Considering lost sale in inventory routing problems for perishable goods

Accepted Manuscript Considering Lost Sale in Inventory Routing Problems for Perishable Goods Samira Mirzaei, Abbas Seifi PII: DOI: Reference:

S0360-8352(15)00223-5 http://dx.doi.org/10.1016/j.cie.2015.05.010 CAIE 4043

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

15 December 2012 22 April 2015 6 May 2015

Please cite this article as: Mirzaei, S., Seifi, A., Considering Lost Sale in Inventory Routing Problems for Perishable Goods, Computers & Industrial Engineering (2015), doi: http://dx.doi.org/10.1016/j.cie.2015.05.010

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Samira Mirzaeia, Abbas Seifib,1, a

Cluster for Operations Research and Logistics, Department of Economics and Business, Aarhus University, Aarhus, Denmark

b

Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran

Abstract This paper presents a mathematical model for an inventory routing problem (IRP). The model is especially designed for allocating the stock of perishable goods. It is assumed that the age of the perishable inventory has a negative impact on the demand of end customers and a percentage of the demand is considered as lost sale. The proposed model balances the transportation cost, the cost of inventory holding and lost sale. In addition to the usual inventory routing constraints, we consider the cost of lost sale as a linear or an exponential function of the inventory age. The proposed model is solved to optimality for small instances and is used to obtain lower bounds for larger instances. We have also devised an efficient

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Corresponding author. E-mail address: [email protected] (A. Seifi) Postal address: Department of Industrial Engineering, Amirkabir University of Technology (Tehran Polytechnic), P.O. Box 15875-4413, Tehran, Iran Tell:+9864545377 Fax:+9866954569

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meta-heuristic algorithm to find good solutions for this class of problems based on Simulated Annealing (SA) and Tabu Search (TS). Computational results indicate that, for small problems, the average optimality gaps are less than 10.9% and 13.4% using linear and exponential lost sale functions, respectively. Furthermore, we show that the optimality gaps found by CPLEX grow exponentially with the problem size while those obtained by the proposed meta-heuristic algorithm increase linearly. Keywords: Inventory routing problem, perishable goods, lost sale, Simulated Annealing, Tabu Search.

1. Introduction Inventory routing problem (IRP) is a major concern in supply chain management. The aim is to integrate the transportation activities and inventory management along the supply chain and avoid the inefficiency caused by solving the underlying vehicle routing and inventory subproblems separately. Recent studies assume that an indefinite number of stock items can be stored to meet future customers' demand. However, the impact of perishability cannot be ignored for certain types of goods that deteriorate over time and may become partially or entirely unsuitable for consumption (Shen et al. 2011). Deteriorating items refer to items that get damaged, spoiled, dried, invalid, or degraded over time (Li et al. 2010) and can be classified into two groups: perishable products and decaying products. Items such as meat, green vegetables, human blood, medicine, flowers, and films that have a maximum usable lifetime are known as perishable products. Commodities like alcohol and gasoline that have no shelf-life are known as decaying products (Goyal and Giri 2001). Although the life of the perishable products can be prolonged by advanced cooling equipment, these products lose their quality over time and this will cause a decrease in the demand for these products. If perishable products are not delivered to retailers on a daily basis, due to the limited lifetime and possible degradation of these products, some customers will refrain from purchasing them. Thus, the demand for the products is affected by the age of the perishable inventory. In IRP literature, some models consider shortage, stock out, or backorder costs (see for instance: Herer and Levy (1997), Jaillet et al. (2002), Abdelmaguid et al. (2006), and 2

Abdelmaguid et al. (2009)). Modeling IRP with deteriorating products and the associated cost of lost sale has not been explicitly considered in the literature. In this paper, we specifically calculate the cost of lost sale in the supply chain for perishable goods to avoid overstocking such items in an attempt to reduce transportation cost. This paper formulates a multi-period inventory routing problem for perishable goods in which the end customers' demand depends on the age of the inventory. The proposed model includes vehicle routing decisions as well as delivery and inventory decisions over a specific planning horizon. It is a non-linear mixed integer programming model which is linearized to be solvable efficiently. The objective function is to minimize the total cost of transportation, lost sale, and holding inventories. In addition to the usual inventory routing constraints, we consider a nonlinear constraint that defines the inventory age as a function of delivery date. The lost sale is assumed to be a linear or an exponential function of the inventory age. The mathematical model with both linear and exponential lost sale functions is solved up to optimality for small to medium size problems. It is also used to find some lower bounds for larger instances. Since such problems are NP-hard due to the underlying vehicle routing problem (VRP), we develop a heuristic algorithm within a metaheuristic framework for solving larger problem instances. The rest of this paper is organized as follows. Section 2 reviews the literature on inventory management and vehicle routing problems for perishable goods. In Section 3, the problem is formulated as a mathematical model using two different lost sale functions. The metaheuristic algorithm is described in Section 4, followed by the results of computational experiments in Section 5. Conclusions and some directions for future research are described in Section 6.

2. Literature Review In this section, we review the literature related to inventory management and vehicle routing problems with regards to perishable goods. Due to the importance of assumption on retailers' orders in modeling the inventory of perishable goods, researchers have studied a variety of constant demand, time-varying demand, stock dependent and price dependent demand (see Raafat (1991), Shah and Shah (2000), Goyal and Giri (2001), and Li et al. (2010)). Many researchers have developed inventory models for deteriorating items with time varying end customer demand using linearly or exponentially decreasing demand (see, for instance: Patel and Dave (1981), Sachan (1984), Chung and Ting (1993), Xu and Wang (1990), Giri and

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Chaudhari (1997), Hariga and Benkherouf (1994), and Wee (1995)). This paper also assumes that the demand of end customers is a linearly or an exponentially decreasing function of the age of perishable goods. Tarantilis and Kiranoudis (2001) proposed a threshold-accepting algorithm to solve a fresh milk distribution problem for a dairy company in Greece. They focused on a routing problem using heterogeneous vehicles. Tarantilis and Kiranoudis (2002) solved the distribution of fresh meat products in Greece using an open multi-depot VRP. They did not consider any additional constraints reflecting the perishable nature of those commodities. Hsu et al. (2007) developed a stochastic VRP model to obtain optimal delivery routes, loads, fleet dispatching and departure times for distribution of perishable food. Their objective function was to minimize the cost of dispatching vehicles transportation as well as the inventory, energy and penalty costs for violating the time window constraints. Osvald and Stirn (2008) developed a heuristic algorithm for the problem of distributing fresh vegetables in which the impact of perishability of the goods was considered as a part of the overall distribution cost. However, since additional cost of distribution of perishable goods is insignificant compared to the transportation cost, we do not consider such details in our modeling of the VRP subproblem . This is in agreement with the practical consideration reported by Tarantilis and Kiranoudis (2001) and by Tarantilis and Kiranoudis (2002). Federgruen and Zipkin (1986) are among the pioneering scholars who integrated the routing and inventory allocation in a single period problem assuming random retailers’ demand for perishable products. They classified perishable products into old and fresh items. Old items would perish in the present period while other goods would be considered fresh for at least one period before becoming outdated. Le et al. (2012) presented a multi-period IRP model for perishable products with a fixed shelf life in which the customer demand was deterministic and unsold goods with no value were discarded. A path flow formulation for this problem was proposed and a lower bound for the problem was obtained using column generation. In addition to the regular IRP constraints, they added another constraint guaranteeing that a retailer should not have an inventory greater than the total demand in the next consecutive time periods based on maximum shelf life of the perishable goods. Furthermore, due to the nature of perishable products, some researchers integrated production planning or facility location problem with the VRP or IRP for perishable products (see Karaesmen et al. (2011)). For example, Chen et al. (2009) considered the production schedule and delivery routes decisions simultaneously in order to maximize the expected profit of the supplier. Seyedhosseini et al. (2014) integrated production and distribution planning in an 4

IRP model to determine product quantities, the number of distribution centers to be visited, and the quantities of perishable products to be delivered. Hiassat et al. (2011) proposed a model for deteriorating items in which the inventory location problem was integrated with the routing decisions for deterministic demand. We do not, however, take the production program or facility location into consideration in this paper and we focus on IRP for perishable goods. We have not found any literature that includes the cost of lost sale explicitly in the formulation of IRP for perishable goods as considered in our paper.

3. Problem Definition We consider a two-echelon supply chain involving a depot, a set of retailers, and a fleet of capacitated homogenous vehicles. A central depot (supplier) serves a set of geographically scattered retailers having deterministic demand. Perishable items are transported from the depot to the retailers in such a way that out-of-stock situations never occur. The problem is a multi-period routing and inventory planning problem with a finite time horizon. Retailers have limited storage capacities and their demand in each period is assumed to be known and should be met by the end of each period. It is also assumed that the end customer’s demand is a linearly or exponentially decreasing function of the age of the perishable goods and any inventory unsold by the time of the next delivery is considered as lost sale. We propose a mixed integer nonlinear programming model whose objective function is the sum of the transportation cost, and the cost of inventory holding and lost sale. In addition to the usual inventory routing constraints, we consider a nonlinear constraint that defines the inventory age as a function of the delivery date. The lost sale is then expressed as either a linear or an exponential function of the inventory age. The assumptions on the VRP subproblem are the same as the ones in the classical models. Each vehicle starts its tour from the central depot and returns to the central depot after delivering the goods to a set of retailers assigned to that vehicle. A retailer is served only once and by one vehicle in each period.

3.1. Model Formulation In this section, we develop a mixed integer non-linear programming model for the IRP with lost sale (named as IRPLS) and linearize it to make it solvable using CPLEX. First, the following sets, constants, parameters, and decision variables are introduced: Index Sets: 5

R : Retailer set D : Set of central depots

N : Set of all points ( R

D)

T : Set of time periods K : Vehicle set.

Constants and Parameters:

cij : Travel cost from retailer/depot i to retailer/depot j for i  N , j  N ; satisfying the triangular inequality: cik  ckj  cij ,  i, j, k  N .

 i : Lost sale cost per period per unit for i  R. hi : Holding cost per period per unit for i  R. d it : Demand of retailer i in period t for t  T , i  R.

Vi : Storage capacity of retailer i for i  R. Si : Time to serve retailer i for i  R.

tij : Travel time from point i to point j for i, j  N . bi : Initial amount of inventory of retailer i for i  R.

 : The rate at which the end customer's demand is decreasing ( 0    1 ). LT : Length of each time period. Q : Vehicle capacity. M : A large positive number.

Decision Variables:

1, if vehicle k travels from point i to point j in period t; X ijkt   otherwise. 0 , 1 , Yikt   0 ,

if vehicle k visits point i in period t ; otherwise.

Wikt : The delivered amount to retailer i by vehicle k in period t ;

Other Variables:

I i ,t : Inventory amount of retailer i in period t for i  R , t  T . 6

api ,t : The age of inventory of retailer i in period t , t  T , i  R. U it : Total amount of goods delivered by a vehicle after visiting retailer i in period t .

The objective function is to minimize the following costs:

1- transportation, 2- holding inventory, and 3- lost sale. The cost of lost sale is a function of the inventory age ( api ,t ) and  . We consider two forms of lost sale functions: a linear and an exponential trend. The linear cost function of lost sale is calculated as i  dit    api ,t and the exponential cost function has the form

i  dit  (1  e  ap ) . i ,t

Therefore, the linear objective function of the proposed multi-period IRP model is formulated as:       Min z1   cij X ijkt    hi I i ,t     i .dit . .api ,t    iR tT   iN jN kK tT   iR tT

(1)

Since customers' demand for most perishable goods have an exponential trend, the second objective function with an exponential lost sale is formulated as follows:        .api ,t  Min z2   cij X ijkt    hi I i ,t     i . di ,t .(1  e ) i  N j  N k  K t  T i  R t  T i  R t  T      

The model constraints are given by (3) to (18) as follows:

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(2)

X

 j  N , k  K , t  T

(3)

 i  N , k  K , t  T

(4)

i  R, t  T

(5)

k  K , t  T

(6)

i  R, t  T

(7)

i  R, t  T

(8)

Ii ,t  Vi ;

 i  R,  t  T ,  k  K

(9)

Wikt  Q  Yikt ;

i  R, k  K , t  T

(10)

W

k  K , t  T

(11)

Uit  U jt  Q  X ijkt  Q  D jt ;

i, j  R, i  j, k  K , t T

(12)

Dit  U it  Q ;

 i  R,  t  T

(13)

X ijkt  0,1 ;

 i, j  N , i  j ,  k  K , t  T (14)

Yikt  0,1 ;

 i  R,  k  K ,  t  T

(15)

Wikt  0;

i  R , k  K ,t  T

(16)

I i ,t  0;

 i  R,  t  T

(17)

api ,t  0,1,..., T  1 ;

 i  R,  t  T

(18)

iN i j

X jN i j

Y

ikt

 Y jkt  0;

ijkt

ijkt

 Yikt  0;

 1;

kK



Si X ijkt  

iR jN i  j

t

ij

X ijkt  LT ;

iN jN i  j

  api ,t   api ,t 1  1 . 1  Yikt  ;  kK  I i ,t 1  Wikt  I i ,t  dit ; kK

ikt

 Q;

iR

Constraints (3) and (4) define the assignment of each node to a tour and the ordinary flow conservation at each node. Constraints (5) specify that each retailer is visited at most once every day. Constraints (6) ensure that the total of service times and transit times in each tour assigned to a vehicle does not exceed the length of working hours in each period. This is a practical constraint since our model is a multi-period one and no distribution task may be left for the next period if not completed in the same period as scheduled. Constraints (7) specify the age of inventory in each period as a function of the period allocated to delivering goods to each retailer. Constraints (8) are the inventory balance equation. Constraints (9) indicate that the inventory of each retailer cannot exceed its storage capacity. Constraints (10) and (11) define upper bounds on the delivery amounts by the vehicle capacity. Constraints (12) to (13) are written based on the Miller-Tucker-Zemlin subtour elimination (Kara et al. (2004)). Since

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there may be some deliveries in different time periods, we have considered the variables U it in each period t. Finally, constraints (14) to (18) are the sign and domain constraints. 3.2. Model Linearization Due to the existence of nonlinear terms in the model including (api ,t 1  1).(1   Yikt ) in kK

constraints (7) and

i .dit .(1  e  .ap ) in the second objective function, we need to linearize i ,t

the model in order to be able to solve its relaxations using linear programming solvers. First, a new variable is defined as follows to linearize constraints (7): SCi ,t  api ,t 1 . Yikt

 i  R,  t  T

(19)

k K

Using (19), constraints (7) are replaced by the following set of constraints: api ,t  api ,t 1  SCi ,t   Yikt  1

 i  R,  t  T

(20)

SCi ,t  api ,t 1;

 i  R, t  T

(21)

SCi ,t  M .Yikt ;

 i  R, t  T ,

(22)

  SCi ,t  api ,t 1  M .  Yikt  1 ;  kK  SCi ,t  0;

 i  R, t  T ,

(23)

 i  R, t  T

(24)

k K

kK

Secondly, using a binary variable i ,t , s as defined below, the second objective function given by (2) is reformulated as (25) and also constraints (26) and (27) are added to the model.

 1,

i ,t ,s  

if Inventory age of retailer i in period t is s,  s  0,1,..., T  1 ;

 0 ,

otherwise.

T 1       Min z2   cij X ijkt    hi I i ,t     i . di ,t .(1   i ,t ,s .e   s )  s 0   iR tT   iN jN kK tT   iR tT T 1

 s 0

i ,t ,s

 1, T 1

api ,t   si ,t ,s  0,

(25)

i  R, t T , s 0,1,..., T  1

(26)

i  R, t T , s 0,1,..., T  1.

(27)

s 0

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4. Solution Approach The proposed model can be considered as a combination of VRP and an inventory control model. Because of the underlying VRP component, the problem belongs to the class of NPhard problems and its complexity grows exponentially with increasing in the number of retailers and time periods. In addition, the age of the products and the limited capacity of retailers make the problem more difficult to solve. A key decision in solving the IRPLS as IRP with backlog is the amount to be delivered to customer i  R in period t  T . In fact, given delivery values Wikt for all customers, periods, and vehicles, with an assumption that (if Wikt  0 then Yikt  1 else Yikt  0 ), the inventory and lost sale values are determined by the constraints (8-9), (16-18), and (19-24) in the model with a linear lost sale function (the first objective function) and by constraints (8-9), (16-18), and (19-27) in the model with an exponential lost sale function (the second objective function). At the same time, the best routing solution given these values of Wikt is obtained by solving a capacitated VRP for each period. Each VRP finds the best possible routes by solving the following problem whenever the delivery amounts are satisfied. The Vehicle Routing subproblem is given by:   Min z   cij X ijkt   iN jN kK tT 

(28)

subject to: Constraints (3-6) and (10-16). Having the optimal values of Wikt , we can calculate the lost sale and inventory. What remains is a routing problem for which various efficient algorithms are available. Therefore, the key decision in solving IRPLS is to identify the optimal delivery amounts of Wikt . In this regard, we develop a suitable heuristic to solve this problem. 4.1. The Proposed Metaheuristic Algorithm The proposed metaheuristic algorithm is based on a hybrid of Simulated Annealing (SA) and Tabu Search (TS). The hybrid algorithm is used to improve the current solution. SA is a meta-heuristic method that is able to escape from local optima by allowing hill-climbing moves. In addition, its ease of implementation and convergence properties has made it a 10

popular technique for logistics applications. TS is a general framework for a variety of iterative local search strategies. The concept of memory is used via a dynamic list of forbidden moves in TS. A main criticism of SA is that it is completely memory less. On the other hand, there is no proof of convergence in the literature for the original TS algorithm (Henderson et al. 2003). In this research, we use a hybrid of SA and TS in an attempt to capitalize on both the asymptotic optimality of SA and the memory feature of TS. The proposed algorithm starts with a predefined delivery pattern. Then, a new delivery pattern for each retailer is generated based on a minimum total approximated cost using the developed heuristic (MTAC) which is described in section 4.1.1. Next, the amounts of inventory and lost sale are determined with respect to the new delivery pattern. Afterwards, the routing subproblem is solved. All these steps take place inside the hybrid algorithm (SATS), to investigate possible improvements to the solutions generated by the developed heuristic. The proposed algorithm has two stopping criteria as follows: 

Is NI greater than MNI?



Is Tr less than T f ?

We initially set NI  0, MNI  40, T0  3000 and T f  10 . If the above criteria are met, we stop; otherwise, we continue to the next steps to improve the current solution. Figure 1 represents the flow chart of the proposed heuristic algorithm.

Fig. 1. here

The parameters of the hybrid SA–TS algorithm are defined below: T0 : Initial temperature;

T f : Final temperature; Tr : Temperature of stage; M : Maximum number of changes for each temperature; e : Maximum number of changes for each period for the equilibrium condition;

 : Temperature reduction coefficient for each stage (a decimal between 0 and 1);

 1 : Small positive integer number for evaluation of equilibrium at temperature; Ce Tr  : Objective function average for all accepted cases in each period at the given temperature;

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Cg Tr  : Objective function average for all previous periods at the given temperature; n : Number of accepted solutions in each period;

nt : Number of accepted solutions at each temperature; | R | : Number of retailers; | T | : Number of planning periods;

MNI : Maximum number of times with no improvement; NI : Number of times with no improvement; W0 : Initial solution;

W : Current solution in the algorithm (initial delivery pattern); Wnh : Solution selected in the neighborhood of W at each iteration;

W * : Best solution obtained in the algorithm; C W  : Objective function value for solution W;

The steps of the proposed hybrid algorithm (SA-TS) are explained by a pseudo code given in Figure 2. The heuristic algorithm (MTAC) devised for generating neighborhood solution is described next in Section 4.2 and the solution method for solving the VRP subproblem is explained in Section 4.3.

Fig. 2. Here

4.2. MTAC Heuristic The heuristic method used in the proposed algorithm is called Minimum Total Approximated Cost (MTAC). The steps of MTAC heuristic are explained by a pseudo code given in Figure 3. The algorithm is intended to find a good delivery pattern to be used in the routing subproblem. It starts with an initial solution in which the total demand of each retailer over all time periods is delivered in the first period, and there is no delivery in the rest of periods. Next, we consider another delivery pattern in which a second delivery is scheduled after the first period. The amount of second delivery is varied based on the retailer’s demand in other periods. Then, the amount of the first delivery is decreased by the amount of the second delivery. The heuristic continues to evaluate the cost of having more deliveries over the remaining time periods so that an out of stock situation does not occur. This process is repeated until the stopping condition is satisfied. Eventually, it generates various delivery 12

patterns for each retailer over the planning horizon with respect to the initial solution and finds the best delivery scenario for each retailer. It is worth nothing that our algorithm is a constructive heuristic which tries to create different delivery patterns and calculate the resulting lost sale and inventory cost and select a delivery pattern that has the least total cost. Fig. 3. Here

4.3. Solving the VRP Subproblem The routing subproblem is one of the key components of IRP. Therefore, an efficient algorithm should be used for solving the underlying VRP. In this study, we employ an efficient saving based algorithm of Clarke and Wright (1964) for solving the VRP. The reason for selecting the Clarke and Wright algorithm is that saving based algorithms have been successfully used for solving the VRP subproblems of IRPs (Golden et al. 1984, Herer & Levy 1997, Abdelmaguid et al. 2009). The solution gets improved by a Nearest Neighborhood Heuristic (NNH) that originates from Gutin et al. (2002). The combination of these algorithms helps us find a good solution for the underlying VRP that is a key part of IRPLS.

5. Computational Experiments In this section, we consider three different scenarios in order to evaluate the performance of the proposed algorithm. The first scenario has been designed to test the impact of a relatively high unit cost of lost sale on the solution and also to test the resulting holding cost, transportation cost, and lost sale. In the second scenario, the parameters are set such that transportation cost is higher than inventory holding cost, which makes incurring higher cost of lost sale economical. The third scenario has been especially designed to evaluate the performance of the proposed algorithm on solving the large scale problems. The algorithm has been coded and compiled in MATLAB and solutions are compared with the lower and upper bounds obtained by CPLEX (version 10.1) in a certain amount of time. We coded the model in Lingo 14.0 to create an MPS data file and run the resulting model in CPLEX environment. All codes were run on a computer with a Pentium dual core processor and the clock speed of 1.8 GHz and 8 GB of RAM.

13

5.1. Problem Sets and Scenarios Three sets of IRP instances were generated according to the three scenarios explained earlier. The problem data reported in Abdelmaguid et al. (2009) was modified and used for the first and second scenarios and random instances were generated for the third scenario. In all the scenarios, retailers’ unit holding cost follows a normal distribution with a mean value of 0.1 and a standard deviation of 0.02, and each retailer has a storage capacity of 120 items. Each data set considers a different scenario for the unit cost of lost sale and transportation cost, retailers’ demand, number of retailers, number of planning periods, number of vehicles, and their capacity as shown in Table 1. Three instances have been generated for a problem size of up to 40 retailers and two instances of a larger problem size with 100 retailers (nodes). It is assumed that an out-of-stock situation never occurs in our models. In the third scenario, retailers' sites are allocated in a square of 20  20 distance units and their coordinates are randomly generated using a uniform distribution within this area. The depot is located in the middle of the square. Each sample is denoted by six digits. The first one represents the scenario number. The second and third ones are reserved for the number of retailers followed by a digit representing the length of the planning horizon. The fifth one stands for the number of vehicles. Finally, the last digit indicates the sample number in that problem class. For example, the code 10552-1 represents the first run of the first scenario with 5 retailers, 5 time periods, and 2 vehicles. Table 1 here 5.2. Computational Results In this section, we report optimality gaps (in percentage) obtained by the difference between CPLEX lower and upper bounds and use it as a performance measure to evaluate the quality of the solutions obtained by the proposed heuristic algorithms. The CPLEX upper bounds were found in a maximum of one-hour running time for the first and second scenarios and in a maximum of 12 hours for the third scenario. Two metaheuristic algorithms used in this research are called SA-MTAC (Simulated Annealing Minimum Total Approximated Cost) and SA-TS-MTAC (Simulated Annealing-Tabu Search-Minimum Total Approximated Cost). We also report the optimality gaps obtained by the differences between the objective values of our solutions and the CPLEX lower bounds (see Tables 2A to 4A and 2B to 4B). It can be seen in Tables 2A and 3A as well as in Tables 2B and 3B that the average optimality gaps 14

obtained by SA-MTAC and SA-TS-MTAC are less than 4.5% for Scenario 1 and 10.9% for Scenario 2 using the first objective function (linear lost sale function), and 5% for Scenario 1 and 13.4% for Scenario 2 using the second objective function (exponential lost sale function). In the first scenario, the average gap obtained by SA-TS-MTAC is 0.46% for the model with first objective function and is 0.31% for the model with second objective function which is better than that of SA-MTAC. This difference is 1.4% for the model with first objective function and is 0.5% for the model with second objective function in Scenario 2. Figures 4A to 6A and Figures 4B to 6B respectively show the average optimality gaps for the six problems sets (three problem instances, each with a linear and an exponential objective functions). For the problem instances of the model with the first objective function, Tables 2A and 3A show that the optimality gap obtained by SA-TS-MTAC from the optimal solution for Scenario 1 is less than 1.54% for the model with 420 variables, and less than 2.4% for the model with 588 variables. For Scenario 2, these values are less than 5.6% and 4.5%, respectively. For the problem instances of the model with exponential lost sale function, Tables 2B and 3B show that the optimality gap obtained by SA-TS-MTAC from the optimal solution for the problems in Scenario 1 is less than 1.1% for the model with 508 variables, and less than 2.8% for the model with 798 variables. For the problems in Scenario 2, these values are less than 7.32% and 7.43%, respectively. The computational times of SA-MTAC for Scenarios 1 and 2 are less than 21 and 27 seconds in all the cases tested in the model with the first and second objective functions. Since we used the concept of memory in SA-TS-MTAC, the computational time increased with the problem size; however, on average, it did not exceed 41 and 51 seconds for the model with the first and second objective functions, respectively (see Tables 5A and 5B). As can be seen in Figures 5A and 5B, with the increase in the number of customers, the optimality gaps obtained by the proposed algorithm are almost constant. When these values are compared with the exponential rate of increase in the optimality gaps obtained by CPLEX upper bound, the efficiency of the heuristic algorithms for larger problem instances is verified. As shown in Figures 6A and 6B, the results of SA-TS-MTAC are also 10.3% and 8.1% closer on average to the lower bounds than those of SA-MTAC for the model with the first and second objective functions. In terms of computational time, SA-MTAC takes less than 11 minutes for larger problems with up to 100 customers in the models with both 15

objective functions. This amount is less than 21 minutes for SA-TS-MTAC (see Tables 5A and 5B).

Figures 4A, 4B, 5A, 5B, 6A and 6B here Table 2A here Table 2B here Table 3Ahere Table 3B here Table 4A here Table 4B here Table 5A here Table 5B here

6. Conclusion Due to the importance of inventory routing problem (IRP) in the operation management of supply chain, this paper develops a non-linear mixed integer programming model for IRP considering the cost of lost sale for perishable goods. The model is designed so as to avoid overstocking perishable goods and take into account the negative impact of the age of perishable inventory on the end customers' demand. A part of the inventory that is not sold by the time of next delivery is considered as lost sale. The nonlinear terms in the model are linearized in order to able to solve the problem using CPLEX. The model is solved to optimality for small instances and obtains lower bounds for larger instances. We also devise a metaheuristic solution method in order to find good solutions for large instances of this class of problems. The proposed algorithm starts with a predefined delivery pattern and uses a neighborhood search to improve the delivery pattern. It then solves the inventory lost sale subproblem and routing subproblem sequentially and tries to improve the solution within the metaheuristic framework. Computational results indicate that for small-sized problems with up to 15 customers, the algorithm can find good solutions in a reasonable time with a maximum average optimality gap of 10.9% using the first objection function and 13.4% with the second objective function. The optimality gap found by CPLEX grows exponentially with the problem size while the ones obtained by the proposed algorithm increase linearly. Therefore, the proposed solution method has shown some promise for solving large instances. Future research could be aimed 16

at finding a tighter lower bound using a relaxed model or an approximating solution method for this class of problems. It would also be interesting to include uncertainty of the end customers' demand in the mathematical models.

References Abdelmaguid, T. F., Dessouky, M. M. (2006).A genetic algorithm approach to the integrated inventory-distribution problem. International Journal of Production Research, 44(21), 4445–4464. Abdelmaguid, T.F., Dessouky, M.M., Ordóñez, F. (2009).Heuristic approaches for the inventory-routing problem with backlogging. Computers & Industrial Engineering, 56(4), 1519–1534. Chen, H. K., Hsueh, Ch. F., Chang. M. Sh. (2009). Production Scheduling and Vehicle Routing with Time Windows for Perishable Food Products. Computers & Operations Research, 36, 2311– 2319. Chung K. J., Ting P.SH. (1993). A Heuristic for Replenishment of Deteriorating Items with a Linear Trend in Demand. The Journal of the Operational Research Society, 44(12), 12351241. Clarke, G., Wright, J. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operations Research, 12, 568–581. Federgruen, A., Prastacos, G., Zipkin, P. (1986). An allocation and distribution model for perishable products. Operations Research, 34, 75-82. Federgruen, A., Zipkin, P. (1984).A combined vehicle routing and inventory allocation problem. Operations Research, 32 (5), 1019-1037. Giri, B.C., and Chaudhuri, K.S.(1997).Heuristic models for deteriorating items with shortages and time-varying demand and costs. International Journal of Systems Science, 28, 53159. Golden, B.L., Assad, A., Dahl, R. (1984).Analysis of a large scale vehicle routing problem with an inventory component. Large Scale Systems, 7, 181–90. Goyal, S.K., Giri, B.C. (2001).Recent trends in modeling of deteriorating inventory. European Journal of Operational Research, 134(1), 1–16. Gutin, G., Yeo, A., Zverovich, A. (2002). Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP. Discrete Applied Mathematics, 117, 81-86. 17

Hariga M., Benkherouf L. (1994). Optimal and heuristic replenishment for deteriorating items with exponential time varying demand. European Journal of operational Research, 79, 123-137. Henderson, D., Jacobson, Sh.H., Johnson, A.W. (2003). The theory and practice of Simulated annealing. In F. Glover &G.A. Kochenberger (Eds.), Handbook of Metaheuristic (p. 287301). Kluwer Academic Publishers. Herer, Y.T., Roundy, R. (1997).Heuristics for a one-warehouse multi retailer distribution problem with performance bounds. Operations Research, 45, 102–15. Hiassat, A., Diabat, A., (2011). A location-inventory-routing problem with perishable products. 41st International Conference on Computers & Industrial Engineering Hsu, C., Hung, S., Li, H. (2007).Vehicle routing problem with time-windows for perishable food delivery. Journal of Food Engineering, 80, 465-475. Jaillet, P., Bard, J.F., Huang, L.,Dror, M. (2002) Delivery cost approximations for inventory routing problems in a rolling horizon framework. Transportation Science, 36, 292–300. Kara, I., Laporte, G., Bektas, T. (2004).A note on the lifted Miller Tucker–Zemlin subtour elimination constraints for the capacitated vehicle routing problem. European Journal of Operational Research,158, 793–795. Karaesmen, I. Z., Scheller-Wolf, A., & Deniz, B. (2011). Managing perishable and aging inventories: review and future research directions. In Planning production and inventories in the extended enterprise, 393-436. Le, T., Diabat, A., Richard, J., Yih, Y. (2012) A column generation-based Algorithm for an inventory routing problem with perishable goods. Optimization Letters, 1481-1502. Li, R., Lan, H., Mawhinney, J.R. (2010). A Review on Deteriorating Inventory Study Journal of Service Science & Management, 3, 117-129. Osvald, A., Stirn, L.Z. (2008). A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food. Journal of Food Engineering, 85, 285–295. Dave, U. and Patel, L.K.(1981). (T, Si) - policy inventory model for deteriorating items with time proportional demand. Journal of Operational Research Society, 32, 137-142. Raafat, F. (1991).Survey of literature on continuously deteriorating inventory model. Journal of the Operational Research Society, 42(1), 27–37. Sachan, R. S., (1984). On (T, Si) - policy inventory model for deteriorating items with time proportional demand, 35, 1013-1019. Seyedhosseini, S. M., and Ghoreyshi, S. M. (2014). An Integrated Model for Production and Distribution Planning of Perishable Products with Inventory and Routing Considerations.

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Mathematical Problems in Engineering, 2014, Article ID 475606, Hindawi Publishing Corporation. Shah, N.H., Shah, Y.K.(2000).Literature survey on inventory model for deteriorating items. Economic Annals, 44, 221-237. Shen, Z., Dessouky, M., Ordonez, F. (2011).Perishable inventory management system with a minimum volume constraint. Operational Research Journal, 62, 2063–2082. Tarantilis, C., Kiranoudis, C. (2001). A metaheuristic algorithm for the efficient distribution of perishable foods. Journal of Food Engineering, 50, 1-9. Tarantilis, C., Kiranoudis, C. (2002). Distribution of fresh meat. Journal of Food Engineering; 51, 85-91. Toth, P., Vigo, D. (2002).The Vehicle Routing Problem, SIAM. Wee H.M. (1995). Deterministic lot size inventory model for deteriorating items with shortages and a declining market. Computers & Operations Research, 22, 345-356. Xu, H., Wang H.P. (2003). An economic ordering policy model for deteriorating items with time proportional demand. European Journal of Operational Research, 46(1), 21-27.

19

Figure(s)

Figures

Start with a predefined initial solution

Determine the Inventory and Lost sale

Solve the Routing sub-problem

No

Are the stopping criteria met? Yes End

Fig.1. Flowchart of the proposed algorithm.

28

SA-TS

Generate a neighborhood solution using MTAC

Step 1. Generate an initial W0 and evaluate C W

0



and set W *  W0 , W  W0 .

While (stopping condition == false) do the following, Step 2. Generate Wnh in the neighborhood of W using MTAC heuristic. Step 3. If the generated Wnh is not in the Tabu list then update the Tabu list and go to Step 4. Else go to Step 2 to generate another solution. Step 4. Compute C  C W   C W  , and nh

4.1) If C  0 , then set change  true , NI  0 . 4.2) Else if C  0 , then If random[0,1)  0.5 , then set change  true , NI  NI  1 ; 4.3) Else 



C If random[0,1)  exp   , then set change  true , NI  NI  1 .  Tr 

Step 5. If (change==true) then set W  Wnh ; n  n  1; If (n  e) then go to step 2; otherwise go to step 6. Step 6. If (Wnh< W*) then set W*= Wnh Else if (NI>MNI) then set Stopping condition=true. Else go to step 7. Step 7. Check the equilibrium condition with n=0; nt=nt+e; If (nt £ M ) If

 Ce Tr   Cg Tr     1    C T   g r  

then go to step 2; otherwise go to step 8.

Else go to step 8. Step 8. If ( Tr  Tf ) then set Stopping condition=true, Else set

Tr1  αTr and go to step 2.

End. Fig.2. Pseudo code of the hybrid SA–TS algorithm

29

Step 1: Evaluate the inventory and lost sale costs for W0 and set CHW W0 . Step 2: Increase the number of deliveries  nd  nd  1 , and determine the period of new delivery

(ct ) .

Step 3: Specify the previous  pt  and next delivery period  nxt  of retailer i  R for the current period

 ct  . If there is no more delivery, set nxt  T . Step 4: For each period t  pt  1 to nxt do 4.1) Define a new delivery pattern by deducting d i ,t from the delivery in previous period and adding this amount to the delivery in the new period. CHWi,k , pt  CHWi,k , pt  di ,t

CHWi,k ,ct  CHWi,k ,ct  di ,t ;

4.2) Evaluate the inventory and lost sale costs for CHW , and calculate the cost change in the delivery pattern and then save it in the DP set. Step 5: Take the best delivery pattern in DP ( CHW * ), and set CHW  CHW * . Step 6: If stopping condition is met ( nd  T ), then stop; otherwise go to step 2. Fig.3. Pseudo code of MTAC heuristic algorithm

18 16 14 12 10

UB-LB diff%

8

SA-LB diff%

6

SATS-LB diff%

4 2 0 420

588

1320 1848 2720 3803

Fig.4A. Average optimality gap for the first scenario problems with the first objective function.

30

14 12 10 UB-LB diff%

8

SA-LB diff%

6

SATS-LB diff%

4 2 0 508

798

1968 2917 3658 3803

Fig.4B. Average optimality gap for the first scenario problems with the second objective function.

25 20 15

UB-LB diff% SA-LB diff%

10

SATS-LB diff%

5 0 420

588

1320 1848 2720 3803

Fig.5A. Average optimality gap for the second scenario problems with the first objective function. 25 20 15

UB-LB diff% SA-LB diff%

10

SATS-LB diff%

5 0 508

798

1968 2917 3658 3803

Fig.5B. Average optimality gap for the second scenario problems with the second objective function.

31

250 200 150 UP-LB diff% 100

SA-LB diff% SATS-LB diff%

50 0

Fig.6A. Average optimality gap for the third scenario problems with the first objective function.

250 200

150 UP-LB diff% 100

SA-LB diff% SATS-LB diff%

50 0

Fig.6B. Average optimality gap for the third scenario problems with the second objective function.

32

Tables Table 1 The Problem sets and Scenarios Scenarios transportation Retailers' cost per unit unit cost of distance lost sale Scenario1 1 N(0.8, 0.4)

Retailers' demand U(25,50)

Number of retailers 5 10 15

Scenario2

2

N(0.4, 0.01)

U(5,50)

5 10 15

Scenario3

1

N(0.4, 0.01)

U(0,50)

20 25 30 40 50 75 100

20

Number of periods 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 7 7 7

Number of Vehicles 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3

Vehicle capacity

200-300 200-300 400-600 400-600 600-900 600-900 300-400 300-400 600-800 600-800 900-1200 900-1200 400-600 400-600 500-700 500-700 600-800 600-800 700-900 700-900 700-800-900 800-1000-1200 900-1200-1500

Table 2A Detailed costs for the first scenario problems (The model with first objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap%   total cost of heuristic  CPLEX lower bound  / CPLEX lower bound  100 CPLEX bounds LB

UB

*Optimality

1-0552-1

225.5

225.5

Gap (%) 0

1-0552-2

143.8

143.8

1-0552-3

204.2

1-0572-1

SA-MTAC

SA-TS-MTAC

Holding cost

Lost sale cost

Transportation cost

Total

**Optimality

Holding cost

Lost sale cost

Transportation cost

Total

**Optimality

7.32

11.4

206.7985

225.5185

Gap (%) 0.023856136

6.78

10.79856

207.8861

225.46

Gap (%) 0

0

4.46

11

129.9324

145.3924

1.137051346

9

13.98236

122.9833

145.97

1.535819274

204.2

0

2.6

5.2

197.5366

205.3366

0.579219325

0

0

204.1541

204.15

0

322.6

322.6

0

33.13

45.331

246.7874

325.2484

0.805768257

3.13

5.331

281.3062

324.25

0.493387169

1-0572-2

273.1

273.1

0

3.567

6.8769

269.1442

279.5881

2.359156812

5.6903

7.9893

265.9442

279.62

2.372226831

1-0572-3 1-1052-1

294.6 318.7

294.6 318.7

0 0

25.45 23.49

31.746 34.4

242.1112 265.4933

299.3072 323.3833

1.582173012 1.460141424

10.25 23.49

10.006002 34.4

279.2094 265.4933

299.47 323.38

1.609535515 1.439128118

1-1052-2

265.7

265.7

1-1052-3 1-1072-1

262.4 419

262.4 426.2

0

3.52

8.6

262.1314

274.2514

3.20865315

5.52

7.6

258.1314

271.25

2.037298241

0 10.68

0 0

0 0

268.6729 444.3040012

268.6729 444.304

2.40155291 6.044354723

4.958 5.5679

6.5689 7.3251905

260.7319 427.3658486

272.26 440.26

3.631434503 4.833414431

1-1072-2

478.9

530

12.89

6.4570186

9.456

499.760304

515.6733

7.671691175

6.4896

9.678

490.6004012

506.77

5.493006876

1-1072-3 1-1552-1

363.9 361.6

367.8 404.6

1.67 7.765

6.95 10.3795

10.90185 16.463

362.5910823 352.4741107

380.443 379.317

4.54678739 4.90144886

4.475 0

7.5670378 0

367.3543138 378.661581

379.4 378.67

4.085187314 4.50752911

1-1552-2

384.8

439.7

4.186

11.7525

15.67

390.231712

417.654

8.52905621

7.132269

8.8967

404.2819121

420.3

8.44117599

1-1552-3 1-1572-1

375.7 458.2

419.8 514.3

2.934 3.29743

29.97572 5.4689

39.2075 10.6628

327.2970655 491.6435

396.480 507.775

5.53358561 10.8239084

28.5057 17.4678

36.2075 27.5376

334.2970655 462.4988528

399.0 507.5

5.84425512 9.71858906

1-1572-2

530

603.7

5.388

8.658

10.456

555.23

574.344

8.37139313

3.6

7.147729

567.3018602

578.1

8.31625696

1-1572-3

520.2

627.8

3.22

26.7689

53.5378

490.3546

570.661

9.69153003

23.7689

50.5378

485.3546

559.6

7.04343859

Max. Optimality Gap (%) Avg. Optimality Gap (%)

12.89

10.8239084

9.71858906

2.8905794

4.42618488

3.96676017

21

Table 2B Detailed costs for the first scenario problems (The model with second objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap%   total cost of heuristic  CPLEX lower bound  / CPLEX lower bound  100 CPLEX bounds LB

UB

SA-MTAC

*Optimality

SA-TS-MTAC

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality Gap (%)

1-0552-1

185.0845

185.0845

0

11.69

23.98867

151.4058

187.0845

1.080571307

11.69

23.98867

151.4058

187.08447

1.08057131

1-0552-2

144.594

144.594

0

4.178

11.51278

129.80989

145.5007

0.627045382

4.369

10.6792

130.5989

145.6471

0.72831514

1-0551-3 1-0571-1

203.3052 319.3576

203.3052 319.3576

0 0

3.1236 13.84

8.13585 36.27962

192.63576 276.1235

203.8952 326.2431

0.290209006 2.156053277

2.62 11.9473

7.3685 34.68593

193.3366 273.23799

203.3251 319.87122

0.00978824 0.16082911

1-0571-2

273.1442

273.1442

0

5.3567

11.53538

270.1442

287.0363

5.085987548

3.5499

7.3682

269.76207

280.68017

2.75897127

1-0571-3 1-1052-1

303.2262 313.1694

303.2262 313.1694

0 0

18.72 0

36.7014 0

254.8048 343.563497

310.2262 343.5635

2.308507642 9.705321465

18.72 1.679

36.7014 3.5789

254.8048 338.23677

310.2262 343.49467

2.30850764 9.6833439

1-1052-2

263.4235

265.7252

0.8737641

0

0

267.7931255

267.7931

1.658783476

0

0

267.7931255

267.793125

1.65878348

1-1052-3 1-1072-1

258.8625 412.4908

262.3719 428.9923

1.3557004 4.0004529

6.6478 0

10.758 0

249.2847298 435.3378074

266.6905 435.3378

3.024010739 5.538791992

6.6478 0

10.758 0

249.2847298 435.3378074

266.69053 435.337807

3.02401074 5.53879199

1-1072-2

479.1749

541.152

12.934129

4.18536

6.4337

486.5535

497.1726

3.755968854

2.536

5.7709434

485.5945535

493.901497

3.07332393

1-1072-3 1-1552-1

357.4743 355.0956

362.6274 406.6707

1.4415302 14.524286

1.5 0

4.032941 0

381.8614161 372.0661581

387.3944 372.0662

8.369848388 4.779151896

2.5 0

5.32623 0

380.1608 372.0661581

387.98703 372.066158

8.53564298 4.7791519

1-1552-2

383.7427

441.0505

14.933913

1.39046

3.27859

430.0505

434.7196

13.28412241

0

0

431.235605

431.235605

12.3762367

1-1552-3 1-1572-1

377.8557 460.8768

417.6322 509.3717

10.526902 10.522313

9.25748 14.6859

14.3579 20.36859

366.3867377 472.4988528

390.0021 507.5533

3.214565163 10.12777011

9.25748 14.6859

14.3579 20.36859

366.3867377 472.4988528

390.002118 507.553343

3.21456516 10.1277701

1-1572-2

528.9464

593.606

12.224225

11.8

17.64919

525.3020277

554.7512

4.878531796

12.679

19.991906

527.771045

560.441951

5.95439368

1-1572-3 Max. Optimality Gap (%)

531.2994

607.2399

14.293353 14.933913

0

0

572.871844

572.8718

7.824673621 13.28412241

2.7895

5.36892

564.6194

572.77782

7.24162468 12.3762367

Avg. Optimality Gap (%)

5.4239205

4.872773004

22

4.56970122

Table 3A Detailed costs for the second scenario problems (The model with first objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap%   total cost of heuristic  CPLEX lower bound  / CPLEX lower bound  100 LB

CPLEX bounds UB *Optimality

Holding cost

Lost sale cost

SA-MTAC Transportation cost

Total

**Optimality

41.21

58.28389

208.4258

307.9197

Gap (%) 0

Holding cost

Lost sale cost

43.21

60.28389

SA-TS-MTAC Transportation cost

Total

**Optimality

208.4258

311.91969

Gap (%) 1.299036729

2-0552-1

307.9197

307.9197

Gap (%) 0

2-0552-2

271.0598

271.0598

0

38.3

55.04846

187.7113

281.0598

3.689208064

38.3

50.04846

197.7113

286.05976

5.533819475

2-0552-3 2-0572-1

320.2839 462.9859

320.2839 462.9859

0 0

40.17 63.04

62.01883 105.1506

228.0951 294.7953

330.2839 462.9859

3.122239363 0

45.17 105.1506

67.01883 63.04

218.0951 294.7953

330.28393 462.9859

3.122239363 0

2-0572-2

482.2626

482.2626

0

74.52

119.5972

298.1454

492.2626

2.073559094

64.52

119.5972

308.1454

492.2626

2.073559094

2-0572-3 2-1052-1

447.4257 482.874

447.4257 487.3498

0 0.92690847

75.79 68.06

100.3415 94.51735

281.2942 360.6717868

457.4257 523.2491

2.235007958 8.361422462

75.79 75.83

110.3415 98.58432

281.2942 344.59704

467.4257 519.011356

4.470015915 7.483806583

2-1052-2

443.6019

455.014

2.57259944

35.568

46.94159

399.57435

482.0839

8.674904164

67.5689

87.89505

327.0740457

482.538

8.777261704

2-1052-3 2-1072-1

507.1981 495.0341

531.3882 544.8615

4.76935935 10.065448

78.11 88.172

93.732 105.8064

393.772385 400.220469

565.6144 594.1989

11.51744949 20.03190669

80.5145 71.7848

93.23297 86.70219

387.5857036 399.300875

561.333175 557.787865

10.6733591 12.67665506

2-1072-2

750.2599

750.6175

0.04766348

95.3569

112.0311

588.785979

796.174

6.119753381

105.99

117.1026

576.965347

800.057911

6.637434708

2-1072-3 2-1552-1

364.6923 540.9541

367.8008 643.3589

0.85236239 18.9304046

46.2874 97.3276

60.17358 126.368

318.296394 420.1530743

424.7573 643.8487

16.47006119 19.02094361

47.784 86.707

65.28045 116.9276

301.416862 445.0724322

414.481313 648.707045

13.65233458 19.91905509

2-1552-2

527.074

573.734

8.852601

47.285

60.36577

522.763436

630.4142

19.60628071

25.75

41.20656

524.00063

590.957188

12.12023879

2-1552-3 2-1572-1

569.773 699.778

698.827 814.623

22.65001 16.4117

74.3576 73.97

102.1895 103.558

510.409671 658.7819

686.9568 836.3099

20.56676564 19.51079607

69.1466 57.02

92.05957 79.53395

539.3259161 687.7028825

700.53209 824.256834

22.94935229 17.78838246

2-1572-2

734.955

769.922

4.757761

74.3933

104.1507

708.638639

887.1826

20.71251151

127.53

157.397

575.9115308

860.838566

12.37030373

2-1572-3 Max. Optimality Gap (%) Avg. Optimality Gap (%)

875.537

953.398

8.893027 22.6500067

116.495

163.093

705.035017

984.623

12.45939113 20.71251151

117.176

134.839

707.893588

959.908538

8.789591387 22.94935229

5.54054659

10.78734447

23

9.463135893

Table 3B Detailed costs for the second scenario problems (The model with second objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap%   total cost of heuristic  CPLEX lower bound  / CPLEX lower bound  100 CPLEX bounds LB

UB

SA-MTAC

*Optimality

SA-TS-MTAC

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality Gap (%)

2-0552-1

285.4864

285.4864

0

65.23

77.7214

163.4041

306.3555

7.310015468

65.23

77.7214

163.4041

306.3555

7.310015468

2-0552-2

253.098

253.098

0

46.44

54.21635

168.9864

269.64275

6.5368948

37.14

45.54265

179.4154

262.09805

3.55595461

2-0551-3 2-0571-1

299.7686 431.0838

299.7686 431.0838

0 0

65.1363 79.25

73.133641 93.03851

180.384 288.17983

318.65394 460.46834

6.299973046 6.816433371

57.63 74.4925

65.33641 86.4051

182.8022 298.7249

305.76861 459.6225

2.001547193 6.620220941

2-0571-2

431.4878

431.4878

0

95.55

117.2148

218.723

431.4878

1.31738E-14

105.45

119.34948

238.7229

463.52238

7.424214543

2-0571-3 2-1052-1

404.2175 450.0641

404.2175 481.8308

0 7.05826126

119.294 63.06

109.93569 88.284

210.1677 344.6794747

439.39739 496.02347

8.70320904 10.21173978

116.09 63.06

107.94894 88.284

208.3849 344.6794747

432.42384 496.0234747

6.978010601 10.21173978

2-1052-2

407.2463

449.5876

10.396976

37.7315

52.45937

380.1383398

470.32921

15.49011243

36.28715

50.460459

379.753521

466.5011304

14.55012123

2-1052-3 2-1072-1

359.7559 546.5529

404.8824 594.4441

12.5436442 8.76240891

36.1409 73.42595

50.636217 98.959547

314.374189 508.557197

401.15131 680.94269

11.50652595 24.58861603

36.1409 70.8926

50.636217 94.395547

314.374189 507.3675897

401.151306 672.6557317

11.50652595 23.07239275

2-1072-2

662.6802

689.7768

4.08894064

68.01

114.58379

593.152789

775.74658

17.06198208

68.01

114.58379

593.152789

775.746577

17.06198208

2-1072-3 2-1552-1

634.2038 514.1678

716.9004 615.2302

13.0394362 19.6555288

100.6808 108.7909

140.60843 128.13092

505.8639395 373.285126

747.15317 610.20695

17.80963272 18.67856097

99.2908 108.7909

130.09343 128.13092

501.1139395 373.285126

730.4981675 610.206946

15.18350529 18.67856097

2-1552-2

515.5174

563.8826

9.38187537

63.234586

86.863023

479.73237

629.82998

22.17433961

63.23459

86.863023

479.73237

629.829979

22.17433961

2-1552-3 2-1572-1

568.0227 699.2883

699.0533 815.4305

23.0678457 16.6086291

58.5694 123.3666

85.823274 153.89355

526.49671 581.0571377

670.88938 858.31729

18.10960794 22.74154904

58.5694 113.2467

85.823274 143.24554

526.49671 583.660631

670.889384 840.152831

18.10960794 20.14398511

2-1572-2

712.6332

760.6842

6.74273946

76.904

102.47577

621.3442942

800.72406

12.36131859

84.1634

113.2866

615.49502

812.94502

14.07622042

2-1572-3 Max. Optimality Gap (%)

827.7093

932.0061

12.6006558 23.0678457

105.1495

136.39207

695.1013253

936.64289

13.16085156 24.58861603

105.1495

136.39207

695.1013253

936.6428923

13.16085156 23.07239275

Avg. Optimality Gap (%)

7.9970523

13.30896458

24

12.87887756

Table 4A Detailed costs for the third scenario problems (The model with first objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap%   total cost of heuristic  CPLEX lower bound  / CPLEX lower bound  100 CPLEX bounds LB

UB

*Optimality

SA-MTAC

SA-TS-MTAC

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality

Lost sale cost

Transportation cost

Total

Gap (%)

Holding cost

**Optimality Gap (%)

3-2072-1

414.2365

466.9803

12.7327746

41.673

46.89823

419.9556585

508.5269

22.76245297

38.781

42.3779

421.4701

502.629

21.33865557

3-2072-2

374.611

473.7326

26.4598744

65.78006

90.91568

340.230085

496.9258

32.65115573

39.479

51.7493

391.1662463

482.3945447

28.77212487

3-2072-3 3-2572-1

431.0994 457.8735

451.1218 519.6296

4.6444973 13.4875899

29.46785 21.189

40.3243 32.98801

443.796491 484.5271031

513.5886 538.7041

19.13462208 17.65348204

24.4962 23.4965

26.57451 31.98801

442.7657705 489.5308352

493.8364808 545.0153482

14.55281098 19.03186103

3-2572-2

493.7252

554.1601

12.2405946

28.79

31.84975

527.482742

588.1225

19.11939948

36.3655

47.13312

487.986638

571.485253

15.74966256

3-2572-3 3-3072-1

516.4524 439.5979

598.6188 652.8093

15.9097721 48.5014601

34.4869 80.4904

54.15135 89.41738

756.8904685 453.51064

845.5287 623.4184

63.71861463 41.8156049

69.32 54.48

48.24303 44.55556

521.1741123 520.0755775

638.7371446 619.1111382

23.6778345 40.83578157

3-3072-2

491.1758

588.9223

19.9005122

33.47804

43.80644

514.7366935

592.0212

20.53142144

37.1934

47.13926

506.8751973

591.2078575

20.36583592

3-3072-3 3-4072-1

530.6719 598.1137

660.2872 963.526

24.4247528 61.0941197

125.4902 58.4064

160.9232 63.94191

661.4651087 815.8952302

947.8785 938.2435

78.61855634 56.86708786

55.2588 66.5024

66.14821 74.72619

759.17082 630.936614

880.5778277 772.1652027

65.93639642 29.10006955

3-4072-2

587.0095

1002.457

70.7735054

35.828

41.42936

698.449658

775.707

32.14556494

32.3496

41.71852

684.0336258

758.1017425

29.14641799

3-4072-3 3-5072-1

628.0276 633.5366

1192.195 1295.355

89.8315615 104.464051

40.1936 144.33333

49.23419 149.9233

715.5127231 618.19695

804.9405 912.4535

28.16960817 44.02538638

38.1552 44.2486

43.80209 53.16175

705.946144 710.2026627

787.9034346 807.6130127

25.45681665 27.47693072

3-5072-2

637.6106

1445.356

126.683182

123.44

140.9386

768.104922

1032.484

61.93010279

55.5014

64.85354

665.61599

785.9709321

23.26817216

3-7573-1

739.2436

1681.247

127.428009

95.4902

95.47795

817.539714

1008.508

36.42429384

82.5895

79.94148

814.2217206

976.7527034

32.1286655

3-7573-2

837.5946

1965.456

134.654844

101.6906

96.74454

927.2441472

1125.679

34.39428696

97.4907

90.43441

912.8918828

1100.816988

31.42598911

3-10073-1

934.1016

2600.236

178.367535

310.174

295.3227

910.84262

1516.339

62.33130742

244.754

323.3123

911.8261995

1479.892472

58.42949755

3-10073-2

915.4921

2830.256

209.151362

196.972

210.3439

1004.92761

1412.243

54.26058728

188.4879

200.3674

1019.820835

1408.676135

35.01046286

Max. Optimality Gap (%) Avg. Optimality Gap (%)

209.151362

78.61855634

65.93639642

71.1527777

40.36408529

30.09466586

25

Table 4B Detailed costs for the third scenario problems (The model with second objective function) *Optimality Gap%   CPLEX upper bound  CPLEX lower bound  / CPLEX lower bound  100 ** Optimality Gap% 

 total cost of

LB

heuristic  CPLEX lower bound  / CPLEX lower bound  100

CPLEX bounds UB *Optimality

Holding cost

Lost sale cost

SA-MTAC Transportation cost

Total

**Optimality

39.177

47.20862

406.6209307

493.0066

Gap (%) 22.79588193

Holding cost

Lost sale cost

39.177

47.20862

SA-TS-MTAC Transportation cost

Total

**Optimality

406.6209307

493.0065554

Gap (%) 22.79588193

3-2072-1

401.4846

463.3035

Gap (%) 15.3975769

3-2072-2

385.5608

477.7824

23.9188216

23.174

22.84268

488.473932

534.4906

38.62680355

21.614

26.90456

481.814357

530.3329122

37.54845207

3-2072-3 3-2572-1

408.4223 448.2363

444.6902 548.1383

8.87999994 22.2877978

26.57899 46.67837

38.31647 53.51445

431.7196491 489.76096

496.6151 589.9538

21.59353481 31.61668981

25.62 49.68932

29.97451 58.26786

434.3657705 515.184564

489.9602808 623.1417461

19.96413536 39.02081248

3-2572-2

485.2115

763.012

57.2534864

26.68909

23.18513

557.9595

607.8337

25.27191076

43.89019

51.44699

486.9397765

582.2769522

20.00477156

3-2572-3 3-3072-1

489.9737 442.9348

594.8484 623.7857

21.4041488 40.8301402

45.7898 53.11

54.21513 55.41184

739.8904685 541.840084

839.8954 650.3619

71.41642564 46.83017004

48.55 51.79958

67.63115 54.27859

506.8663798 545.87285

623.0475298 651.9510225

27.15938219 47.18893673

3-3072-2

482.7557

884.0507

83.1258958

29.9038

38.98006

541.1614321

610.0453

26.36728849

35.82

40.62024

540.659321

617.0995564

27.82853861

3-3072-3 3-4072-1

514.3885 591.7316

770.5179 1512.3422

49.7929872 155.579083

134.5679 89.7856

147.2179 92.97556

649.5184 612.369944

931.3042 795.1311

81.05074627 34.37360858

55.7988 84.9856

76.04821 108.9976

759.7817082 604.369944

891.6287159 798.3531

73.33760687 34.91811152

3-4072-2

582.3546

892.35782

53.2327245

40.3536

51.23419

702.127231

793.715

36.2941105

53.2352

50.83354

668.87749

772.9462328

32.72776291

3-4072-3 3-5072-1

615.2378 621.2165

834.2897 1198.329

35.6044281 92.9003818

49.57936 136.8333

45.23419 185.3815

797.527231 519.976614

892.3408 842.1914

45.03997999 35.57132615

38.89152 136.8333

43.80906 185.3815

707.946144 519.976614

790.646728 842.1914473

28.51075275 35.57132615

3-5072-2

634.7069

1396.3629

120.00121

115.1844

142.1939

769.604922

1026.983

61.80431818

57.38014

68.85421

672.8761599

799.1105139

25.90228874

3-7573-1

727.4808

1643.3648

125.898033

163.88

163.9452

819.31215

1147.137

57.68627447

163.88

163.9452

819.31215

1147.137371

57.68627447

3-7573-2

834.3862

2355.3648

182.287123

91.2966

98.79814

904.941472

1095.036

31.2385338

87.6707

110.8434

893.918828

1092.432969

30.92653839

26

3-10073-1

922.6368

2900.253

214.343954

196.772

190.3439

1018.12761

1405.243

52.30733155

215.07

168.9775

1021.845762

1405.893281

52.37775915

3-10073-2

916.4528

2864.3892

212.551743

238.17

247.6504

1498.661017

1984.481

116.5393981

128.2807

147.977

1358.322824

1634.580499

78.35948553

Max. Optimality Gap (%)

214.343954

116.5393981

78.35948553

Avg. Optimality Gap (%)

84.182752

46.46801848

38.4349343

27

Table 5A CPU Time of Problems in Scenarios (The model with first objective function) Problems SASA-TS- Problems SASA-TSProblems MTAC MTAC MTAC MTAC 1-0552-1 3.637299 2.869922 2-0552-1 1.186414 2.94362195 3-2072-1

SAMTAC 11.65345

SA-TSMTAC 31.3410217

1-0552-2

4.64264

3.861535

2-0552-2

1.211732

3.1535347

3-2072-2

9.764207

38.1860464

1-0552-3

3.87274

4.15878

2-0552-3

2.173911

3.4825878

3-2072-3

13.80086

37.335387

1-0572-1

4.82734

5.930935

2-0572-1

2.880102

6.4029350

3-2572-1

24.77431

40.28005

1-0572-2

7.14376

4.053665

2-0572-2

2.644711

5.0536652

3-2572-2

25.52421

39.3078165

1-0572-3

6.75370

6.670697

2-0572-3

2.937630

6.326707

3-2572-3

20.32724

38.1792792

1-1052-1

4.1479

5.164786

2-1052-1

2.718892

6.1647861

3-3072-1

23.12307

65.2027658

1-1052-2

5.2659

5.07187

2-1052-2

3.883769

5.1869717

3-3072-2

30.38811

68.6796717

1-1052-3

5.96308

5.932234

2-1052-3

3.001306

5.4932234

3-3072-3

24.62145

63.69336

1-1072-1

7.13326

13.48429

2-1072-1

3.204409

14.429019

3-4072-1

64.13213

128.609323

1-1072-2

7.08946

11.99494

2-1072-2

4.633342

13.949425

3-4072-2

59.71561

132.674941

1-1072-3

6.94610

11.77068

2-1072-3

3.939917

11.706836

3-4072-3

71.17567

123.269311

1-1552-1

7.94885

15.74080

2-1552-1

5.153067

15.574080

3-5072-1

55.17393

838.438244

1-1552-2

10.2135

13.61853

2-1552-2

7.009839

13.118525

3-5072-2

68.55044

786.002970

1-1552-3

9.23294

12.55925

2-1552-3

5.799907

12.924956

3-7573-1

298.1277

985.165491

1-1572-1

20.5023

33.8403

2-1572-1

10.50857

40.87403

3-7573-2

486.8835

1128.0762

1-1572-2

9.00479

40.74885

2-1572-2

8.891952

35.488497

3-10073-1

402.8773

1103.3847

1-1572-3

13.6805

36.08583

2-1572-3

7.200438

31.858277

3-10073-2

602.0730

1155.0918

Table 5B CPU Time of Problems in Scenarios (The model with second objective function) Problems SASA-TSProblems SASA-TS- Problems MTAC MTAC MTAC MTAC

SAMTAC

SA-TSMTAC

1-0552-1

6.28905

4.1482588

2-0552-1

2.1765

1.987946

3-2072-1

17.36924

36.27949

1-0552-2

6.1249691

3.9448268

2-0552-2

2.17848

1.49874

3-2072-2

32.47295

48.48204

1-0552-3

2.789939

2.7209404

2-0552-3

2.628967

2.8743

3-2072-3

28.37475

45.2472

1-0572-1

7.45885

5.326707

2-0572-1

3.47178

3.3958

3-2572-1

32.372469

53.8245

1-0572-2

4.67955

4.9742166

2-0572-2

4.7897

5.48922

3-2572-2

30.36478

46.94022

1-0572-3

3.796479

4.1645572

2-0572-3

3.48456

4.379402

3-2572-3

28.391645

51.9536

1-1052-1

7.78945

6.7493223

2-1052-1

5.1757836

8.428754

3-3072-1

39.3489

74.47895

1-1052-2

7.236804

6.373683

2-1052-2

6.7968

10.46274

3-3072-2

53.2846

92.24785

1-1052-3

6.10087

7.3973683

2-1052-3

6.25415

9.294872

3-3072-3

67.382964

81.4275

1-1072-1

11.27629

13.44290

2-1072-1

5.8425

20.3795

3-4072-1

80.27389

138.4975

1-1072-2

10.4589

12.99494

2-1072-2

6.35785

8.38745

3-4072-2

108.3749

163.9526

1-1072-3

9.167843

13.087068

2-1072-3

6.18736

12.75934

3-4072-3

96.72499

164.2964

1-1552-1

15.24648

15.77408

2-1552-1

6.876002

13.2468

3-5072-1

69.38463

163.2495

1-1552-2

13.04896

14.11852

2-1552-2

8.456264

17.1480

3-5072-2

72.37047

186.2964

1-1552-3

9.45633

13.92496

2-1552-3

7.372844

16.4792

3-7573-1

367.3694

386.3275

1-1572-1

25.34562

50.47403

2-1572-1

8.18302

30.483

3-7573-2

334.4759

433.3328

1-1572-2

26.89100

35.74885

2-1572-2

10.62849

25.6489

634.3287

1121.63

1-1572-3

25.55457

34.74899

2-1572-3

6.423075

28.6986

3-100731 3-100732

592.3247

1304.068

28

Research Highlight     

We present a mathematical model of inventory routing problem for perishable goods. The lost sale cost is integrated in the inventory routing problem. The demand for perishable goods is assumed to be a function of the inventory age. The model is linearized in order to make it solvable using CPLEX. A hybrid meta-heuristic algorithm based on SA-TS is devised to solve larger instances.

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