Balancing Common Treatment and Epidemic Control in Medical Procurement during COVID-19: Transform-and-Divide Evolutionary Optimization
Yu-Jun Zheng, Xin Chen, Tie-Er Gan, Min-Xia Zhang, Wei-Guo Sheng, Ling Wang
11 Balancing Common Treatment and EpidemicControl in Medical Procurement during COVID-19:Transform-and-Divide Evolutionary Optimization
Yu-Jun Zheng,
Senior Member, IEEE,
Xin Chen, Tie-Er Gan, Min-Xia Zhang, Wei-Guo Sheng,
Member, IEEE, and Ling Wang
Abstract —Balancing common disease treatment and epidemiccontrol is a key objective of medical supplies procurement inhospitals during a pandemic such as COVID-19. This problemcan be formulated as a bi-objective optimization problem forsimultaneously optimizing the effects of common disease treat-ment and epidemic control. However, due to the large number ofsupplies, difficulties in evaluating the effects, and the strict budgetconstraint, it is difficult for existing evolutionary multiobjectivealgorithms to efficiently approximate the Pareto front of theproblem. In this paper, we present an approach that firsttransforms the original high-dimensional, constrained multiob-jective optimization problem to a low-dimensional, unconstrainedmultiobjective optimization problem, and then evaluates eachsolution to the transformed problem by solving a set of simplesingle-objective optimization subproblems, such that the problemcan be efficiently solved by existing evolutionary multiobjectivealgorithms. We applied the transform-and-divide evolutionaryoptimization approach to six hospitals in Zhejiang Province,China, during the peak of COVID-19. Results showed that theproposed approach exhibits significantly better performance thanthat of directly solving the original problem. Our study hasalso shown that transform-and-divide evolutionary optimizationbased on problem-specific knowledge can be an efficient solutionapproach to many other complex problems and, therefore,enlarge the application field of evolutionary algorithms.
Index Terms —Multiobjective optimization, evolutionary al-gorithms, medical supplies procurement, epidemic control,transform-and-divide.
I. I
NTRODUCTION T O mount an effective response to COVID-19, hospitalsmust procure medical supplies for epidemic control.However, the total budget of any hospital is limited: if a hospi-tal procures too many supplies for epidemic control, it has toreduce supplies for common disease treatment, which woulddamage its medical services. Consequently, it is importantfor a hospital to balance between common disease treatmentand epidemic control in medical supplies procurement. Theproblem of determining the purchase quantity of each supplycan be formulated as a bi-objective optimization problem for
Y.J. Zheng, X. Chen, and W.G. Sheng are with the School of InformationScience and Engineering, Hangzhou Normal University, Hangzhou 311121,China (e-mail: [email protected]; [email protected]).T.E. Gan is with the First Affiliated Hospital, Zhejiang Chinese MedicalUniversity, Hangzhou 310006, China (e-mail: [email protected]).M.X. Zhang is with the College of Computer Science & Technol-ogy, Zhejiang University of Technology, Hangzhou 310023, China (e-mail:[email protected]).L. Wang is with the Department of Automation, TSinghua University,Beijing 100084, China (e-mail:[email protected]). simultaneously optimizing the effect of epidemic control andeffect of common disease treatment. There are three mainchallenges to solving this problem. The first is to evaluate theeffects of epidemic control and common disease treatment ina relatively accurate manner. The second is to meet the budgetconstraint, which is often strict in the pandemic. The third isto approximate the Pareto front of the problem in an efficientmanner. For the first challenge, we develop a procedure tosimulate the arrival and treatment of cases of infection andcases of common diseases according to a general principleof disease treatment and medical supplies usage. However,this also makes the objective functions expensive. Moreover,a major hospital often involves tens of thousands of medicalsupplies, which makes the dimension of the solution space toohigh. The combination of these reasons makes the problemvery difficult to solve.Decomposition is a general approach to solving largecomplex problems that are beyond the reach of standardtechniques. Decomposition in optimization appears in earlywork on large-scale linear programming problems from the1960s [1]. Many problems with separable objective functionsare trivial to solve by mathematical methods. For additivelydecomposed functions that are not able to optimize by standardgenetic algorithms, M¨uhlenbein and Mahnig [2] proposed thefactorized distribution algorithm that factors the distributioninto conditional and marginal distributions based on functionstructures. Strasser et al. [3] proposed factored evolutionaryalgorithms, which factors an optimization problem by cre-ating overlapping subpopulations that optimize over subsetsof variables. Many combinatorial optimization problems canbe solved by using efficient methods to solve subproblemsand combining the results to obtain solutions to the originalproblems. Zheng and Xue [4] utilized these characteristicsto automatically derive efficient problem-solving algorithms,including evolutionary algorithms (EAs) that are mainly usedfor NP-hard problems. Unfortunately, the medical suppliesprocurement problem considered in this paper does not satisfythe basic conditions of decomposition, because it is quitecommon that one supply can be used in multiple diseases,and the treatment of a disease can involve many supplies.For complex problems whose subcomponents interact andaffect each other, Potter and De Jong [5] proposed a coop-erative coevolution architecture that decomposes a probleminto subcomponents, which are then evolved as a collectionof cooperating species. Different decomposition methods have a r X i v : . [ c s . N E ] A ug different performance in cooperative coevolution for differentproblems. Yang et al. [6] proposed a cooperative coevolutionframework that uses random grouping and adaptive weightingin problem decomposition and coevolution for optimizinglarge nonseparable problems. Chandra [7] presented a com-petitive cooperative coevolution method for training recurrentneural networks, where different decomposition methods com-pete with different features they have in terms of diversityand degree of non-separability. To capture the interdependencyamong variables, Hu et al. [8] proposed a fast interdepen-dency identification algorithm, which avoids obtaining thefull interdependency information of nonseparable variables forcooperative coevolution. Yang et al. [9] proposed a data-drivenapproach, which exploits historical data to mine the evolutionconsistency among variables for dynamic variable grouping.Gomes et al. [10] extended cooperative coevolution with oper-ators that put the number of coevolving subpopulations underevolutionary control. Wang et al. [11] proposed a formula-based grouping strategy for grouping variables directly basedon the separable and nonseparable operations in the formula ofan objective function. Mahdavi et al. [12] proposed an incre-mental cooperative coevolution algorithm in which subcompo-nents are dynamically added to an integrated subcomponentbeing evolved. For multimodal optimization problems, Penget al. [13] proposed a method that concurrently searches formultiple optima as informative representatives to be exchangedamong subcomponents for compensation in coevolution. Fordynamic optimization problems, Peng et al. [14] used multiplepopulations in cooperative coevolution to compensate infor-mation in dynamic landscapes. Zhang et al. [15] proposed adynamic cooperative coevolution framework, which allocatescomputational resources to elitist subcomponents with superiorvariables. Although cooperative coevolution has shown itsefficiency in solving many engineering optimization problems[16]–[20], for most nonseparable problems, decompositioncauses the loss of a great deal of information, and thealgorithms easily gravitate towards sub-optima represented byNash equilibria rather than global optima [21].A multiobjective optimization problem is much more diffi-cult than its single-objective counterpart, and decompositionis also a basic strategy in multiobjective optimization. Zhangand Li [22] proposed a multiobjective evolutionary algorithmsbased on decomposition (MOEA/D), which decomposes amultiobjective optimization problem into a set of single-objective optimization subproblems using decomposition ap-proaches such as weighted sum, weighted Tchebycheff, andpenalty-based boundary interaction. However, for some prob-lems, these decomposition approaches may not be suitablefor balancing the diversity and convergence. Wang et al. [23]revolved this difficulty by imposing a constraint to an uncon-strained subproblem, where the improvement region of eachsubproblem is determined by an adaptive control parameter.MOEA/D makes an assumption that two neighboring sub-problems have similar optimal solutions, but some problemsdo not satisfy this assumption. Mei et al. [24] proposed adecomposition-based memetic algorithm with neighborhoodsearch for multiobjective capacitated arc routing problem,which combines decomposition-based and domination-based techniques for solution selection. An EA proposed by Cai etal. [25] also combined domination-based sorting and decom-position, the former for evolving an internal population andthe latter for maintaining an external archive. Jan and Zhang[26] introduced a penalty function to MOEA/D for multi-objective constrained optimization. Konstantinidis and Yang[27] adapted MOEA/D to solve a K -connected deploymentand power assignment problem by introducing a problem-specific repair heuristic for infeasible solutions. Zhang et al.[28] extended MOEA/D for big optimization problems byembedding a gradient-based local search. Chen et al. [29]extended MOEA/D by assigning each subproblem with anupper bound vector based on (cid:15) -constraint for constrainedoptimization. Qiao et al. equipped MOEA/D with an angle-based adaptive penalty scheme. Fang et al. [30] proposedcoordinate transformation in the objective space to acceleratethe convergence process of multiobjective EA. Jiang et al.[31] studied the effect of scalarizing functions and presentedtwo new functions for improving decomposition in MOEA/D.There have been many other extensions and applications ofMOEA/D in recent years [32]. The cooperative coevolutionarchitecture that decomposes a problem in the decision spacehas also been extended for multiobjective optimization [33]–[36]. To our knowledge, there is only one study, by He etel. [37], combining decomposition in both the decision spaceand objective space, which is similar to our work. The keydifference is that our method utilizes problem-specific knowl-edge to ensure the separability of the transformed problem.Unfortunately, we found that, although using decomposition-based strategies in MOEA/D and other similar algorithms[38]–[42] can reduce the complexity to a certain degree, theperformance of those algorithms is still far from satisfactory insolving the medical supplies procurement problem in practice.In this study, we present a transform-and-divide approach toefficiently solve the problem. First, we transform the originalproblem of determining the purchase quantity of each supplyto a new problem of distributing the budget to epidemic controland all common diseases. In our case studies, the dimension ofthe transformed problem is only one to two percent of that ofthe original problem. However, evaluating each solution to thetransformed problem is itself a nontrivial optimization prob-lem. Second, we divide the evaluation problem into a set oflow-dimensional, single-objective optimization subproblems.We propose a hybrid evolutionary optimization approach,which employs a multiobjective EA to evolve a populationof main solutions to the transformed problem and uses a tabusearch algorithm to solve the subproblems. During the peakof COVID-19, we applied the proposed approach to six hos-pitals in Zhejiang Province, China. Results demonstrated thatthe transform-and-divide evolutionary optimization approachexhibits significantly better performance than that of directlyusing multiobjective EAs to solve the original problem. Themain contributions of this paper are twofold: • We propose a transform-and-divide evolutionary op-timization approach to medical supplies procurementand demonstrate its practicability and efficiency duringCOVID-19. • We show that using problem-specific knowledge to trans-form and divide a complex optimization problem canlead to competitive EAs for the problem. This approachcan be extended to many other problems and enlarge theapplication field of EAs.The remainder of this paper is organized as follows. SectionII presents the medical supplies procurement problem. SectionIII simply describes how to directly use basic multiobjectiveEAs to solve the original problem. Section IV proposesthe transform-and-divide evolutionary optimization approach.section V presents the computational results. Section VI con-cludes with a discussion.II. P
ROBLEM D ESCRIPTION
A. Supplies for Epidemic Control and Common Treatment
We consider a medical supplies procurement problem for-mulated as follows. In a pandemic, a hospital plans to pro-cure medical supplies, including a set S = { S , S , . . . , S n } of n supplies for epidemic control, and a set S (cid:48) = { S n +1 , S n +2 , . . . , S n + n (cid:48) } of n (cid:48) supplies for normal diseasetreatment. For each supply S k , the current inventory is a k ,the unit price is c k , and the unit volume is v k . The problemis to determine the purchase quantity x k of each supply( ≤ k ≤ n + n (cid:48) ), such that the effects of epidemic controland normal disease treatment are simultaneously optimized.The supplies for epidemic control can be divided into twoclasses. The first class consists of supplies such as latex glovesand normal saline that must be used in the treatment of asuspected case of infection; we use Ψ to denote the setof these supplies, and use q ,k to denote the quantity ofeach S k ∈ Ψ required to treat a case. The second classconsists of supplies that are alternative in some treatmentitems. Table I presents six treatment items and their alternativesupplies used for COVID-19 control in this study. The sixsets of supplies are denoted by Ψ , Ψ , . . . , Ψ , respectively,and the quantity of each alternative S k ∈ Ψ j required totreat a case is denoted by q j,k . Different alternatives havedifferent treatment effects. The treatment effect of using eachalternative S k ∈ Ψ j is estimated as e j,k . For example, theeffects of peroxide, impermeable gown, and normal gownin “body protection” item are estimated as 1, 0.9 and 0.7,respectively. If we choose the S k j ∈ Ψ j for the j -th treatmentitem ( ≤ j ≤ ), the corresponding epidemic control effect onthe case is empirically estimated as: E ( k , ...k ) = (0 . e ,k +0 . e ,k ) e ,k (0 . e ,k +0 . e ,k ) e ,k (1)The hospital is capable of treating a set D = { D , D , . . . , D m } of m diseases. Similarly, for each disease D i , the set of supplies that must be used is denoted by Ψ i, ,and the set of supplies that are alternative in J i treatmentitems are denoted by Ψ i, , Ψ i, , . . . , Ψ i,J i , respectively. Thequantity of each S k ∈ Ψ i, required to treat a case is q i, ,k ,and the quantity of each alternative S k ∈ Ψ i,j required to treata case is q i,j,k . Different alternatives have different treatmenteffects. The treatment effect of using S k ∈ Ψ i,j is estimatedas e i,j,k . If we choose the S k j ∈ Ψ i,j for the j -th treatment item ( ≤ j ≤ J i ), the corresponding treatment effect on thecase is empirically estimated by a therapeutic effect function E i ( k , k , ..., k J i ) . Like Eq. (1), the typical expression of E i is a weighted sum or product of e i,j,k [43]. B. Number of Cases
Let T be the procurement decision cycle. In our case studies,the hospital procures medical supplies every 15 days. Thesupply quantities are determined based on the estimation ofthe number of hospital visits in the next decision cycle. Forthe number of cases of each disease D i , we estimate threevalues: the expected value r i , lower limit (optimistic value) r i , and upper limit (pessimistic value) r i . The values can beobtained based on historical morbidity data and environmentalinfluence factors [44]–[47].The number of suspected cases of epidemic infection isestimated based on the number of hospital visits of differentdiseases. For each disease D i , we estimate a probability p i that a patient of D i is a suspected case of COVID-19. Ingeneral, a disease having more similar symptoms with theepidemic has a higher p i . For example, an acute respiratoryinfectious disease has a high p i . For a disease (such as fracture)that is unrelated to the epidemic, we set p i to the currentincidence p e of infection (including suspected infection) inthe local region. We also estimate an average number r (cid:48) i ofaccompanying persons of a patient of D i ; in general, a criticaldisease has a large r (cid:48) i . The probability that an accompanyingperson of a patient of D i is a suspected case of COVID-19is p (cid:48) i , which is set to . p i if D i has similar symptoms withthe epidemic and p e otherwise. The total number of suspectedcases of infection in the next decision cycle is estimated asfollows (we use r i as we take a serious or pessimistic view ofepidemic control): r = m (cid:88) i =1 ( p i + p (cid:48) i r (cid:48) i ) r i (2) C. Objective Function Evaluation
A solution to the medical supplies procurement problemcan be represented by a ( n + n (cid:48) ) -dimensional vector x = { x , . . . , x n , x n +1 , . . . , x n + n (cid:48) } . The fitness of x is evaluated bytwo objective functions: (1) the epidemic control effect Υ( x ) ,which is the sum of treatment effects of all suspected casesof infection; (2) the common disease treatment effect Υ (cid:48) ( x ) ,which is the weighted sum of treatment effects of all commoncases, where the weight of each D i is w i . It is assumed that thearrival time of cases follows a uniform distribution. That is, foreach disease D i , as the expected number of cases in a decisioncycle of 15 days is r i , then there is a case arriving every (15 × h w ) /r i hours, while h w is the daily working hours (24for emergency diseases and 8 for non-emergency diseases inour study). Moreover, there is one suspected case of infectionin every / ( p i + p (cid:48) i r (cid:48) i ) cases of D i .A general principle of disease treatment is “focusing onthe current patient”, i.e., whenever a new case arrives, thephysician always chooses the most effective supply fromthe available alternatives, as he does not know how many TABLE IA
LTERNATIVE SUPPLIES FOR EPIDEMIC CONTROL IN THIS STUDY .Items Body protection Face protection Detection Oxygen therapy Antivirus DisinfectantSupplies protective clothing face shield nucleic acid kit high-flow nasal cannula α -interferon peroxideimpermeable gown N95 mask+goggle antibody kit nasal cannula lopinavir chlorine-containingnormal gown surgical mask+goggle oxygen mask chloroquine phosphate alcoholsarbidol cases would come later. Based on this first-come-first-serveddiscipline, we sort supplies in Ψ j ( ≤ j ≤ ) or Ψ i,j ( ≤ i ≤ m ; 1 ≤ j ≤ J i ) in nonincreasing order of e j,k or e i,j,k ,and simulate the arrival and treatment of all cases accordingto the procedure shown in Algorithm 1 to calculate the valuesof Υ( x ) and Υ (cid:48) ( x ) . In Algorithm 1, the boolean variable tr denotes whether the remaining supplies are capable of treatinga suspected case of epidemic infection, and tr i denotes whetherthe remaining supplies are capable of treating a case of D i . Algorithm 1:
Procedure for evaluating the effects ofepidemic control and disease treatment for the originalproblem. Initialize
Υ = 0 , tr = true ; for i = 1 to m do initialize Υ i = 0 , tr i = true ; ; for k = 1 to n + n (cid:48) do a k ← a k + x k ; ; Start timing simulation; while the decision cycle is not complete do if a new case of D i arrives and tr i = true then foreach S k ∈ Ψ i, do a k ← a k − q i, ,k ; if a k < q i, ,k then tr i ← false ; ; for j = 1 to J i do let S k be the first supply in Ψ i,j ; while a k < q i,j,k do Remove S k from Ψ i,j ; if Ψ i,j = ∅ then tr i ← false ; ; let k j = k ; a k ← a k − q i,j,k ; Υ i ← Υ i + E i ( k , . . . , k J i ) ; if the case is a suspected infected case and tr = true then foreach S k ∈ Ψ do a k ← a k − q ,k ; if a k < q ,k then tr ← false ; ; for j = 1 to do let S k be the first supply in Ψ j ; while a k < q j,k do Remove S k from Ψ j ; if Ψ j = ∅ then tr ← false ; ; a k ← a k − q j,k ; let k j = k ; Υ ← Υ + E ( k , . . . , k ) ; return Υ( x ) = Υ and Υ (cid:48) ( x ) = (cid:80) mi =1 w i Υ i . D. Constraints
A procurement solution x must satisfy problem constraints.First, the total procurement cost cannot exceed the budget C : n + n (cid:48) (cid:88) k =1 c k x k ≤ C (3)The hospital should perform its normal functions. In thisstudy, it is required that the hospital is able to treat r i (thelower number) cases of each disease D i . These constraintscan be tested by simulating the arrival and treatment of thelower numbers of cases in Algorithm 1: if any new case cannotbe treated, i.e., whenever the condition tr i = false (Line 6 ofAlgorithm 1) is triggered while the number of simulated casesof D i is less than r i , the constraint is violated.It is also required that the hospital is able to treat r suspected cases of infection. Whenever the condition tr = false (Line 18) in Algorithm 1 is triggered, the constraint is violated.III. B ASIC E VOLUTIONARY O PTIMIZATION M ETHODS
For the above ( n + n (cid:48) ) -dimensional, constrained bi-objectiveoptimization problem, we can use evolutionary constrainedmultiobjective algorithms to search for the Pareto optimalsolutions. The search range of each dimension k is [ x k , x k ] .The lower limit x k is set to the total quantity of S k requiredin non-alternative treatment items for all cases: x k = (cid:26) max(0 , rq ,k − a k ) , ≤ k ≤ n max(0 , (cid:80) mi =1 r i q i, ,k − a k ) , n +1 ≤ k ≤ n + n (cid:48) (4)The upper limit x k can be set to the total required quantityof S k under the assumption that S k is always chosen whenever S k is an alternative. That is, if ≤ k ≤ n , we set x k = (cid:0) (cid:88) j (cid:48) ∈{ j | ≤ j ≤ ∧ S k ∈ Φ j } rq j (cid:48) ,k (cid:1) − a k (5)Otherwise, we set x k = (cid:0) (cid:88) ( i (cid:48) ,j (cid:48) ) ∈{ ( i,j ) | ≤ i ≤ m ∧ ≤ j ≤ J i ∧ S k ∈ Φ i,j } r i (cid:48) q i (cid:48) ,j (cid:48) ,k (cid:1) − a k (6)We adopt the following five well-known evolutionary con-strained multiobjective algorithms to solve the problem: • The nondominated sorting genetic algorithm II (NSGA-II) with the constrained-domination principle [48]. • The constrained multiobjective evolutionary algorithm(CMOEA) based on an adaptive penalty function and adistance measure [49]. • The multiobjective evolutionary algorithm based on de-composition (MOEA/D) [22] with a penalty function forconstrain handling [26]. • The differential evolution with self-adaptation and lo-cal search for constrained multiobjective optimiza-tion (DECMOSA) [50], which combines constrained-domination and penalty function for constrain handling. • The multi-objective particle swarm optimizer based ondominance with decomposition (D MOPSO) [38].CMOEA, MOEA/D, and DECMOSA employ penalty func-tions for constrain handling. Violation of constraint (3) iscalculated as max(0 , (cid:80) n + n (cid:48) k =1 c k x k − C ) . For constraints thatall suspected cases of infection and the lower number of casesof each common disease must be treated, we set the violationof each constraint equal to the budget C , i.e., the violation is C times the number of false tr i and tr in Algorithm 1.Nevertheless, the performance of all the above algorithms isnot satisfying, mainly because the dimension ( n + n (cid:48) ) is veryhigh (approximately 10,000 ∼ EW T RANSFORM - AND -D IVIDE E VOLUTIONARY O PTIMIZATION M ETHOD
In this section, we propose a new transform-and-divideapproach to efficiently solve the problem. First, we trans-form the original high-dimensional, constrained bi-objectiveoptimization problem to a low-dimensional, unconstrained bi-objective optimization problem, which can be solved usingevolutionary (unconstrained) multiobjective algorithms. Theevaluation of each solution to the transformed problem canbe divided into a set of low-dimensional, single-objectiveoptimization subproblems, which can be solved using a tabusearch algorithm.
A. Problem Transformation
We transform the original problem of determining thepurchase quantity of each supply to a problem of determiningthe purchase budget for epidemic control and the purchasebudget for each disease. First of all, we calculate the cost forpurchasing the supplies that must be used in the non-alternativetreatment items and, therefore, obtain the remaining budget as: C (cid:48) = C − n + n (cid:48) (cid:88) k =1 c k x k (7)Consequently, the transformed problem is to distribute C (cid:48) to m +1 components, denoted by { y , y , . . . , y m } , where y is the budget for purchasing alternative supplies for epidemiccontrol, and y i is the budget for purchasing alternative suppliesfor treating disease D i ( ≤ i ≤ m ). The dimension of thetransformed problem is m (approximately 200 ∼
600 in a majorhospital in our case studies), which is significantly smaller thanthe dimension n + n (cid:48) of the original problem.Moreover, the search range of each dimension of the trans-formed problem is also much smaller than that of the originalproblem. For epidemic control, the lower limit y of budget y can be obtained using the following steps:1) Use supplies in storage to treat as many suspected casesof infection as possible; 2) If there is no remaining case, set y = 0 ;3) Else, for each remaining case, always purchase the cheap-est supply among the alternatives, and set y to the totalpurchase cost.And the upper limit y of budget y can be obtained usingthe following steps:1) Treat each suspected case in the most effective way, i.e,always select the supply with the maximum treatmenteffect e j,k among the alternatives, and calculate the totalrequired quantity of each supply;2) Calculate the purchase quantity of each supply, and set y to the total purchase cost.Therefore, the search range of y is limited to [ y , y ] . Wecan obtain the search range [ y i , y i ] of each y i in a similarmanner. In our case studies, the average value of ( y i − y i ) isapproximately 95 (in unit of 100 RMB), while that of ( x k − x k ) is approximately 1100 (in minimum order quantity). B. Problem Division
The solution space of the transformed problem is signifi-cantly smaller than that of the original problem. But how toevaluate a solution y = { y , y , . . . , y m } to the transformedproblem? The task can be divided into m +1 optimization sub-problems. The first subproblem is to determine the purchasequantities under the budget y so as to maximize the epidemiccontrol effect. Each of the remaining m subproblems is todetermine the purchase quantities under the budget y i so as tomaximize the treatment effect of disease D i ( ≤ i ≤ m ) .However, the division leads to difficulty in allocatingsupplies in storage to different diseases. We overcome thisdifficulty by employing a procedure similar to Algorithm 1to simulate the arrival and treatment of all cases. But theprocedure has two differences from Algorithm 1: • Initially, we only consider supplies in storage, i.e., Line3 of Algorithm 1 is not executed. • If there is no supply in storage that can be used for atreatment item (i.e., the condition in Line 14 or Line 26is satisfied), we temporarily purchase “in advance” thecheapest alternative supply for the item.The procedure also produces the “cheapest” solution to eachsubproblem, which can be evolved to an optimal or near-optimal solution, as described in the next subsection.
C. Hybrid Evolutionary Optimization
The proposed method employs an evolutionary multiobjec-tive algorithm to evolve a population of main solutions to thetransformed problem, and employs a tabu search algorithm tosolve the subproblems for evaluating each main solution.For the first subproblem, each solution z can be representedby six vectors as follows (the vector lengths do not need tobe the same): { z , , z , , . . . , z , | Ψ | }{ z , , z , , . . . , z , | Ψ | } ... { z , , z , , . . . , z , | Ψ | } where z j,k denotes the number of cases that use the k -thalternative supply for the j -th treatment item, and each vectorsatisfies (cid:0) (cid:80) | Ψ j | k =1 z j,k (cid:1) = r .The procedure described in Sec. IV-B produces the cheapestsolution to the subproblem, denoted by z † . First, we con-tinually use the following steps to improve z † by replacingan alternative supply to a more effective alternative for arandomly selected case until z † cannot be further improved:1) Randomly selecting two components z j,k and z j,k (cid:48) in avector satisfying z j,k (cid:48) > ;2) Set z j,k (cid:48) = z j,k (cid:48) − and z j,k = z j,k +1 if doing so wouldnot violate the budget constraint.Starting from the improved z † , the tabu search algorithmcontinually uses the following steps to search around andimprove z † until the stopping condition is satisfied:1) Generate k N neighboring solutions of the current z † , eachbeing obtained by randomly selecting two components z j,k and z j (cid:48) ,k (cid:48) satisfying k < | Ψ j | , k (cid:48) < | Ψ j (cid:48) | , z j,k > , and z j (cid:48) ,k (cid:48) +1 > , and setting z j,k = z j,k − , z j,k +1 = z j,k +1 +1 , z j (cid:48) ,k (cid:48) = z j (cid:48) ,k (cid:48) +1 , and z j (cid:48) ,k (cid:48) +1 = z j (cid:48) ,k (cid:48) +1 − , if doing sowould not violate the budget constraint;2) Select the best neighbor that is not tabued or is betterthan the current z † , make z † move to this neighbor, andadd this move to the tabu list.The remaining m subproblems can be solved by tabu searchin a similar way. As demonstrated by the experiments, thetabu search algorithm can quickly obtain optimal solutionsfor most subproblem instances, given that the dimensions ofthe subproblems are relatively small. For example, as we canobserve from Table I, the dimension of the first subproblemis 18 (note that the last dimension of each vector can bedetermined by other dimensions of the vector, and the actualdimension in the solution space is only 12). Therefore, thetabu search algorithm is very suitable for the subproblems, asit will be invoked many times to evaluate main solutions.For the main transformed problem, we adopt the followingevolutionary multiobjective algorithms to evolve main solu-tions and invoke the tabu search algorithm: • NSGA-II [48]. • MOEA/D [22]. • A differential evolution for multiobjective optimizationwith self-adaptation (DEMOwSA) [51]. • A multiobjective particle swarm optimization (MOPSO)algorithm [52] which extends comprehensive learning[53] for multiobjective optimization.
D. Complexity Analysis
In this subsection, we theoretically compare the complex-ities of the original problem and the transformed problem.For notational simplicity, we regard suspected infection asa disease with subscript i = 0 . The number of all possiblesolutions to the original problem is N = n + n (cid:48) (cid:89) k =1 ( x k − x k ) (8) And the algorithm 1 for evaluating each solution to theoriginal problem has a time complexity O ( f ) = m (cid:88) i =0 J i (cid:88) j =0 r i | Φ i,j | (9)After transformation, the number of all possible solutionsto the new problem is N (cid:48) = m (cid:89) i =0 ( y i − y i ) (10)To evaluate each solution to the transformed problem, weneed to solve m + 1 subproblems. The number of possiblesolutions to the i -th subproblem is N i = J i (cid:89) j =0 | Φ i,j | (cid:89) k =1 ( z j,k − z j,k ) (11)where z j,k and z j,k denote the upper and lower limits ofdecision variable z j,k .And the time complexity of evaluating each solution to the i -th subproblem is O ( f i ) = J i (cid:88) j =0 r i | Φ i,j | (12)Consequently, the total time complexity of the transformedproblem is N (cid:48) (cid:0) (cid:80) mi =0 N i O ( f i ) (cid:1) , while that of the originalproblem is N · O ( f ) , and the complexity reduction ratio oftransformation is R c = log (cid:0) N · O ( f ) (cid:1) log (cid:0) N (cid:48) ( (cid:80) mi =0 N i O ( f i )) (cid:1) (13)As the expressions of N , N (cid:48) , N i , O ( f ) and O ( f i ) arecomplicated, in practice, we can use the average range (cid:98) x ofall ( x k − x k ) in Eq. (8), the average (cid:99) | Φ | of all (cid:0) (cid:80) J i j =0 | Φ i,j | (cid:1) in Eqs. (9) and (12), the average (cid:98) y of all ( y i − x i ) in Eq. (10),and the average (cid:98) z of all ( z j,k − z j,k ) in Eq. (11). Let r = m (cid:80) i =0 r i be the number of all cases, we have R c ≈ log (cid:0)(cid:98) x n + n (cid:48) r (cid:99) | Φ | (cid:1) log (cid:0)(cid:98) y m (cid:98) z (cid:99) | Φ | r (cid:99) | Φ | (cid:1) = ( n + n (cid:48) ) log (cid:98) x + log( r (cid:99) | Φ | ) m log (cid:98) y + (cid:99) | Φ | log (cid:98) z + log( r (cid:99) | Φ | ) (14)In our case studies, the average values are (cid:98) x ≈ , (cid:98) y ≈ , (cid:98) z ≈ , (cid:99) | Φ | ≈ , n + n (cid:48) ≈ , and m ≈ . Therefore,the average complexity reduction ratio on the instances isapproximately 58.V. C OMPUTATIONAL R ESULTS
A. Problem Instances
We use the proposed method for medical supplies procure-ment in Zhejiang Hospital of Traditional Chinese Medicine(ZJHTCM) from 15 Feb to 15 Apr, 2020, the peak of COVID-19 in Zhejiang Province, China. Since 15 Mar, we also extendthe application to other five hospitals (denoted by H1–H5).
Therefore, there are 14 real-world instances of the medicalsupplies procurement problem. Table II summarizes the maincharacteristics of the instances, where (cid:80) i r i denotes the totalexpected number of cases of all common diseases, J denotesthe average treatment items per disease, | Φ | denotes theaverage number of alternatives per treatment item, and thebudget C is in RMB. The instances are solved on a workstationwith an i7-6500 2.5GH CPU, 8GB DDR4 RAM, and anNVIDIA Quadro M500M card. B. Performance for Solving the Subproblem
Before testing the algorithms for solving the main problem,we first test the performance of the tabu search algorithmfor subproblems. From the above real-world main probleminstances, we select 16 subproblem instances, the dimensions D of which range from 12 to 72. For the algorithm, we setthe neighborhood size k N to D , tabu length to 12, and themaximum number of iterations to D . On each instance, werun the algorithm 50 times to test whether and how long itcan obtain the exact optimal solution (validated by an exactbranch-and-bound algorithm [54]).Fig. 1 presents the convergency curves (averaged over the 50runs) of the tabu search algorithm on the subproblem instances.The algorithm reaches the optima within 100 iterations (10 msin our computing environment) when the problem dimensionis smaller than 24, within 200 iterations (30 ms) when thedimension is smaller than 40, and within 400 iterations (120ms) on all instances. In our case studies, the average dimensionof the instances is approximately 37, which can be solvedusing approximately 160 iterations (25 ms); the dimensionof the largest instance is 72, which can be solved using 369iterations (116 ms). Using multithreading and GPU accelera-tion, the average CPU time for evaluating a main solution toa problem of 400 diseases is approximately 600 ms. C. Performance for Solving the Original and TransformedProblems
For each main problem instance, we use five evolution-ary constrained multiobjective algorithms, including NSGA-II with constraint handling (denoted by NSGA-II-C) [48],CMOEA [49], MOEA/D with constraint handling (denotedby MOEA/D-C) [26], DECMOSA [50], and D MOPSO [38],to solve the original problem; we also use four evolutionarymultiobjective algorithms, including NSGA-II [48], MOEA/D[22], DEMOwSA [51], and MOPSO [52], all combined withtabu search, to solve the transformed problem. The controlparameters of all algorithms are tuned on the whole set ofinstances. For a fair comparison, all the algorithms use thesame stopping criterion that the CPU time does not exceed90 minutes, which is also applied in our practice. On eachinstance, each algorithm is run 30 times.Fig. 2 compares the hyperarea (the area under the Pareto-approximated front in objective space, also known as thehypervolume) [55], [56] obtained by each algorithm on eachmain problem instance. It is clear that the last four algorithmsusing transform-and-divide exhibit significant performance ad-vantages over the first five algorithms. On large-size instances of ZJHTCM and H5, the median hyperareas of the fourtransform-and-divide EAs are approximately eight to ten timesof those of the basic EAs; on the other instances, the medianhyperareas of the transform-and-divide EAs are approximatelysix to eight times of those of the basic EAs. In general,the performance of the transform-and-divide EAs is mainlyaffected by m (the number of diseases) and C (the totalbudget), while that of the basic EAs is mainly affected by n + n (cid:48) (the number of supplies). This is why all algorithms obtainrelatively high hyperareas on the instances of H1 and H2,where m , n + n (cid:48) , and C are relatively small. Nevertheless, theperformance advantages of the transform-and-divide EAs overthe basic EAs are very significant on these relatively small-size instances. On the last two instances of H5, the values ofthese parameters are large, and the performance advantages ofthe transform-and-divide EAs over the basic EAs are not sosignificant. On each instance, the minimum hyperareas of thetransform-and-divide EAs are still significantly larger (aboutfour to five times) than the maximum hyperareas of the basicEAs. The results demonstrate that the proposed transform-and-divide method can greatly reduce the difficulty of solving thecomplex original problem.We also make a pairwise comparison between eachtransform-and-divide EA and its constrained version for theoriginal problem, e.g., NSGA-II vs. NSGA-II-C, NSGA-II vs. CMOEA, MOEA/D vs. MOEA/D-C, DEMOwSA vsDECMOSA, and MOPSO vs D MOPSO. That is, in eachpair, the first is an unconstrained multiobjective EA used intransform-and-divide, and the second is a multiobjective EAadding constraint handling mechanisms (such as constrained-domination or penalty functions) to the first one. The compar-ison is made based on the ratio of the hyperarea obtainedby the first algorithm to the maximum hyperarea obtainedby the second algorithm. Fig. 3 presents the changes ofhyperarea ratios over the CPU time on each problem instance.In general, it takes about five minutes for a transform-and-divide EA to reach the maximum hyperarea obtained byits counterpart after 90 minutes; on large-size instances ofZJHTCM and H5, the ratio is between 3 and 4 at tenminutes and exceeds 5 before twenty minutes; on the otherinstances, the ratio is approximately 2 at ten minutes andbetween 3 and 4 at twenty minutes. The high hyperarea ratiosdemonstrate that the transform-and-divide EAs approximatethe Parato front of the problem significantly more efficientlythan the basic EAs, mainly because the transform-and-dividestrategy significantly reduces the solution space, and alsobecause the unconstrained EAs for the transformed problemcan explore the solution space more effectively than theirconstrained versions. Comparatively, the ratios of NSGA-IIvs. its counterparts (NSGA-II-C using constrained-dominationand CMOEA using a penalty) grow more slowly than those ofDEMOwSA and MOPSO vs. their counterparts in early stagesof evolution. As there is no significant difference among thefive basic EAs for the original problem, the results indicatethat DEMOwSA and MOPSO are more efficient in exploringthe solution space to approximate the Parato front of thetransformed problem.Besides the hyperarea metric, we also compare the al-
TABLE IIS
UMMARY OF THE REAL - WORLD INSTANCES OF THE MEDICAL SUPPLIES PROCUREMENT PROBLEM .Hospital Period m n + n (cid:48) (cid:80) i r i r J | Φ | C ZJHTCM 2 nd half Feb 476 32,535 71,196 64 5.84 7.27 3,516,0001 st half Mar 476 32,416 76,580 38 5.84 7.27 3,378,0002 nd half Mar 479 32,628 78,331 34 5.86 7.29 3,022,0001 st half Apr 479 32,628 90,459 36 5.86 7.32 3,698,000H1 2 nd half Mar 162 17,522 8,208 4 7.46 5.41 521,0001 st half Apr 162 17,510 13,640 3 7.46 5.41 830,000H2 2 nd half Mar 193 15,666 17,353 24 8.06 5.25 785,0001 st half Apr 193 15,681 19,309 14 8.06 5.25 902,500H3 2 nd half Mar 328 24,469 32,052 14 7.84 5.87 1,682,0001 st half Apr 328 24,469 42,667 17 7.84 5.97 2,127,000H4 2 nd half Mar 393 27,600 35,733 50 6.90 6.13 2,415,5001 st half Apr 399 27,215 38,452 28 6.87 6.14 2,607,200H5 2 nd half Mar 573 35,906 60,900 27 6.66 5.36 3,920,0001 st half Apr 573 34,902 75,393 30 6.66 5.48 4,818,000 f * msiter (a) D =12 f * msiter (b) D =15 f * msiter (c) D =18 f * msiter (d) D =21 f * msiter (e) D =24 f * msiter (f) D =27 f * msiter (g) D =30 f * msiter (h) D =34 f * msiter (i) D =38 f * msiter (j) D =42 f * msiter (k) D =46 f * msiter (l) D =50 f * msiter (m) D =55 f * msiter (n) D =60 f * msiter (o) D =66 f * msiter (p) D =72 Fig. 1. Convergency curves of the tabu search algorithm on subproblem instances. The bottom horizontal axis is the number of iterations, the top horizontalaxis is the CPU time (in milliseconds), the vertical axis is the objective function value, and f ∗ is the exact optimal objective function value. gorithms in terms of the coverage ( Cov ) metric [55], i.e.,
Cov ( X, X (cid:48) ) is the fraction of solution set X obtained by analgorithm that are strictly dominated by at least one solution of set X (cid:48) obtained by another algorithm. The results are clearthat 100% solutions obtained by the basic EA are dominatedby at least a solution obtained by its transform-and-divide (a) ZJHTCM, 1 st half Mar (b) ZJHTCM, 2 nd half Mar (c) ZJHTCM, 1 st half Apr (d) ZJHTCM, 1 st half Apr (e) H1, 2 nd half Mar (f) H1, 1 st half Apr (g) H2, 2 nd half Mar (h) H2, 1 st half Apr (i) H3, 2 nd half Mar (j) H3, 1 st half Apr (k) H4, 2 nd half Mar (l) H4, 1 st half Apr (m) H5, 2 nd half Mar (n) H5, 1 st half AprFig. 2. Comparison of hyperareas obtained by the algorithms on main problem instances. Each box plot shows the maximum, minimum, median, first quartile(Q1), and third quartile (Q3) of hyperareas over the 30 runs of an algorithm. counterpart, while none of solutions obtained by the transform-and-divide EA is dominated by at least a solution obtained by the corresponding basic EA. Consequently, decision-makersalways prefer to adopt solutions produced by transform-and- NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (a) ZJHTCM, 1 st half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (b) ZJHTCM, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (c) ZJHTCM, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (d) ZJHTCM, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (e) H1, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (f) H1, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (g) H2, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (h) H2, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (i) H3, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (j) H3, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (k) H4, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (l) H4, 1 st half Apr NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (m) H5, 2 nd half Mar NSGA-II vs NSGA-II-C NSGA-II vs CMOEAMOEA/D vs MOEA/D-C DEMOwSA vs DECMOSAMOPSO vs D2MOPSO (n) H5, 1 st half AprFig. 3. Pairwise comparison between each unconstrained multiobjective EA used in transform-and-divide and its constrained counterpart for the originalproblem. The vertical axis is the ratio of the hyperarea obtained by the first algorithm to the maximum hyperarea obtained by the second algorithm, and thehorizontal axis is the CPU time (in minutes). divide EAs, while solutions obtained by the basic EAs canhardly provide reference. In practice, decision-makers choosefinal solutions for implementation as follows: • The best solutions of MOEA/D on the fourth instance ofZJHTCM and the second instance of H4. • The best solutions of DEMOwSA on the second instanceof ZJHTCM, the second instance of H1, the secondinstance of H2, the second instance of H3, and the secondinstance of H5. • The best solutions of MOPSO on the remaining seveninstances. VI. C
ONCLUSION
This paper presents a transform-and-divide evolutionaryoptimization approach to medical supplies procurement underthe background of COVID-19. Our approach first transformsthe original high-dimensional, constrained multiobjective op-timization problem to a low-dimensional, unconstrained mul-tiobjective optimization problem, and then evaluates eachsolution to the transformed problem by solving a set ofsimple single-objective optimization subproblems, such thatthe problem can be efficiently solved by existing evolution-ary multiobjective algorithms. We applied the transform-and-divide evolutionary optimization approach to six hospitalsin Zhejiang Province, China, during the peak of COVID-19. Results showed that our approach exhibits significantlybetter performance than that of directly solving the originalproblem. Decision-makers of the hospitals always choose thebest solutions produced by the transform-and-divide methodfor implementation and achieve promising results in balancingepidemic control and common disease treatment in practice.The proposed transform-and-divide evolutionary optimiza-tion based on problem-specific knowledge can be an efficientsolution approach to many other complex problems. Forexample, considering a problem of personalizing healthcaresolutions for a large number of residents. As the numberof candidate healthcare medicines and treatment items arelarge, the dimension of the problem is high. However, byclustering the residents based on their health status and limitthe medicines and treatment items to each cluster, the problemdimension can be significantly reduced, and we can solvethe subproblem of personalizing healthcare solutions for eachcluster much more efficiently. Another example is to distribute m types of disaster relief supplies from n suppliers to n demanders, we need to determine the quantity of each supplyfrom each supplier to each demander. The dimension ofthe problem is mn n . However, we can establish a virtual“intermediary” and transform the problem to a new problemof determining the quantity of each supply from each supplierto the intermediary and that from the intermediary to each de-mander. Therefore, the dimension of the transformed problemis m ( n + n ) , but we need to solve additional subproblems ofdetermining at the intermediary which parts of supplies shouldbe sent to each demander. In many cases, the transform-and-divide strategy can greatly reduce the difficulty of problem-solving, but it often requires effective discovery and utilizationof problem-specific knowledge for problem transformation and division. Although EAs are regarded as robust problem-independent search heuristics for a large variety of optimiza-tion problems [57], [58], we argue that proper exploitation of problem-dependent knowledge can significantly improve theefficiency of EAs in solving highly complex problems and,therefore, enlarge the application field of EAs.R EFERENCES[1] G. B. Dantzig and P. Wolfe, “Decomposition principle for linear pro-grams,”
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