A HYBRID SIMULATED ANNEALING ALGORITHM TO ESTIMATE A BETTER UPPER BOUND OF THE MINIMAL TOTAL COST OF A TRANSPORTATION PROBLEM WITH VARYING DEMANDS AND SUPPLIES

. Minimizing the total cost of transportation of a homogeneous product from multiple sources to multiple destinations when demand at each source and supply at each destination are deterministic and constant is commonly addressed in the literature. However, in practice, demand and supply may fluctuate within a certain range due to variations of the global economy. Subsequently, finding the upper bound of the minimal total cost of this transportation problem with varying demands and supplies (TPVDS) is NP hard. Yet, bounding the minimal total cost is of prime importance for financial sustainability. Although the lower bound of the minimal total cost can be methodologically attained, determining the exact upper bound is challenging. Herein, we demonstrate that existing methods may underestimate this upper minimal total cost bound. We therefore propose an alternative efficient and robust method that is based on the hybridization of simulated annealing and steepest descent. We provide theoretical evidence of its good performance in terms of solution quality and prove its superiority in comparison to existing techniques. We further validate its performance on benchmark and newly generated instances. Finally, we exemplify its utility on a real-world TPVDS.


Introduction
Minimizing the total cost of transporting a homogeneous product from multiple sources (e.g., warehouses) to multiple destinations (e.g., supermarkets) subject to supply availability and demand satisfaction is a wellestablished transportation problem (TP).When the supplies and demands are known and deterministic, TP is solved using a special type of simplex method, warm started from feasible initial solutions that are very easy to construct (e.g., obtained via Vogel's approximation method [6]) or are more sophisticated but efficient [5].However, in practice, the demand and supply of a product may vary, albeit within a certain range, in response to global economic or political events.By simply incorporating interrelated inventory costs incurred during transportation and at destinations into the unitary transport cost, TP oversimplifies the real-life problem.In fact, the inventory costs do also vary within a certain range.In such circumstances, transporters, clients, and suppliers are not merely interested in a deterministic estimate of the total transport cost but are keen in assessing the risks they may be incurring.That is, they wish to bind the minimal total cost so that they better plan their return on investment.
Despite its practical relevance, the TP with non-deterministic supply, demand and unit transport cost has received little attention [1,11].Kaur and Kumar [8] approximately solved a TP with fixed demands and supplies but uncertain imprecise values of transportation cost.Their algorithm represented the transportation costs by generalized trapezoidal fuzzy numbers.Its complexity was later further enhanced [2].Liu et al. [10] approximated the minimal transport cost of solids using a fuzzy simulation-based tabu search.Liu [9] proposed a heuristic  L that calculates the minimal total cost bounds  L and  L of the transportation problem with demand and supply quantities varying within their respective ranges using a pair of mathematical programs.Juman and Hoque [4] demonstrated that  L may, in some instances, fail to identify a correct upper minimal total cost bound.They further extended the transportation problem to include the interrelated inventory costs during transportation and at destinations.They then developed heuristic  JH that finds a lower bound  JH and an upper bound  JH to the minimal total cost of their extended model.For all tested instances,  JH =  L .For small and medium sized instances,  JH ≤  L .This advantage did not however extend to large sized instances.
In this paper, we consider a TP with varying demands and supplies including inventories (TPVDSII) where the objective is to minimize total cost.Because the supply and demand vary within a predefined range, the problem reduces to estimating the lower bound  and upper bound  of the minimal total cost.Although  can be found methodologically, determining  is challenging: It involves an NP-hard problem.Estimating  heuristically is possible but may have short comings as in the cases of  L and  JH .In this paper, we overcome this shortcoming of worst-case performance by tightening  JH .We propose a new efficient and robust method that either matches or improves existing upper bounds.It incorporates more combinatorial choices of supply and demand than  JH while it avoids the exhaustive enumeration of the ( + )! ∏︀  =1 (∆  + 1) ∏︀  =1 (∆  + 1) potential combinations of choices of supplies and demands within their respective ranges of the TPVDSII, where  and  are the numbers of suppliers and buyers respectively, ∆  is the range of the supply quantity of the th supplier, and ∆  is the range of the demand quantity of the th buyer.In a real-life yoghurts' distribution problem [7], the number of combinations is 1.678110 157 .Enumerating all these supply-demand scenarios to estimate the exact upper bound cost is impractical.So, the development of a streamline method to tackle these choices of supply-demand would be challenging.Our new approach, labelled hereafter , incorporates more combinatorial choices of supply and demand than  JH .
This paper addresses a pertinent real-life supply chain logistics' problem that occurs in many circumstances, such as the distribution of yoghurt, tea, raw material, etc.It proves that existing methods  L and  JH underestimate the upper bound , and propose a better near-optimal estimation of the worst-case realization on the least aggregated expenses of the TPVDSII.It applies the estimation technique  to a real-world yoghurt show case.It further undertakes an extensive numerical experiment using benchmark and newly generated random instances to assess the performance of .
Section 2 describes the problem.Section 3 details  JH and illustrates its shortcoming.Section 4 details the new efficient robust method  that finds a tighter upper minimal total cost bound.Section 5 compares the performance of  to existing methods.Section 6 concludes the paper and gives potential extensions.

Problem definition
TPVDSII is a transportation problem such that the following assumptions apply.
(1) A homogeneous product is transported from  suppliers to  buyers.
(2) The demand d of buyer ,  = 1, . . ., , for the product is deterministic and integer but variable over time within a bounded interval: Consequently, total demand is integer and varies over time in a bounded interval: (3) The supply ŝ of supplier ,  = 1, . . ., , for the product is deterministic and integer but variable over time within the a bounded interval: ŝ ∈ [︀   ,   ]︀ .
Consequently, the total supply is integer and varies over time in a bounded interval: (5) Each buyer ,  = 1, . . ., , has enough storage capacity to accommodate the required inventory.The unit inventory holding cost per period at buyer ,  = 1, . . ., , is ℎ  .(6) A transport vehicle is available to transport the required shipment quantities.The unit transportation cost and time of delivery from a supplier ,  = 1, . . ., , to a buyer ,  = 1, . . ., , are   and   , respectively.(7) The quantity   shipped from a supplier ,  = 1, . . ., , to a buyer ,  = 1, . . ., , reaches  within the same period.
Using the above notation and assumptions, we model TPVDSII using the integer decision variables   , which denote the quantities shipped from supplier ,  = 1, . . ., , to buyer ,  = 1, . . ., . where The model ( 1)-( 4) corresponds to an integer linear program.It has at least one feasible solution if the sum of supply quantities exceeds the sum of demand quantities: It can be solved as a linear program with at least one of its optimal solutions being integer when ŝ ,  = 1, . . ., , d ,  = 1, . . ., , and cost coefficients   are integer.In such a case, the model ( 1)-( 4) can be easily solved using the simplex method for transportation problems.
To obtain an upper bound  to the minimal total cost z, we treat the supply ŝ ,  = 1, . . ., , and demand d ,  = 1, . . ., , as decision variables that vary within their respective ranges. is therefore the solution value of the linear program Equation ( 1)-( 4) For small-sized instances and ranges of supply and demand, the model ( 1)-( 6) can be solved using a linear programming off-the-shelf solver such as LINGO.However, this becomes challenging as the problem size increases.
The application of  JH to the large-scale problem 4 of Table 3 in Juman and Hoque [4] yields )︀ .When the problem size is large, LINGO neither converges to the optimum  nor proves its optimality.For those TPVDS instances, an efficient and robust heuristic to find a tighter upper minimal total cost bound is a viable alternative.

An efficient robust heuristic to estimate the upper minimal total cost bound
The proposed heuristic  starts from the solution (x, ŝ, d) 0 = (x, ŝ, d) JH and initial solution value  0 =  JH .Its iterative steps proceed depending on which of the three cases i-iii occurs.

When
∑︀  =1 ŝ = ∑︀  =1 d and  0 =  JH ,  considers two actions.First, it keeps all   ,  = 1, . . ., , fixed but increases one   value and decreases another   ′ by the same amount such that the equality ∑︀  =1 ŝ = ∑︀  =1 d still holds.Second, it keeps all   ,  = 1, . . ., , fixed but increases one   value and decreases another   ′ value by the same amount so that the equality ∑︀  =1 ŝ = ∑︀  =1 d still holds.Either way, the resulting feasible solutions may have a minimal total cost greater than  0 .Algorithms 1 and 2 address each of these two actions, respectively.

When
∑︀  =1 ŝ < ∑︀  =1 d and  0 =  JH ,  undertakes three actions, all of which maintain the inequality.The first keeps all   ,  = 1, . . ., , fixed but increases one   value and decreases another   ′ by the same amount.The second keeps all   ,  = 1, . . ., , fixed but increases one   value and decreases another   ′ value by the same amount.The third keeps all   ,  = 1, . . ., , fixed but decreases one   value by one.Any of these actions may generate feasible solutions whose minimal total cost greater than  0 .Algorithms 1-3 address each of these respective actions.Decrease d by 1 and increase d ′ by 1. 9: Solve model ( 1)-( 4 values.For each , it calculates the minimal total cost   ( d ).It updates   to max{  ,   ( d )} and retains the values d  that correspond to   .It continues its iterations until either d =   or d =   .If   > , Algorithm 3 updates the upper bound  setting it to   .It then iterates over all possible  before it stops.The updated  value will be the upper minimal total cost bound.

Proposed approach
Our efficient and robust method  addresses the above three cases, using Algorithm 4, which is a steepest descent (SD) that estimates an upper bound to the minimal total cost bound.
The following theorem highlights the good quality of the upper bound  in comparison to  JH .
Theorem 4.1.Algorithm 4 yields an upper bound estimate  that is at least as good as or better than  JH .
Proof.Algorithm 4 uses  JH to generate its initial solution (x, ŝ, d) 0 and initial bound  0 ; in fact, it sets  =  0 and (x, ŝ, d) = (x, ŝ, d) 0 , where (x, ŝ, d) 0 and  0 are the output of  JH .At each iteration, it generates new solutions by varying the supply and demand within their respective ranges, calculating the resulting total cost and updating  whenever a higher cost solution value is encountered.Thus,  ≥  0 .
Algorithm 4 searches for an upper bound of the total transport and inventory cost.It improves existing bounds of benchmark instances.Yet, it is an iterative SD; thus, has a limited capability of diversifying the Algorithm 2. Iterating over supply for case i. Set  ′ =  ′ + 1. 5: end if 6:  = 1.7: if ŝ ̸ =   and ŝ ′ ̸ =   then 8: Decrease ŝ by 1 and increase ŝ ′ by 1. 9: Solve model ( 1)-( 4 search.As such, it is coupled with a stochastic exploration that allows it to mimic the behaviour of a simulated annealing (SA).Specifically, we generate solutions using SD and subject them to SA, which estimates the upper minimal total cost bound of the TPVDSII.Algorithm 5 details the steps of the proposed approach, labelled .

Computational results
We compare our proposed approach  against  L and  JH for both small and large size instances, and for a real-world application.The methods are coded in MATLAB and run on a personal computer with Intel Core (TM) 2 Duo CPU T7700, 2.40 GHz, and RAM 1.99 GB.

Small sized problems
First, we compare  to  JH and  L on fifteen small sized numerical problems: problems 1-7 are randomly generated new instances whereas problems 8-15 are benchmark instances [4].Data for problems 1-5 are given in Table  Table 3 compares (i) the bounds  to  JH and  L and (ii) the run time RT needed to calculate  to the run times RT L and RT JH needed to compute  L and  JH , respectively, where all run times are in seconds.The last two columns of Table 3 report the percent improvement ∆ * brought  with respect to heuristic *, where * =  L ,  JH and ∆ * = −* * 100%.Table 3 and specifically its last two columns indicate that  improves the upper bounds obtained by  L and  JH by as much as 94.90% and 90.50%, respectively.This improvement occurs for problem 3, where both Algorithm 3. Iterating over supply for case ii.Decrease ŝ by 1.

5:
Solve model ( 1)-( 4  L and  JH underestimate , most likely because  L and  JH miss the scenarios that cause such high total costs during their partial enumeration.This improvement occurs at a slighter increase of run time.However, the difference, which is of the order of 0.35 s, is negligible compared to the sizeable impact of a better estimation of the upper bound.For all tested instances, LINGO obtains the same solution value as  when allocated 3600 s of run time.Algorithm 5. Heuristic . 1: Input: A problem instance.2: Set  = 0. 3: Apply SD to obtain an initial solution (x, ŝ, d) 0 and its optimal solution value (x  ).4: Set the initial temperature  0 = 0.9, the cooling factor  = 0.5, k the length of the SA plateau, and  the number of plateaus.5: Set  =  0 , the best feasible solution x * = x 0 , and its cost value  * = (x 0 ).6: while  ≤ k do 7: while  ≤  do 8: while  ≤  do 9: Set )  ⌉, where   Uniform (0, 1).10: Set )  ⌉, where   Uniform (0, 1).11: Substitute  (+1)  and  (+1)  in equations ( 1)-( 4)-and obtain x (  + 1) and (x (+1) ).

Large sized problems
Second, we compare  to  JH and  L on ten large sized problems: problems 1-6 are randomly generated new instances whereas problems 7-10 are benchmark instances [4].Data for problems 1-6 are given in Table 4. Table 5 reports the same results as Table 3 but for large instances.
Table 5 suggests that the upper minimal total cost bounds to each of problems 8-10 found by all three methods are identical, whereas  outperforms  JH for problems 1-7 and  L for problems 1-4.Moreover, for the large size instances, the largest percent increases ∆ L and ∆ JH , which are 39.90% and 41.70%, are smaller than for the first set of instances, where they reached 94.90% and 90.50%, respectively.This is most likely because the search space is much larger and reaching areas that contain the global maximum bound becomes more challenging.The average CPU time of , over the ten large size instances, is almost twice as large as the average run time of  JH , further indicating the size of the search space.It is indeed generally difficult for any exact optimal technique to find the exact upper minimal total cost bound for large size instance of the TPVDS even when allocated hours of run time.

A real world case
To evaluate the performance of our proposed algorithm , we solve a real-world yoghurt distribution problem [7].The yoghurt factory has two warehouses located at Ibbankatuwa and Dambulla, Sri Lanka.Yoghurt products are distributed island wide to all 25 distribution centres.Table 6 reports the total costs and run times of the solutions of ,  L , and  JH along with the best known bound  best reported in Juman et al. [7] and its run time RT best .
For this real-world problem, the new proposed method H finds the largest near-optimal upper minimal total cost bound in comparison  L and  JH .It further improves  best .This improvement occurs at the small CPU time of 78.6 s.

Sensitivity analysis
Sensitivity analysis helps us understand the impact of variations  in the demand and variations  in supply range on the upper minimal total cost bound .We undertake such analysis on the real-world yoghurt distribution problem, where a range of 0 indicates no variation.Figure 1 illustrates the behavior of the upper bound

Conclusion
In this paper, we investigated the transportation problem with varying demands and supplies (TPVDS).Although, the lower bound of the minimal total costs can be found methodologically, finding the upper minimal total cost bound to the TPVDSII (or TPVDS) within their respective ranges becomes an NP-hard problem.Recent models include transportation and inventory costs at destinations, as they are interrelated factors.For such models, two heuristic techniques calculate the lower and the upper minimal total cost bounds.Herein, we propose a method that matches existing bounds or improves them.Better upper minimal total cost bounds to TPVDS are invaluable to decision makers particularly at a time of fluctuating demand and logistics disruptions.Similarly, the highlighted impact of increments in the demand ranges and supply ranges on the upper minimal total cost bound are of particular importance.
In developing the new method, the demand is assumed to be variable.However, variation in the demand may follow a particular trend.Hence future research might be carried out in extending the extended model, to include an appropriate trend of demand distribution.We intend to devote ourselves in this direction in our future research.

Figure 1 .
Figure 1.Upper bound cost versus increment  of demand range.
Figure 2 reflect the same behavior when the supply range increases, specifically when  increases from 0 to 40.It confirms that the wider the range the more flexibility the supply chain has in channeling the flow; thus, the lower costs.

Figure 2 .
Figure 2. Upper bound cost versus increment  of supply range.

Table 1 .
Three example where  JH underestimates .
the minimal total cost   ′ ( d , d ′ ).It updates   ′ setting it to max{  ′ ,   ′ ( d , d ′ )} and retains the values ( d , d ′ ) that correspond to   ′ .It continues its iterations until either d =   or d ′ =   ′ .If   ′ > , then Algorithm 1 updates the upper bound  setting it to   ′ .Algorithm 1 then iterates over all possible (,  ′ ) combinations.It stops when it has considered all pairs (,  ′ ) of buyers.The updated  value will be the upper minimal total cost bound.Algorithm 2 considers consecutive pairs (,  ′ ) of vendors.

Table 2 .
Data for small-sized instances.

Table 3 .
Comparing upper bounds and run times of ,  L and  JH for small instances.

Table 4 .
Data for large-sized instances.

Table 6 .
Comparative upper bound solutions of a real-world TPVDS.