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Joint pricing and lot-sizing problem with variable capacity
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Available online at www.sciencedirect.com Control 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 106–111
pricing and lot-sizing problem with Joint Joint and problem Joint pricing pricingvariable and lot-sizing lot-sizing problem with with capacity variable capacity variable capacity ∗,∗∗ ∗∗ ∗∗
Paulin Couzon ∗,∗∗ Yassine Ouazene ∗∗ Farouk Yalaoui ∗∗ ∗∗ ∗∗ Paulin Couzon ∗,∗∗ ∗,∗∗ Yassine Ouazene ∗∗ Farouk Yalaoui ∗∗ Paulin Couzon Yassine Ouazene Farouk Yalaoui ∗ ∗ e de Technologie de Troyes, ∗ Chaire Connected Innovation, Universit´ Connected Innovation, Universit´ eTroyes de Technologie de Troyes, ∗ Chaire 12 rue Marie Curie, CS 42060 10004 CEDEX, France Chaire Connected Innovation, Universit´ de Technologie de Troyes, ∗∗12 rue MarieUniversit´ Curie, CS 42060 10004 eTroyes CEDEX, France ∗∗ ICD-LOSI, e de Technologie de Troyes, 12 rue Marie ∗∗12 rue Marie Curie, CS 42060 10004 Troyes CEDEX, France ICD-LOSI, Universit´ e de Technologie de Troyes, 12 rue Marie Curie, CS 42060 10004 Troyes CEDEX, France ∗∗ ICD-LOSI, e de Technologie de Troyes, 12 rue Marie Curie,Universit´ CS 42060 10004 Troyes CEDEX, France Curie, CS 42060 10004 Troyes CEDEX, France Abstract: This paper presents an extension to the classical capacitated lot-sizing problem. Abstract: presents an extension to themodelized classical capacitated lot-sizing problem. Demand forThis the paper products is price-dependent and as an isoelastic function. The Abstract: paper presents an extension to themodelized classical capacitated lot-sizing problem. Demand is forThis theconsidered products is price-dependent and as an isoelastic function. capacity not as a parameter anymore, it becomes a decision variable for The the Demand for the products is price-dependent and modelized as an isoelastic function. capacity isas not considered asinventory a parameter anymore, it becomes a variable decisioncapacity variable leads for The the company the production, and pricing decisions. This to capacity isas not considered asinventory a parameter anymore, it becomes a variable decisioncapacity variable leads for the company the production, and pricing decisions. This to two variants of the problem: one with a time-dependent capacity and the other one a uniform company as the production, inventory and pricing decisions. This variable capacity leads to two variants of the problem: one with a time-dependent capacity and the other one a uniform capacity overofthe periods. We propose heuristic methods, which arethebased on Lagrangian two variants the problem: one with a time-dependent capacity and other one a uniform capacity overto the periods. We propose heuristic The methods, which are based on Lagrangian multipliers, solve variations of the problem. methods provide high-quality results. capacity overto the periods. We propose heuristic The methods, which are based on Lagrangian multipliers, solve variations of the problem. methods provide high-quality results. c Copyright 2019 IFAC multipliers, solve variations of of the problem. The Hosting methods high-quality results. c to Copyright 2019 IFAC © 2019, IFAC (International Federation Automatic Control) by provide Elsevier Ltd. All rights reserved. c 2019 IFAC Copyright Keywords: Lot-sizing; Dynamic pricing; Variable capacity; Mathematical programming; Keywords: Lot-sizing; Dynamic pricing; Heuristic method optimization; IsoelasticVariable demandcapacity; Mathematical programming; Keywords: Lot-sizing; Dynamic pricing; Heuristic method optimization; IsoelasticVariable demandcapacity; Mathematical programming; Heuristic method optimization; Isoelastic demand 1. INTRODUCTION ones. Few articles in the literature dealt with this kind 1. INTRODUCTION ones. Few articles in the problems. literature Kim dealtand withLee this(1998) kind of capacitated lot-sizing 1. INTRODUCTION ones. Few articles in the problems. literature Kim dealtand withLee this(1998) kind of capacitated lot-sizing Lot-sizing problems consist of deciding the quantities to of dealt with a lot-sizing problem with Kim pricing decisions and capacitated lot-sizing problems. and Lee (1998) Lot-sizing problems deciding the quantities to dealt with a lot-sizing problem with pricing decisions and produce and stock toconsist satisfy of a given demand. These proba variable capacity. However, their problem considered Lot-sizing problems consist of deciding the quantities to with acapacity. lot-sizingHowever, problem with pricing decisions and produce stock to satisfy given These prob- dealt a single variable problem considered lems wereand introduced in 1913a by Forddemand. Harris, as described product without time their considerations. Deng and produce and stock to satisfy aby given demand. These prob- aaa single variable capacity. However, their problem considered lems were introduced in 1913 Ford Harris, as described product without time considerations. Deng and in thewere historical surveyinpublished by Andriolo et al. (2014). aYano (2006) worked also with single product,Deng but they lems introduced 1913 by Ford Harris, et as al. described single product without time aaconsiderations. and in the historical survey published by Andriolo (2014). Yano (2006) worked also with single product, but they A survey on Lot-sizing problems has been written by(2014). Drexl Yano had several time periods. They studied the characteristics in the historical survey published by Andriolo et al. (2006) worked also with a single product, but they A survey on Lot-sizing problemsthe hasmost been popular written by Drexl had several time periods. They studied the characteristics and Kimms (1997), detailing variants of the optimal solutions, and the impactthe that the capacity A survey on Lot-sizing problemsthe hasmost been popular written by Drexl had several time periods.and They characteristics and Kimms (1997), detailing variants of thestudied impact that the capacity and models for lot-sizing problems. Wagner and Whitin hasthe onoptimal prices. solutions, Due to the setup costs, the conclusion of and Kimms (1997), detailing the most popular variants of the optimal solutions, and the impact that the capacity and models for lot-sizing problems. Wagner and Whitin has on prices. Due totothe setupet costs, the conclusion of (1958) proposed an exact method to solve a special lotthis article is similar Bajwa al. (2016b): there is no and models for lot-sizing problems.toWagner and Whitin on prices. Due totothe setupet costs, the conclusion of (1958) proposed an exact solve a in special lot- has this article is similar Bajwa al. (2016b): there is no sizing problem. The use ofmethod pricing to decisions lot-sizing smooth relationship between theetcapacity and the price. (1958) proposed an exact method solve a special lotthis article is similar to Bajwa al. (2016b): there is no sizing use pricing decisions in lot-sizing relationship between the capacity and the price. modelsproblem. date backThe from theofwork of Thomas (1970). Unlike smooth sizing problem. use pricing decisions in lot-sizing smooth relationship the capacity andthe thedemand price. models date backThe from theofwork ofauthor Thomas (1970). Unlike For a large part ofbetween the literature, when classical lot-sizing problem, the assumed a variFor a product large part of the literature, when the function demand models date back from the work ofauthor Thomas (1970). aUnlike classical lot-sizing problem, the assumed varifor a is modelized by a closed-form large part of the literature, when the function demand able demand, depending on the price of the product. No For aa product is modelized a closed-form classical lot-sizing problem, the price author a variable demand, depending onwas the of assumed the product. No for depending on the price, the by function is a linear one. for a product is modelized by a closed-form function mathematical formulation developed, but an optimal depending on the price, the function is a linear one. able demand, depending onwas thedeveloped, price of the No This function is a convenient one, because it sometimes mathematical formulation butproduct. an optimal depending on the price, the function is a linear one. policy for the price was derived, based on production and a convenient one, because it sometimes mathematical formulation was developed, but an optimal policy for Haugen the price based on production and This allowsfunction to get a is closed-form solution, but it doesn’t always This function isclosed-form a convenient one, because it sometimes demand. etwas al. derived, (2007) considered a capacitated allows to get a solution, but it doesn’t always policy for the price was derived, based on production and demand. Haugen et al. (2007) considered a capacitated represent economic reality. Huang et al. (2013) surveyed to get a closed-form solution, it(2013) doesn’t always lot-sizing problem, and added pricing considerations. They allows represent economic Huang etbut al.the surveyed demand. Haugen et al.added (2007) considered a capacitated the different demandreality. functions used in literature for lot-sizing problem, and pricing considerations. They represent economic reality. Huang et al. (2013) surveyed compared their solutions with the solutions obtained for a the different demand functions used in the literature for lot-sizing problem, and added pricing considerations. They decision modeling, and of the price-dependent functions. compared solutions with the solutions obtained a the different demand functions used in the literature for capacitatedtheir lot-sizing problem without pricing. Gonz´afor lezmodeling, andmost of the price-dependent functions. compared their solutions with the solutions obtained a decision capacitated lot-sizing problem without pricing. Gonz´afor lezIn this survey, the two popular and accepted demand decision modeling, andmost of the price-dependent functions. Ram´ ırez et al. (2011) developed an algorithm based on a In this survey, the two popular and accepted demand capacitated lot-sizing problem without pricing. Gonz´ a lezRam´ ırez et al. (2011) developed an algorithm based on a function are the linear one and the iso-elastic one, which this survey,the thelinear two most popular and accepted demand Dantzig-Wolfe decomposition to an solve a capacitated lot-a In function oneeffects and the one, which Ram´ ırez et al. (2011) developed algorithm based on allows to are model nonlinear on iso-elastic the demand. Dantzig-Wolfe decomposition to solve a capacitated lotfunction are the linear one and the iso-elastic one, which sizing problem decomposition with multiple products and pricing deciDantzig-Wolfe to solve aand capacitated lot- allows to model nonlinear effects on the demand. sizing multiple developed products pricingsimilar deciallows model nonlinear effects thetwo demand. sions. problem Bajwa etwith al. (2016b) a model Withintolot-sizing literature, thereonare distincts time sizing problem with multiple products and pricing decisions. Bajwa et al. (2016b) developed a model similar Within lot-sizing literature, there are two distincts time to the one of Haugen et al. (2007). They compared the modelizations. The first one considers ”small buckets”, sions. Bajwa et al. (2016b) developed a model similar Within lot-sizing literature, there are two distincts time to the one ofpricing Haugenand et al. (2007). They compared the modelizations. The first one considers ”small buckets”, coordinated production approach with the only one product can be produced during a time period. to the one of Haugen et al. (2007). They compared the modelizations. The first one considers ”small buckets”, coordinated pricing and production approach with the the only one product can be produced during a time period. uncoordinated one. Ouazene et al. (2017) extended The second one considers ”big buckets”, several products coordinated pricing and production approach with the the only one product can be produced duringseveral a timeproducts period. uncoordinated one.considering Ouazene et al. (2017) extended The second one considers ”big buckets”, previous work by a multiple selling channels can be produced during a period. Besides, it is possible uncoordinated one. Ouazene et al. (2017) extended the The second one considers ”big buckets”, several products previous work by considering a multiple selling channels can be produced during a period. it is problem possible problem. to model a dynamic pricing withBesides, production previous be produced during a period. Besides, it is problem possible problem. work by considering a multiple selling channels can to model a dynamic pricing with production with a continuous time. Li et al. (2015) developed a problem. model a dynamic pricing with production problem Nowadays, some companies have the ability to modify to with a continuous time. Liand et dynamic al. (2015)pricing developed a Nowadays, some companies have the ability to modify continous-time production model. a continuous time. Liand et dynamic al. (2015)pricing developed a their production lines to adapthave them to the actual demand. continous-time production model. Nowadays, somelines companies ability todemand. modify with their production to adapt themthe to the actual They represented inventory level variation by a differential continous-time production and dynamic pricing model. They are able to increase or decrease their energy capacity, represented inventory level variation by a differential their production lines to adapt them to theenergy actual capacity, demand. They equation and derived optimal pricing policy. They arealso ableusing to increase or decrease represented inventory level variation by a differential they are external contractstheir to face a production equation and derived optimal pricing policy. They arealso ableusing to increase or decrease their energy capacity, They they are external contracts to face a production equation and derived optimal pricing policy. peak. These situations can be represented as a capacitated they are alsosituations using external contracts to face production The remainder of this paper is organised as follows. First peak. These be represented as This aa capacitated remainder of model this paper is organised follows. First lot-sizing problem, withcan a variable capacity. capacity The the mathematical and the notations as used to define it peak. These situations can be represented as a capacitated The remainder of model this paper is organised follows. First lot-sizing problem, with a variable capacity. Thisand capacity the mathematical and the notations as used to define it becomes a decision as the pricing, production stock lot-sizing a variable Thisand capacity becomes aproblem, decisionwith as the pricing,capacity. production stock the mathematical model and the notations used to define it becomes decision the pricing, production and stock 2405-8963 ©a 2019, IFAC as (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 108Control. Copyright © 2019 IFAC 108 10.1016/j.ifacol.2019.11.160 Copyright © 2019 IFAC 108
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are presented. Then the resolution method for the general model and the special case with a capacity constant over time periods are detailed. In section 4, the results obtained with our method are compared with the ones obtained with LINGO software. Finally, we present some concluding remarks and future perspectives for this research.
J j=1
The decisions variables used in the mathematical model are the following ones: • • • • • •
Pjt : Price for product j during period t; Sjt : Quantity of product j sold during period t; Xjt : Quantity of product j produced during period t; Ijt : Inventory of product j at the end of period t; Ct : Production capacity during period t; Yjt : Binary, equal to 1 if there is a setup to produce j during period t.
The first five variables are non-integer and greater or equal to zero. The last one is a binary variable. The following input parameters are used for the mathematical model: • • • • • • • • • •
J: Number of products; T : Number of time periods; cjt : Production cost for product j in period t; hjt : Inventory cost for product j in period t; ajt : Setup cost to produce j during period t; ut : Unit capacity cost in period t; vj : Production capacity used by product j; αj : Demand of product j for a price equal to 1; βj : Elasticity of product j demand; γjt : Demand seasonality of product j in period t .
J T j=1 t=1
−
(1) ut C t
t=1
−βj
≤ 0, ∀j, t
T t=1
(5)
SjT − IjT −1 − XjT = 0, ∀j
(6)
vj Xjt − Ct Yjt ≤ 0, ∀j, t
(7)
1 βj βj ) t−1 (mint0 ∈{1,..,t} (cjt0 + k=t0
αj γjt (1 −
Xjt ≤
Pjt ≥
Sj1 + Ij1 − Xj1 = 0, ∀j
hjk ))βj
1 βj βj ) t−1 k=t0 t=1 (mint0 ∈{1,..,t} (cjt0 +
T
αj γjt (1 −
(mint0 ∈{1,..,t} (cjt0 + 1−
1 βj
t−1
k=t0
, ∀j, ∀t
hjk ))βj
hjk ))
, ∀j, ∀t
(8)
, ∀j (9)
(10)
Xjt , Sjt , Pjt , Ijt ≥ 0, Yjt ∈ {0, 1}, ∀j, t
(11)
Ct ≥ 0, ∀t
(12)
(1) aims at maximizing the total profit, by maximizing the revenue minus all the different costs. Constraints (2) limit sales by the demand, and allow lost sales. Constraints (3) are the production capacity constraints. Constraints (4)(6) link the inventory level to the production and sales variables, with initial and final inventory equal to zero. (7) incur a setup cost for each production period. The constraints (8)-(10) are restricting the solution domain by cutting off nonoptimal values. These constraints are derived from mathematical analysis of the problem with a fixed setup configuration.
First of all, this model considers multiple products, but there is no dependency between the products apart from the capacity. Then, the model can be separated in several single product problems. Thus, the j indexes are ommited in the entire section. This separation is possible only because the capacity cost depend linearly of the capacity. To solve this problem, it may be useful to reformulate it. By setting setup variables to fixed values, the problem become a nonlinear model, which is easier to solve than the mixed-integer nonlinear one. The reformulation is described as follows:
subject to Sjt − γjt αj Pjt
Sjt ≤
(4)
3.1 General model
(Pjt Sjt − cjt Xjt − hjt Ijt − ajt Yjt ) T
(3)
3. OPTIMIZATION APPROACH
The model is denoted by (P1 ) and is defined by the following relations:
max z =
vj Xjt ≤ Ct , ∀t
Sjt + Ijt − Ijt−1 − Xjt = 0, ∀j, ∀t ∈ {2, ..., T − 1}
2. MATHEMATICAL FORMULATION The problem studied in this work is a lot-sizing problem faced by a company wanting to produce and sell itself its products in order to maximize its profits. The company is able to modify its production capacity and set its prices. The time here is discretized by using the ”big-bucket” periods, meaning that several products can be produced during a period. Using a continuous time here could be interesting but it is beyond the scope of the paper. There is no competitor in the market, so that the demand for a product is only price-dependent. The demand function is the iso-elastic one, but the whole work can be extended to a linear one. This demand is written as Djt (Pjt ) = −β γjt αj Pjt j , with γ the seasonality of the product, α the parameter managing the range of the demand and β the price elasticity of the demand.
107
(2) 109
Let us consider Xmn as the quantity of product produced during period m to be sold during period n.
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This variable allows us to remove variables It , Xt and St . It T is included whitin the definition of Xmn , Xt = n=t Xtn t and St = m=1 Xmt .
Besides, constraints (2) allow to have lost sales, but it is always possible to reduce the production to match the demand and obtain a better solution (if one except the case with no setup before the sale). Therefore, within constraints (2), inequalities are replaced by equalities. Finally, the following notations are used: • N={(m,n) such that Ym = 1}. n−1 • Amn = cm + t=m ht • Bn = αγn
1
(Xmn ((Bn ) β (
− such that:
n l=1
(m,n)∈N
T
n=m
T
m=1
The necessary KKT conditions are verified for a solution of the model (RP1 ). Besides, the objective function (13) is a concave function and the constraints are affine. Therefore, the KKT conditions are also sufficient conditions, and a solution verifying the KKT conditions is an optimal solution of (RP1 ). However, the resolution of (RP1 ) is even easier in the following case: speculative Remark 1: If ut = u, ∀t and there is no n−1 incentive to keep inventory (that is cm + t=m ht ≥ cn , ∀m, n).
Then with these notations, the model (noted (RP1 )) is:
max zP =
again for each m until the solution converge. At each step, the positivity of the Xmn variables is verified. Besides, the capacity variables are determined from the Xmn variables and constraints (14).
Then for each n, there is only one m such that Xmn > 0.
1
Xln )− β − Amn ))
um C m −
T
This remark means that for each n, we just need to find the m such that Xmn has the best impact on the objective value.
am Ym
m=1
vXmn ≤ Cm , ∀m
Xmn ≥ 0, ∀m, n Cm ≥ 0, ∀m
(13)
(14) (15) (16)
In order to avoid enumerating every possible setup configuration, it is interesting to develop heuristic to solve the problem in a small amount of time. In that case, two different heuristics have been tested to provide a setup configuration to the model (RP1 ). The first one is based on a local search, and is denoted LSH. The search starts with an initial setup vector with each value set to 1. Then for each value, the local search try to set it to 0 and keep it if the solution is improved.
T Within the objective function (13), the expression m=1 am YmThe second heuristic implemented is a modified Silveris constant, and therefore doesn’t affect the optimization Meal heuristic (Silver and Meal (1973)) and is denoted after as mSM. This heuristic was initially develprocess. oped to solve a single product uncapacitated lot-sizing The Lagrange function of the model is defined as: problem. This problem was introduced in Wagner and Whitin (1958) who developed an exact algorithm to solve it. The heuristic decides when a setup should be asT T signed, and how many periods the production should λm ( vXmn − Cm ) L(Xmn , Cm ; λm ) = −zP + cover by finding the period k minimizing the value n=m m=1 k ajt +
In order to solve the problem to optimality, we need to find a solution verifying the Karush-Kuhn-Tucker (KKT) ∂L = 0 and conditions. This solution have to verify ∂X mn ∂L ∂Cm = 0. We have the following derivatives: n 1 1 1 ∂L = Amn − Bnβ (1 − )( Xln )− β + λm v ∂Xmn β
(17) (18)
Then, by setting the derivatives to zero to get the KKT conditions, the previous equations become: n Amn + vum −β ) − Xmn = ( 1 Xln (19) Bnβ (1 − β1 ) l=1,l =m λm = um
Djt2 (cjt2 +vj ut2 +hjt2 (t2 −t−1)) k
.
Remark 2: The resolution of (RP1 ) can be straight away extended when there are several variable capacities (for instance energy resources, labor resources, machines...). 3.2 Special case: Capacity is constant over time
l=1
∂L = um − λm ∂Cm
t2 =t
(20)
The optimal production values can be easily found by determining each Xmn for a fixed value of m and start 110
In most companies, the modification of the production capacity takes time to occur, and it cannot always be achieved on a hourly (or daily) basis. Then, it can be useful to study the problem where a company can choose its production capacity, but it cannot then be changed during the whole time horizon. Then, in the model (P1 ), the variables Ct are replaced by the variable C, and the same goes for parameters ut which are replaced by u. This second model is denoted as (P2 ). Unlike the model (P1 ), the model here cannot be separated into independent single product problems. The objective function of (P2 ) can be written as:
2019 IFAC MIM Berlin, Germany, August 28-30, 2019
max z =
J T j=1 t=1
Paulin Couzon et al. / IFAC PapersOnLine 52-13 (2019) 106–111
1: 2:
(Pjt Sjt − cjt Xjt − hjt Ijt − ajt Yjt ) −
T
(21) uC
t=1
The last term can be simplified as uCT . In order to ease the notation, we decide to replace uT by u, the result at the end is the same since u and T are both constant. By using the same notations with a j index to denote the product, the reformulation of the model (P2 ), denoted as (RP2 ) is the following one:
max zP =
1
(Xjmn ((Bjn ) βj (
n
Xjln )
j
l=1
(j,m,n)∈N
−uC −
such that:
− β1
J T
j=1 n=m
− Ajmn ))
J T
ajm Yjm
j=1 m=1
vj Xjmn ≤ C, ∀m
(22)
(23)
Xjmn ≥ 0, ∀j, m, n C≥0
(24) (25)
Then the Lagrangian becomes:
L(Xjmn , C; λm ) = −zP +
T
λm (
J T
j=1 n=m
m=1
vj Xjmn − C) (26)
As before, we differentiate L with respect to Xjmn and C and get: n 1 1 ∂L − 1 β = Ajmn −Bjnj (1− )( Xjln ) βj +λm vj (27) ∂Xjmn βj l=1
T ∂L =u− λm ∂C m=1
(28)
Finally, by setting these derivatives to 0 to verify the KKT conditions, the following relations arise:
Xjmn = (
Ajmn + vj λm 1 βj
Bjn (1 −
1 βj )
)−βj −
n
Xjln
(29)
l=1,l =m
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:
109
Compute the optimal solution with unlimited capacity Sort by nonincreasing order the values J T j=1 n=m vj Xjmn for each m ∈ {1, ..., T }. These values are denoted as Q1 ,...,QM . U = Q1 L=0 t=2 while t < M do Compute the values of λ from equation (29) with capacity Qt T if m=1 λm > u then U = Qt−1 L = Qt break else Compute the values of λ with MRA algorithm with capacity Qt T if m=1 λm > u then U = Qt−1 L = Qt break else t=t+1 end if end if end while while U − L > do C = U +L 2 Use MRA algorithm with the capacity C. T if m=1 λm > u then L=C else U=C end if end while Update the objective value with capacity cost uC.
Algorithm 1. Pseudocode of algorithm 1
In order to find the initial interval, the optimal (for the uncapacitated problem) production quantities are set by nonincreasing order. The initial capacity is set at the minimum required to satisfy all the production. Then, this capacity is decreased so that it is able to satisfy all the production but the first one, and so on. At each capacity level, the different λm are determined, and if they exceed u, it means that the capacity interval has been found. U is the upper bound of the interval and L is its lower bound. For each fixed capacity, all the λm are determined by using the MRA algorithm developed by Bajwa et al. (2016b).
Here we developed a modified bisection method to solve the problem of finding the optimal λm , Xjmn and C values.
Since this algorithm is -optimal for a given setup configuration, it needs to be provided the best setup configuration to find the optimal solution of the global problem. We choose to provide it by a local search. The Silver-Meal heuristic cannot be applied to this problem because of the multiple products sharing the production capacity.
The method is described by algorithm 1. The idea of the method is first to determine the interval where the optimal capacity lies, and then divide the interval at each further iteration.
The local search begins with all setups equal to 1. Then for each product j and each period t, the algorithm tries to replace the setup Yjt by 0. If the solution is improved, the setup value is kept, if not the setup value returns to 1.
T
λm = u
(30)
m=1
111
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4. NUMERICAL RESULTS
the computation time is 0.01 second on average, with no significant difference between the parameters set tested.
The methods are tested over the instances proposed by Bajwa et al. (2016b) (denoted as I1) and Bajwa et al. (2016a) (denoted as I2), these instances are based on realworld data. For the second instance, setup costs have been added to adapt the data to our problem. For both instances, J is equal to 3 and T is equal to 6.
I1
The data related to each instance are reported in Table 1. The two instances are tested among 4 demand scenarios (denoted from Sc1 to Sc4), given by the Table 2, these scenarios impact the demand seasonality γjt for each product. The first scenario represents the case without any seasonality. The second scenario has a low seasonality at the beginning of the horizon, and a high seasonality at the end. The third scenario is the opposite of the second one, with a decreasing seasonality. Finally, the fourth scenario is a mix between second and third scenarios, with the seasonality depending on the product.
I1
I2
αj 500 400 600 20000 18000 800
βj 1.9 1.6 2.5 3.5 4.0 5.5
vj 1.0 1.0 1.0 1.0 1.0 1.0
cjt 1.6 1.3 1.5 3.0 3.0 1.1
hjt 0.02 0.05 0.04 0.035 0.035 0.013
ajt 8.5 4.5 7.5 10.5 4.5 3.5
Sc1
Sc2
Sc3
Sc4
1 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.3 0.1
2 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.1
3 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.1
4 0.1667 0.1667 0.1667 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.2
5 0.1667 0.1667 0.1667 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.2
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
Decreasing ut values time LSH time 300 1.41% 0.01 300 3.55% 0.02 300 0.00% 0.01 300 0.58% 0.01 300 6.90% 0.02 300 15.02% 0.01 300 0.00% 0.01 300 3.33% 0.01
mSM 1.06% 6.74% 0.00% 2.44% 1.39% 2.99% 0.00% 1.28%
time 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
Table 3. Model (P1 ): Percentage deviation and computational time (in seconds) between LINGO and methods developed. Table 4 presents the results obtained for the model (P2 ), with the tolerance in the bisection method equal to 0.01. Since the capacity is the same during the time horizon for the second model, there is only one unit capacity cost needed. The values u = 1 and u = 0.5 are considered here. First it should be noticed that LINGO software is not always better than our approach. For a high unit capacity cost, our developed algorithm is able to find better solutions. Besides, LINGO is not able to prove the optimality of its solutions, within 300 seconds, the upper bound found by LINGO is always at least twice the value of the best solution.
Table 1. Data for each instance (Bajwa et al. (2016b), Bajwa et al. (2016a)) t
I2
LINGO 171.895 173.690 171.444 171.120 127.691 137.035 121.340 125.279
6 0.1667 0.1667 0.1667 0.3 0.3 0.3 0.1 0.1 0.1 0.1 0.1 0.3
I1
I2
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
u=1 Gap time 4.82% 9 2.96% 12 6.33% 7 3.68% 8 -0.18% 7 -1.10% 8 -0.89% 7 0.14% 8
u = 0.5 Gap time 3.42% 7 7.27% 9 2.31% 6 2.84% 6 10.50% 6 4.90% 7 0.00% 8 -1.85% 7
Table 4. Model (P2 ): Percentage deviation and computational time (in seconds) between LINGO and the method developed.
Table 2. Demand scenarios (Bajwa et al. (2016b)) For both models, the results of the methods are compared with the results obtained by LINGO software. The resolution with the solver has been limited to 300 seconds, and only the global solver setting is used. The results presented in the tables are the percentage deviation, gap = zLIN GO −zmethod . zLIN GO For the model (P1 ), the instances are tested based on different demand scenarios, and with either a unit constant capacity cost ct = 0.1, ∀t, or with a decreasing unit capacity cost ct from 0.6 to 0.1 with a 0.1 step. With a fixed cost capacity, results for LSH, mSM and LINGO are the same, that’s why they are not showed here. The results for the decreasing cost capacity are presented within Table 3. The increasing unit capacity cost is not developed either because the obtained results are the same as the ones with fixed capacity cost. For both heuristic methods, 112
Finally, the parameter used for the optimality tolerance in the bisection method have a great impact on the computation time of the method. Besides, the quality of the solution depends on the setup configuration, it is then possible to have a large tolerance gap for the bisection method during the time used to find a good setup configuration. After the algorithm, it is then possible to reevaluate the best obtained solution with a smaller . With u = 1 and = 0.01, the algorithm runs in 8 seconds on average, whereas with = 0.1 and a final reevaluation with = 0.01, all the obtained solutions are the same, and the average computation time is 6 seconds. 5. CONCLUSION AND FUTURE RESEARCH The two mathematical models presented in this paper are solved by two similar algorithms. These algorithms use a decomposition of the initial model into a model with a fixed setup configuration and a procedure to find these configurations. The reformulated problems are solved to
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optimality thanks to a Lagragian relaxation and the use of KKT conditions. A heuristic method is then used to provide good setup configuration to the algorithm. For the first model, it was possible to modify the Silver-Meal heuristic to decide about the setup decisions. Unfortunately, this heuristic is not suitable to solve the second model because of the multiple products and the non-fixed demand. A local search procedure is applied instead, and provides promising results on both models. The capacity costs in this work were assumed to be linear, but it is not always relevant in industrial system. A direct extension of this work would be to consider concave capacity cost, Atamt¨ urk and Hochbaum (2001) presented some complexity results when using concave cost within a lot-sizing problem. A future extension of this work would be to consider multiple resources limitation. This expansion makes the problem with a variable and uniform capacity over the horizon much more difficult to solve than the single resource one, it would then be challenging to develop a specialized heuristic to solve it. ACKNOWLEDGEMENTS This research was supported by the Industrial Chair Connected-Innovation. REFERENCES Andriolo, A., Battini, D., Grubbstr¨ om, R.W., Persona, A., and Sgarbossa, F. (2014). A century of evolution from harris s basic lot size model: Survey and research agenda. International Journal of Production Economics, 155, 16–38. doi:10.1016/j.ijpe.2014.01.013. Atamt¨ urk, A. and Hochbaum, D.S. (2001). Capacity acquisition, subcontracting, and lot sizing. Management Science, 47(8), 1081–1100. doi: 10.1287/mnsc.47.8.1081.10232. Bajwa, N., Fontem, B., and Sox, C.R. (2016a). Optimal product pricing and lot sizing decisions for multiple products with nonlinear demands. Journal of Management Analytics, 3(1), 43–58. doi: https://doi.org/10.1080/23270012.2015.1121118. Bajwa, N., Sox, C.R., and Ishfaq, R. (2016b). Coordinating pricing and production decisions for multiple products. Omega, 64, 86–101. doi: https://doi.org/10.1016/j.omega.2015.11.006. Deng, S. and Yano, C.A. (2006). Joint production and pricing decisions with setup costs and capacity constraints. Management Science, 52(5), 741–756. doi: 10.1287/mnsc.1050.0491. Drexl, A. and Kimms, A. (1997). Lot sizing and schedulingsurvey and extensions. European Journal of operational research, 99(2), 221–235. doi:10.1016/S03772217(97)00030-1. Gonz´ alez-Ram´ırez, R.G., Smith, N.R., and Askin, R.G. (2011). A heuristic approach for a multi-product capacitated lot-sizing problem with pricing. International Journal of Production Research, 49(4), 1173–1196. doi: 10.1080/00207540903524482. Haugen, K.K., Olstad, A., and Pettersen, B.I. (2007). The profit maximizing capacitated lot-size (pclsp) problem. 113
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European Journal of Operational Research, 176(1), 165– 176. doi:https://doi.org/10.1016/j.ejor.2005.08.001. Huang, J., Leng, M., and Parlar, M. (2013). Demand functions in decision modeling: A comprehensive survey and research directions. Decision Sciences, 44(3), 557– 609. doi:https://doi.org/10.1111/deci.12021. Kim, D. and Lee, W.J. (1998). Optimal joint pricing and lot sizing with fixed and variable capacity. European Journal of Operational Research, 109(1), 212–227. doi: 10.1016/S0377-2217(97)00100-8. Li, S., Zhang, J., and Tang, W. (2015). Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products. International Journal of Production Research, 53(10), 2937–2950. doi: https://doi.org/10.1016/j.ejor.2017.10.026. Ouazene, Y., Yalaoui, F., Kelly, R., and Idjeraoui, T. (2017). Coordination and optimization of dynamic pricing and production decisions. In Computational Intelligence (SSCI), 2017 IEEE Symposium Series on, 1–6. IEEE. doi:https://doi.org/10.1109/SSCI.2017.8285252. Silver, E.A. and Meal, H.C. (1973). A heuristic for selecting lot size quantities for the case of a deterministic time-varying demand rate and discrete opportunities for replenishment. Prod. Inventory Manage., 2, 64–74. Thomas, J. (1970). Price-production decisions with deterministic demand. Management Science, 16(11), 747– 750. doi:10.1287/mnsc.16.11.747. Wagner, H.M. and Whitin, T.M. (1958). Dynamic version of the economic lot size model. Management science, 5(1), 89–96. doi:https://doi.org/10.1287/mnsc.5.1.89.
ScienceDirect
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pricing and lot-sizing problem with Joint Joint and problem Joint pricing pricingvariable and lot-sizing lot-sizing problem with with capacity variable capacity variable capacity ∗,∗∗ ∗∗ ∗∗
Paulin Couzon ∗,∗∗ Yassine Ouazene ∗∗ Farouk Yalaoui ∗∗ ∗∗ ∗∗ Paulin Couzon ∗,∗∗ ∗,∗∗ Yassine Ouazene ∗∗ Farouk Yalaoui ∗∗ Paulin Couzon Yassine Ouazene Farouk Yalaoui ∗ ∗ e de Technologie de Troyes, ∗ Chaire Connected Innovation, Universit´ Connected Innovation, Universit´ eTroyes de Technologie de Troyes, ∗ Chaire 12 rue Marie Curie, CS 42060 10004 CEDEX, France Chaire Connected Innovation, Universit´ de Technologie de Troyes, ∗∗12 rue MarieUniversit´ Curie, CS 42060 10004 eTroyes CEDEX, France ∗∗ ICD-LOSI, e de Technologie de Troyes, 12 rue Marie ∗∗12 rue Marie Curie, CS 42060 10004 Troyes CEDEX, France ICD-LOSI, Universit´ e de Technologie de Troyes, 12 rue Marie Curie, CS 42060 10004 Troyes CEDEX, France ∗∗ ICD-LOSI, e de Technologie de Troyes, 12 rue Marie Curie,Universit´ CS 42060 10004 Troyes CEDEX, France Curie, CS 42060 10004 Troyes CEDEX, France Abstract: This paper presents an extension to the classical capacitated lot-sizing problem. Abstract: presents an extension to themodelized classical capacitated lot-sizing problem. Demand forThis the paper products is price-dependent and as an isoelastic function. The Abstract: paper presents an extension to themodelized classical capacitated lot-sizing problem. Demand is forThis theconsidered products is price-dependent and as an isoelastic function. capacity not as a parameter anymore, it becomes a decision variable for The the Demand for the products is price-dependent and modelized as an isoelastic function. capacity isas not considered asinventory a parameter anymore, it becomes a variable decisioncapacity variable leads for The the company the production, and pricing decisions. This to capacity isas not considered asinventory a parameter anymore, it becomes a variable decisioncapacity variable leads for the company the production, and pricing decisions. This to two variants of the problem: one with a time-dependent capacity and the other one a uniform company as the production, inventory and pricing decisions. This variable capacity leads to two variants of the problem: one with a time-dependent capacity and the other one a uniform capacity overofthe periods. We propose heuristic methods, which arethebased on Lagrangian two variants the problem: one with a time-dependent capacity and other one a uniform capacity overto the periods. We propose heuristic The methods, which are based on Lagrangian multipliers, solve variations of the problem. methods provide high-quality results. capacity overto the periods. We propose heuristic The methods, which are based on Lagrangian multipliers, solve variations of the problem. methods provide high-quality results. c Copyright 2019 IFAC multipliers, solve variations of of the problem. The Hosting methods high-quality results. c to Copyright 2019 IFAC © 2019, IFAC (International Federation Automatic Control) by provide Elsevier Ltd. All rights reserved. c 2019 IFAC Copyright Keywords: Lot-sizing; Dynamic pricing; Variable capacity; Mathematical programming; Keywords: Lot-sizing; Dynamic pricing; Heuristic method optimization; IsoelasticVariable demandcapacity; Mathematical programming; Keywords: Lot-sizing; Dynamic pricing; Heuristic method optimization; IsoelasticVariable demandcapacity; Mathematical programming; Heuristic method optimization; Isoelastic demand 1. INTRODUCTION ones. Few articles in the literature dealt with this kind 1. INTRODUCTION ones. Few articles in the problems. literature Kim dealtand withLee this(1998) kind of capacitated lot-sizing 1. INTRODUCTION ones. Few articles in the problems. literature Kim dealtand withLee this(1998) kind of capacitated lot-sizing Lot-sizing problems consist of deciding the quantities to of dealt with a lot-sizing problem with Kim pricing decisions and capacitated lot-sizing problems. and Lee (1998) Lot-sizing problems deciding the quantities to dealt with a lot-sizing problem with pricing decisions and produce and stock toconsist satisfy of a given demand. These proba variable capacity. However, their problem considered Lot-sizing problems consist of deciding the quantities to with acapacity. lot-sizingHowever, problem with pricing decisions and produce stock to satisfy given These prob- dealt a single variable problem considered lems wereand introduced in 1913a by Forddemand. Harris, as described product without time their considerations. Deng and produce and stock to satisfy aby given demand. These prob- aaa single variable capacity. However, their problem considered lems were introduced in 1913 Ford Harris, as described product without time considerations. Deng and in thewere historical surveyinpublished by Andriolo et al. (2014). aYano (2006) worked also with single product,Deng but they lems introduced 1913 by Ford Harris, et as al. described single product without time aaconsiderations. and in the historical survey published by Andriolo (2014). Yano (2006) worked also with single product, but they A survey on Lot-sizing problems has been written by(2014). Drexl Yano had several time periods. They studied the characteristics in the historical survey published by Andriolo et al. (2006) worked also with a single product, but they A survey on Lot-sizing problemsthe hasmost been popular written by Drexl had several time periods. They studied the characteristics and Kimms (1997), detailing variants of the optimal solutions, and the impactthe that the capacity A survey on Lot-sizing problemsthe hasmost been popular written by Drexl had several time periods.and They characteristics and Kimms (1997), detailing variants of thestudied impact that the capacity and models for lot-sizing problems. Wagner and Whitin hasthe onoptimal prices. solutions, Due to the setup costs, the conclusion of and Kimms (1997), detailing the most popular variants of the optimal solutions, and the impact that the capacity and models for lot-sizing problems. Wagner and Whitin has on prices. Due totothe setupet costs, the conclusion of (1958) proposed an exact method to solve a special lotthis article is similar Bajwa al. (2016b): there is no and models for lot-sizing problems.toWagner and Whitin on prices. Due totothe setupet costs, the conclusion of (1958) proposed an exact solve a in special lot- has this article is similar Bajwa al. (2016b): there is no sizing problem. The use ofmethod pricing to decisions lot-sizing smooth relationship between theetcapacity and the price. (1958) proposed an exact method solve a special lotthis article is similar to Bajwa al. (2016b): there is no sizing use pricing decisions in lot-sizing relationship between the capacity and the price. modelsproblem. date backThe from theofwork of Thomas (1970). Unlike smooth sizing problem. use pricing decisions in lot-sizing smooth relationship the capacity andthe thedemand price. models date backThe from theofwork ofauthor Thomas (1970). Unlike For a large part ofbetween the literature, when classical lot-sizing problem, the assumed a variFor a product large part of the literature, when the function demand models date back from the work ofauthor Thomas (1970). aUnlike classical lot-sizing problem, the assumed varifor a is modelized by a closed-form large part of the literature, when the function demand able demand, depending on the price of the product. No For aa product is modelized a closed-form classical lot-sizing problem, the price author a variable demand, depending onwas the of assumed the product. No for depending on the price, the by function is a linear one. for a product is modelized by a closed-form function mathematical formulation developed, but an optimal depending on the price, the function is a linear one. able demand, depending onwas thedeveloped, price of the No This function is a convenient one, because it sometimes mathematical formulation butproduct. an optimal depending on the price, the function is a linear one. policy for the price was derived, based on production and a convenient one, because it sometimes mathematical formulation was developed, but an optimal policy for Haugen the price based on production and This allowsfunction to get a is closed-form solution, but it doesn’t always This function isclosed-form a convenient one, because it sometimes demand. etwas al. derived, (2007) considered a capacitated allows to get a solution, but it doesn’t always policy for the price was derived, based on production and demand. Haugen et al. (2007) considered a capacitated represent economic reality. Huang et al. (2013) surveyed to get a closed-form solution, it(2013) doesn’t always lot-sizing problem, and added pricing considerations. They allows represent economic Huang etbut al.the surveyed demand. Haugen et al.added (2007) considered a capacitated the different demandreality. functions used in literature for lot-sizing problem, and pricing considerations. They represent economic reality. Huang et al. (2013) surveyed compared their solutions with the solutions obtained for a the different demand functions used in the literature for lot-sizing problem, and added pricing considerations. They decision modeling, and of the price-dependent functions. compared solutions with the solutions obtained a the different demand functions used in the literature for capacitatedtheir lot-sizing problem without pricing. Gonz´afor lezmodeling, andmost of the price-dependent functions. compared their solutions with the solutions obtained a decision capacitated lot-sizing problem without pricing. Gonz´afor lezIn this survey, the two popular and accepted demand decision modeling, andmost of the price-dependent functions. Ram´ ırez et al. (2011) developed an algorithm based on a In this survey, the two popular and accepted demand capacitated lot-sizing problem without pricing. Gonz´ a lezRam´ ırez et al. (2011) developed an algorithm based on a function are the linear one and the iso-elastic one, which this survey,the thelinear two most popular and accepted demand Dantzig-Wolfe decomposition to an solve a capacitated lot-a In function oneeffects and the one, which Ram´ ırez et al. (2011) developed algorithm based on allows to are model nonlinear on iso-elastic the demand. Dantzig-Wolfe decomposition to solve a capacitated lotfunction are the linear one and the iso-elastic one, which sizing problem decomposition with multiple products and pricing deciDantzig-Wolfe to solve aand capacitated lot- allows to model nonlinear effects on the demand. sizing multiple developed products pricingsimilar deciallows model nonlinear effects thetwo demand. sions. problem Bajwa etwith al. (2016b) a model Withintolot-sizing literature, thereonare distincts time sizing problem with multiple products and pricing decisions. Bajwa et al. (2016b) developed a model similar Within lot-sizing literature, there are two distincts time to the one of Haugen et al. (2007). They compared the modelizations. The first one considers ”small buckets”, sions. Bajwa et al. (2016b) developed a model similar Within lot-sizing literature, there are two distincts time to the one ofpricing Haugenand et al. (2007). They compared the modelizations. The first one considers ”small buckets”, coordinated production approach with the only one product can be produced during a time period. to the one of Haugen et al. (2007). They compared the modelizations. The first one considers ”small buckets”, coordinated pricing and production approach with the the only one product can be produced during a time period. uncoordinated one. Ouazene et al. (2017) extended The second one considers ”big buckets”, several products coordinated pricing and production approach with the the only one product can be produced duringseveral a timeproducts period. uncoordinated one.considering Ouazene et al. (2017) extended The second one considers ”big buckets”, previous work by a multiple selling channels can be produced during a period. Besides, it is possible uncoordinated one. Ouazene et al. (2017) extended the The second one considers ”big buckets”, several products previous work by considering a multiple selling channels can be produced during a period. it is problem possible problem. to model a dynamic pricing withBesides, production previous be produced during a period. Besides, it is problem possible problem. work by considering a multiple selling channels can to model a dynamic pricing with production with a continuous time. Li et al. (2015) developed a problem. model a dynamic pricing with production problem Nowadays, some companies have the ability to modify to with a continuous time. Liand et dynamic al. (2015)pricing developed a Nowadays, some companies have the ability to modify continous-time production model. a continuous time. Liand et dynamic al. (2015)pricing developed a their production lines to adapthave them to the actual demand. continous-time production model. Nowadays, somelines companies ability todemand. modify with their production to adapt themthe to the actual They represented inventory level variation by a differential continous-time production and dynamic pricing model. They are able to increase or decrease their energy capacity, represented inventory level variation by a differential their production lines to adapt them to theenergy actual capacity, demand. They equation and derived optimal pricing policy. They arealso ableusing to increase or decrease represented inventory level variation by a differential they are external contractstheir to face a production equation and derived optimal pricing policy. They arealso ableusing to increase or decrease their energy capacity, They they are external contracts to face a production equation and derived optimal pricing policy. peak. These situations can be represented as a capacitated they are alsosituations using external contracts to face production The remainder of this paper is organised as follows. First peak. These be represented as This aa capacitated remainder of model this paper is organised follows. First lot-sizing problem, withcan a variable capacity. capacity The the mathematical and the notations as used to define it peak. These situations can be represented as a capacitated The remainder of model this paper is organised follows. First lot-sizing problem, with a variable capacity. Thisand capacity the mathematical and the notations as used to define it becomes a decision as the pricing, production stock lot-sizing a variable Thisand capacity becomes aproblem, decisionwith as the pricing,capacity. production stock the mathematical model and the notations used to define it becomes decision the pricing, production and stock 2405-8963 ©a 2019, IFAC as (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 108Control. Copyright © 2019 IFAC 108 10.1016/j.ifacol.2019.11.160 Copyright © 2019 IFAC 108
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are presented. Then the resolution method for the general model and the special case with a capacity constant over time periods are detailed. In section 4, the results obtained with our method are compared with the ones obtained with LINGO software. Finally, we present some concluding remarks and future perspectives for this research.
J j=1
The decisions variables used in the mathematical model are the following ones: • • • • • •
Pjt : Price for product j during period t; Sjt : Quantity of product j sold during period t; Xjt : Quantity of product j produced during period t; Ijt : Inventory of product j at the end of period t; Ct : Production capacity during period t; Yjt : Binary, equal to 1 if there is a setup to produce j during period t.
The first five variables are non-integer and greater or equal to zero. The last one is a binary variable. The following input parameters are used for the mathematical model: • • • • • • • • • •
J: Number of products; T : Number of time periods; cjt : Production cost for product j in period t; hjt : Inventory cost for product j in period t; ajt : Setup cost to produce j during period t; ut : Unit capacity cost in period t; vj : Production capacity used by product j; αj : Demand of product j for a price equal to 1; βj : Elasticity of product j demand; γjt : Demand seasonality of product j in period t .
J T j=1 t=1
−
(1) ut C t
t=1
−βj
≤ 0, ∀j, t
T t=1
(5)
SjT − IjT −1 − XjT = 0, ∀j
(6)
vj Xjt − Ct Yjt ≤ 0, ∀j, t
(7)
1 βj βj ) t−1 (mint0 ∈{1,..,t} (cjt0 + k=t0
αj γjt (1 −
Xjt ≤
Pjt ≥
Sj1 + Ij1 − Xj1 = 0, ∀j
hjk ))βj
1 βj βj ) t−1 k=t0 t=1 (mint0 ∈{1,..,t} (cjt0 +
T
αj γjt (1 −
(mint0 ∈{1,..,t} (cjt0 + 1−
1 βj
t−1
k=t0
, ∀j, ∀t
hjk ))βj
hjk ))
, ∀j, ∀t
(8)
, ∀j (9)
(10)
Xjt , Sjt , Pjt , Ijt ≥ 0, Yjt ∈ {0, 1}, ∀j, t
(11)
Ct ≥ 0, ∀t
(12)
(1) aims at maximizing the total profit, by maximizing the revenue minus all the different costs. Constraints (2) limit sales by the demand, and allow lost sales. Constraints (3) are the production capacity constraints. Constraints (4)(6) link the inventory level to the production and sales variables, with initial and final inventory equal to zero. (7) incur a setup cost for each production period. The constraints (8)-(10) are restricting the solution domain by cutting off nonoptimal values. These constraints are derived from mathematical analysis of the problem with a fixed setup configuration.
First of all, this model considers multiple products, but there is no dependency between the products apart from the capacity. Then, the model can be separated in several single product problems. Thus, the j indexes are ommited in the entire section. This separation is possible only because the capacity cost depend linearly of the capacity. To solve this problem, it may be useful to reformulate it. By setting setup variables to fixed values, the problem become a nonlinear model, which is easier to solve than the mixed-integer nonlinear one. The reformulation is described as follows:
subject to Sjt − γjt αj Pjt
Sjt ≤
(4)
3.1 General model
(Pjt Sjt − cjt Xjt − hjt Ijt − ajt Yjt ) T
(3)
3. OPTIMIZATION APPROACH
The model is denoted by (P1 ) and is defined by the following relations:
max z =
vj Xjt ≤ Ct , ∀t
Sjt + Ijt − Ijt−1 − Xjt = 0, ∀j, ∀t ∈ {2, ..., T − 1}
2. MATHEMATICAL FORMULATION The problem studied in this work is a lot-sizing problem faced by a company wanting to produce and sell itself its products in order to maximize its profits. The company is able to modify its production capacity and set its prices. The time here is discretized by using the ”big-bucket” periods, meaning that several products can be produced during a period. Using a continuous time here could be interesting but it is beyond the scope of the paper. There is no competitor in the market, so that the demand for a product is only price-dependent. The demand function is the iso-elastic one, but the whole work can be extended to a linear one. This demand is written as Djt (Pjt ) = −β γjt αj Pjt j , with γ the seasonality of the product, α the parameter managing the range of the demand and β the price elasticity of the demand.
107
(2) 109
Let us consider Xmn as the quantity of product produced during period m to be sold during period n.
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This variable allows us to remove variables It , Xt and St . It T is included whitin the definition of Xmn , Xt = n=t Xtn t and St = m=1 Xmt .
Besides, constraints (2) allow to have lost sales, but it is always possible to reduce the production to match the demand and obtain a better solution (if one except the case with no setup before the sale). Therefore, within constraints (2), inequalities are replaced by equalities. Finally, the following notations are used: • N={(m,n) such that Ym = 1}. n−1 • Amn = cm + t=m ht • Bn = αγn
1
(Xmn ((Bn ) β (
− such that:
n l=1
(m,n)∈N
T
n=m
T
m=1
The necessary KKT conditions are verified for a solution of the model (RP1 ). Besides, the objective function (13) is a concave function and the constraints are affine. Therefore, the KKT conditions are also sufficient conditions, and a solution verifying the KKT conditions is an optimal solution of (RP1 ). However, the resolution of (RP1 ) is even easier in the following case: speculative Remark 1: If ut = u, ∀t and there is no n−1 incentive to keep inventory (that is cm + t=m ht ≥ cn , ∀m, n).
Then with these notations, the model (noted (RP1 )) is:
max zP =
again for each m until the solution converge. At each step, the positivity of the Xmn variables is verified. Besides, the capacity variables are determined from the Xmn variables and constraints (14).
Then for each n, there is only one m such that Xmn > 0.
1
Xln )− β − Amn ))
um C m −
T
This remark means that for each n, we just need to find the m such that Xmn has the best impact on the objective value.
am Ym
m=1
vXmn ≤ Cm , ∀m
Xmn ≥ 0, ∀m, n Cm ≥ 0, ∀m
(13)
(14) (15) (16)
In order to avoid enumerating every possible setup configuration, it is interesting to develop heuristic to solve the problem in a small amount of time. In that case, two different heuristics have been tested to provide a setup configuration to the model (RP1 ). The first one is based on a local search, and is denoted LSH. The search starts with an initial setup vector with each value set to 1. Then for each value, the local search try to set it to 0 and keep it if the solution is improved.
T Within the objective function (13), the expression m=1 am YmThe second heuristic implemented is a modified Silveris constant, and therefore doesn’t affect the optimization Meal heuristic (Silver and Meal (1973)) and is denoted after as mSM. This heuristic was initially develprocess. oped to solve a single product uncapacitated lot-sizing The Lagrange function of the model is defined as: problem. This problem was introduced in Wagner and Whitin (1958) who developed an exact algorithm to solve it. The heuristic decides when a setup should be asT T signed, and how many periods the production should λm ( vXmn − Cm ) L(Xmn , Cm ; λm ) = −zP + cover by finding the period k minimizing the value n=m m=1 k ajt +
In order to solve the problem to optimality, we need to find a solution verifying the Karush-Kuhn-Tucker (KKT) ∂L = 0 and conditions. This solution have to verify ∂X mn ∂L ∂Cm = 0. We have the following derivatives: n 1 1 1 ∂L = Amn − Bnβ (1 − )( Xln )− β + λm v ∂Xmn β
(17) (18)
Then, by setting the derivatives to zero to get the KKT conditions, the previous equations become: n Amn + vum −β ) − Xmn = ( 1 Xln (19) Bnβ (1 − β1 ) l=1,l =m λm = um
Djt2 (cjt2 +vj ut2 +hjt2 (t2 −t−1)) k
.
Remark 2: The resolution of (RP1 ) can be straight away extended when there are several variable capacities (for instance energy resources, labor resources, machines...). 3.2 Special case: Capacity is constant over time
l=1
∂L = um − λm ∂Cm
t2 =t
(20)
The optimal production values can be easily found by determining each Xmn for a fixed value of m and start 110
In most companies, the modification of the production capacity takes time to occur, and it cannot always be achieved on a hourly (or daily) basis. Then, it can be useful to study the problem where a company can choose its production capacity, but it cannot then be changed during the whole time horizon. Then, in the model (P1 ), the variables Ct are replaced by the variable C, and the same goes for parameters ut which are replaced by u. This second model is denoted as (P2 ). Unlike the model (P1 ), the model here cannot be separated into independent single product problems. The objective function of (P2 ) can be written as:
2019 IFAC MIM Berlin, Germany, August 28-30, 2019
max z =
J T j=1 t=1
Paulin Couzon et al. / IFAC PapersOnLine 52-13 (2019) 106–111
1: 2:
(Pjt Sjt − cjt Xjt − hjt Ijt − ajt Yjt ) −
T
(21) uC
t=1
The last term can be simplified as uCT . In order to ease the notation, we decide to replace uT by u, the result at the end is the same since u and T are both constant. By using the same notations with a j index to denote the product, the reformulation of the model (P2 ), denoted as (RP2 ) is the following one:
max zP =
1
(Xjmn ((Bjn ) βj (
n
Xjln )
j
l=1
(j,m,n)∈N
−uC −
such that:
− β1
J T
j=1 n=m
− Ajmn ))
J T
ajm Yjm
j=1 m=1
vj Xjmn ≤ C, ∀m
(22)
(23)
Xjmn ≥ 0, ∀j, m, n C≥0
(24) (25)
Then the Lagrangian becomes:
L(Xjmn , C; λm ) = −zP +
T
λm (
J T
j=1 n=m
m=1
vj Xjmn − C) (26)
As before, we differentiate L with respect to Xjmn and C and get: n 1 1 ∂L − 1 β = Ajmn −Bjnj (1− )( Xjln ) βj +λm vj (27) ∂Xjmn βj l=1
T ∂L =u− λm ∂C m=1
(28)
Finally, by setting these derivatives to 0 to verify the KKT conditions, the following relations arise:
Xjmn = (
Ajmn + vj λm 1 βj
Bjn (1 −
1 βj )
)−βj −
n
Xjln
(29)
l=1,l =m
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:
109
Compute the optimal solution with unlimited capacity Sort by nonincreasing order the values J T j=1 n=m vj Xjmn for each m ∈ {1, ..., T }. These values are denoted as Q1 ,...,QM . U = Q1 L=0 t=2 while t < M do Compute the values of λ from equation (29) with capacity Qt T if m=1 λm > u then U = Qt−1 L = Qt break else Compute the values of λ with MRA algorithm with capacity Qt T if m=1 λm > u then U = Qt−1 L = Qt break else t=t+1 end if end if end while while U − L > do C = U +L 2 Use MRA algorithm with the capacity C. T if m=1 λm > u then L=C else U=C end if end while Update the objective value with capacity cost uC.
Algorithm 1. Pseudocode of algorithm 1
In order to find the initial interval, the optimal (for the uncapacitated problem) production quantities are set by nonincreasing order. The initial capacity is set at the minimum required to satisfy all the production. Then, this capacity is decreased so that it is able to satisfy all the production but the first one, and so on. At each capacity level, the different λm are determined, and if they exceed u, it means that the capacity interval has been found. U is the upper bound of the interval and L is its lower bound. For each fixed capacity, all the λm are determined by using the MRA algorithm developed by Bajwa et al. (2016b).
Here we developed a modified bisection method to solve the problem of finding the optimal λm , Xjmn and C values.
Since this algorithm is -optimal for a given setup configuration, it needs to be provided the best setup configuration to find the optimal solution of the global problem. We choose to provide it by a local search. The Silver-Meal heuristic cannot be applied to this problem because of the multiple products sharing the production capacity.
The method is described by algorithm 1. The idea of the method is first to determine the interval where the optimal capacity lies, and then divide the interval at each further iteration.
The local search begins with all setups equal to 1. Then for each product j and each period t, the algorithm tries to replace the setup Yjt by 0. If the solution is improved, the setup value is kept, if not the setup value returns to 1.
T
λm = u
(30)
m=1
111
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Paulin Couzon et al. / IFAC PapersOnLine 52-13 (2019) 106–111
4. NUMERICAL RESULTS
the computation time is 0.01 second on average, with no significant difference between the parameters set tested.
The methods are tested over the instances proposed by Bajwa et al. (2016b) (denoted as I1) and Bajwa et al. (2016a) (denoted as I2), these instances are based on realworld data. For the second instance, setup costs have been added to adapt the data to our problem. For both instances, J is equal to 3 and T is equal to 6.
I1
The data related to each instance are reported in Table 1. The two instances are tested among 4 demand scenarios (denoted from Sc1 to Sc4), given by the Table 2, these scenarios impact the demand seasonality γjt for each product. The first scenario represents the case without any seasonality. The second scenario has a low seasonality at the beginning of the horizon, and a high seasonality at the end. The third scenario is the opposite of the second one, with a decreasing seasonality. Finally, the fourth scenario is a mix between second and third scenarios, with the seasonality depending on the product.
I1
I2
αj 500 400 600 20000 18000 800
βj 1.9 1.6 2.5 3.5 4.0 5.5
vj 1.0 1.0 1.0 1.0 1.0 1.0
cjt 1.6 1.3 1.5 3.0 3.0 1.1
hjt 0.02 0.05 0.04 0.035 0.035 0.013
ajt 8.5 4.5 7.5 10.5 4.5 3.5
Sc1
Sc2
Sc3
Sc4
1 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.3 0.1
2 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.1
3 0.1667 0.1667 0.1667 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.1
4 0.1667 0.1667 0.1667 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.2
5 0.1667 0.1667 0.1667 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.2
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
Decreasing ut values time LSH time 300 1.41% 0.01 300 3.55% 0.02 300 0.00% 0.01 300 0.58% 0.01 300 6.90% 0.02 300 15.02% 0.01 300 0.00% 0.01 300 3.33% 0.01
mSM 1.06% 6.74% 0.00% 2.44% 1.39% 2.99% 0.00% 1.28%
time 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
Table 3. Model (P1 ): Percentage deviation and computational time (in seconds) between LINGO and methods developed. Table 4 presents the results obtained for the model (P2 ), with the tolerance in the bisection method equal to 0.01. Since the capacity is the same during the time horizon for the second model, there is only one unit capacity cost needed. The values u = 1 and u = 0.5 are considered here. First it should be noticed that LINGO software is not always better than our approach. For a high unit capacity cost, our developed algorithm is able to find better solutions. Besides, LINGO is not able to prove the optimality of its solutions, within 300 seconds, the upper bound found by LINGO is always at least twice the value of the best solution.
Table 1. Data for each instance (Bajwa et al. (2016b), Bajwa et al. (2016a)) t
I2
LINGO 171.895 173.690 171.444 171.120 127.691 137.035 121.340 125.279
6 0.1667 0.1667 0.1667 0.3 0.3 0.3 0.1 0.1 0.1 0.1 0.1 0.3
I1
I2
Sc1 Sc2 Sc3 Sc4 Sc1 Sc2 Sc3 Sc4
u=1 Gap time 4.82% 9 2.96% 12 6.33% 7 3.68% 8 -0.18% 7 -1.10% 8 -0.89% 7 0.14% 8
u = 0.5 Gap time 3.42% 7 7.27% 9 2.31% 6 2.84% 6 10.50% 6 4.90% 7 0.00% 8 -1.85% 7
Table 4. Model (P2 ): Percentage deviation and computational time (in seconds) between LINGO and the method developed.
Table 2. Demand scenarios (Bajwa et al. (2016b)) For both models, the results of the methods are compared with the results obtained by LINGO software. The resolution with the solver has been limited to 300 seconds, and only the global solver setting is used. The results presented in the tables are the percentage deviation, gap = zLIN GO −zmethod . zLIN GO For the model (P1 ), the instances are tested based on different demand scenarios, and with either a unit constant capacity cost ct = 0.1, ∀t, or with a decreasing unit capacity cost ct from 0.6 to 0.1 with a 0.1 step. With a fixed cost capacity, results for LSH, mSM and LINGO are the same, that’s why they are not showed here. The results for the decreasing cost capacity are presented within Table 3. The increasing unit capacity cost is not developed either because the obtained results are the same as the ones with fixed capacity cost. For both heuristic methods, 112
Finally, the parameter used for the optimality tolerance in the bisection method have a great impact on the computation time of the method. Besides, the quality of the solution depends on the setup configuration, it is then possible to have a large tolerance gap for the bisection method during the time used to find a good setup configuration. After the algorithm, it is then possible to reevaluate the best obtained solution with a smaller . With u = 1 and = 0.01, the algorithm runs in 8 seconds on average, whereas with = 0.1 and a final reevaluation with = 0.01, all the obtained solutions are the same, and the average computation time is 6 seconds. 5. CONCLUSION AND FUTURE RESEARCH The two mathematical models presented in this paper are solved by two similar algorithms. These algorithms use a decomposition of the initial model into a model with a fixed setup configuration and a procedure to find these configurations. The reformulated problems are solved to
2019 IFAC MIM Berlin, Germany, August 28-30, 2019
Paulin Couzon et al. / IFAC PapersOnLine 52-13 (2019) 106–111
optimality thanks to a Lagragian relaxation and the use of KKT conditions. A heuristic method is then used to provide good setup configuration to the algorithm. For the first model, it was possible to modify the Silver-Meal heuristic to decide about the setup decisions. Unfortunately, this heuristic is not suitable to solve the second model because of the multiple products and the non-fixed demand. A local search procedure is applied instead, and provides promising results on both models. The capacity costs in this work were assumed to be linear, but it is not always relevant in industrial system. A direct extension of this work would be to consider concave capacity cost, Atamt¨ urk and Hochbaum (2001) presented some complexity results when using concave cost within a lot-sizing problem. A future extension of this work would be to consider multiple resources limitation. This expansion makes the problem with a variable and uniform capacity over the horizon much more difficult to solve than the single resource one, it would then be challenging to develop a specialized heuristic to solve it. ACKNOWLEDGEMENTS This research was supported by the Industrial Chair Connected-Innovation. REFERENCES Andriolo, A., Battini, D., Grubbstr¨ om, R.W., Persona, A., and Sgarbossa, F. (2014). A century of evolution from harris s basic lot size model: Survey and research agenda. International Journal of Production Economics, 155, 16–38. doi:10.1016/j.ijpe.2014.01.013. Atamt¨ urk, A. and Hochbaum, D.S. (2001). Capacity acquisition, subcontracting, and lot sizing. Management Science, 47(8), 1081–1100. doi: 10.1287/mnsc.47.8.1081.10232. Bajwa, N., Fontem, B., and Sox, C.R. (2016a). Optimal product pricing and lot sizing decisions for multiple products with nonlinear demands. Journal of Management Analytics, 3(1), 43–58. doi: https://doi.org/10.1080/23270012.2015.1121118. Bajwa, N., Sox, C.R., and Ishfaq, R. (2016b). Coordinating pricing and production decisions for multiple products. Omega, 64, 86–101. doi: https://doi.org/10.1016/j.omega.2015.11.006. Deng, S. and Yano, C.A. (2006). Joint production and pricing decisions with setup costs and capacity constraints. Management Science, 52(5), 741–756. doi: 10.1287/mnsc.1050.0491. Drexl, A. and Kimms, A. (1997). Lot sizing and schedulingsurvey and extensions. European Journal of operational research, 99(2), 221–235. doi:10.1016/S03772217(97)00030-1. Gonz´ alez-Ram´ırez, R.G., Smith, N.R., and Askin, R.G. (2011). A heuristic approach for a multi-product capacitated lot-sizing problem with pricing. International Journal of Production Research, 49(4), 1173–1196. doi: 10.1080/00207540903524482. Haugen, K.K., Olstad, A., and Pettersen, B.I. (2007). The profit maximizing capacitated lot-size (pclsp) problem. 113
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