800
actual maximum profit
excepted profit
minimum acceptable profit
3000
3600
2500
Profit
3400
3200
3000
5
10
15
Negotiation rounds
20
25
30
Fig.3 The profit changes of manufacturer
2800
2600
actual maximum profit
expected profit
minimum acceptable profit
2400
2200
0
0
5
10
15
Negotiation Rounds
20
25
30
Fig.2 The profit changes of retailer
From Fig.2, we can see that in the 30th negotiation
round, the retailer’s expected profit is lower than the
actual maximum profit he can achieve from the
manufacturer’s proposals, and then the deal is done. For
further research, we analyzed each curve separately:
(1) From the curve of minimum acceptable profit:
In the first seven rounds, the minimum acceptable profit
doesn’t change. The reason is that the manufacturer’s
proposal in the first seven rounds didn’t exceed the initial
boundary value that the retailer persists. But, the
acceptable minimum value of the retailer is growing,
which reflects the “greedy psychology” of negotiator
who expect the profit the higher the better. In addition,
we have to notice that the minimum acceptable profit
value is always lower than the actual maximum profit.
Because updating the minimum acceptable profit is the
latest stage in negotiation process and the updated value
is used in the next negotiation round;
(2) From the curve of expected profit value, we can
see that the curve is in a downward trend. Also, in the
first seven rounds, the curve presented obvious
“concave”, which is accordance with the description of
anxious transaction type strategy. However, in the latter,
the concave trend is becoming blurred, and the updating
boundary value is the reason;
(3) From the curve of actual maximum profit, an
obvious phenomenon can be found. The actual maximum
profit value is higher in the latter round than the previous
one, which satisfy the requirement of Pareto
improvement.
Similarly to the way of retailer’s record, there are
also three curves for manufacturer to record the changes
of actual maximum profit that the manufacturer achieves,
expected profit and the minimum acceptable profit in the
interactive process. These changes are shown in Fig3.
From Fig.3, we can see that in the 30th negotiation
round, the retailer accepts the proposal that the
manufacturer made. Therefore, there are only 29 records.
Analyze the three curves separately as follows: 1) from
the curve of minimum acceptable profit: In the first six
rounds, the minimum acceptable profit doesn’t change.
The reason is the same as retailer. What’s more, the
manufacturer has a “greedy psychology” too. The curve
is exhibiting a growing trend after the six rounds; 2) the
manufacturer chooses the stable transaction type as his
concession strategy. Therefore, in the first six rounds, the
curve is showing a clear “linear downward” trend. As the
minimum tolerance boundary changes, the curve presents
intermittent decline, which fits the description of stable
transaction type strategy; 3) the actual maximum profit
curve is also growing as the negotiation going, which
meets the Pareto improved condition.
In conclusion, the retailer and the manufacturer are
on the equal status, and they both have the “greedy
psychology” in the negotiation process. In addition, the
negotiation process is the result of expected profit of
both partners decreased step by step. The concession
magnitude is determined by the negotiator’s attitude, and
the obtained profit in each round is higher than the last
round the opponent’s proposal makes him achieve.
Those are in line with the actual business situation.
4.3 Mode comparative analysis
Compared with the current mainstream research on
Stackelberg mode and centralized decision-making mode,
the profit that the buyer, the seller and the whole supply
chain achieve is shown in Tab.1.
From the Tab.1, we can see that the deal can’t be
done in Stackelberg coordination mode. Because the
retailer got profit value is less than the minimum
acceptable boundary. Generally, in this situation, the
partners will negotiate with the other. Otherwise, they
will neither get the benefit. In addition, the biggest
drawback of the Stackelberg mode is the constraint of the
status, whose range of application is narrow. While the
centralized decision-making mode is an ideal running
state: both sides will fully share the information and they
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Coordination mode
Tab.1 Profit analysis in different modes
profit
Manufacturer
Retailer
Supply chain
Stackelberg
Centralized decision-making
Collaborative planning based on negotiation
4622.75
Negotiation
4450.5
overall profit depends on the negotiating capacity.
However, the enterprises in the supply chain are rational
and selfish. Achieving the complete information and all
the parties make decision for the whole supply chain’s
profit is almost impossible in real life. So, in the
negotiation mode, the status between supplier and
demander is relatively equal and they make a deal under
asymmetric information conditions and the whole supply
chain profit is close to the centralized decision-making
mode. All of those are in accordance with the practice
and the enforce ability is higher. Furthermore, the
negotiating attitude depends on personal preference with
a certain degree of autonomy. The different attitudes the
partners choose may lead to a failed result, which
complies with the reality. Above all, this simulation
research has a very important practical significance.
5 Conclusions
In this paper, a lot sizing order model is founded
and we propose a win-win negotiation algorithm. In the
collaborative planning mode based on negotiation, the
relationship between supplier and demander is equal.
Also, this mode doesn’t need the complete information.
It breaks the restriction of status between the seller and
buyer in Stackelberg mode and is more practice than the
centralized decision-making mode. Through the
numerical example, the profit reduction process and a
“greedy psychology” are described in the negotiation,
which simulates the realistic negotiation circumstance
better.
Furthermore,
compared
with
other
decision-making mode, the strategy proposed in this
paper is much more effective and executive. But there
are still areas to improve: one-to-one negotiation is only
considered in this paper, one-to many or many-to-many
negotiation is worth of further study. In addition, many
psychological factors affect the interactive process in
negotiation, such as other negotiation attitude and the
market environment changes and so on.These aspects
need to be further excavated.
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