SIMULATED ANNEALING AND JOINT MANUFACTURING BATCH-SIZING. Ruhul SARKER. Xin YAO

Transcription

1 Yugoslav Journal of Oerations Research 13 (003), Number, SIMULATED ANNEALING AND JOINT MANUFACTURING BATCH-SIZING Ruhul SARKER School of Comuter Science, The University of New South Wales, ADFA, Canberra, Australia. Xin YAO School of Comuter Science, The University of Birmingham, Edgbaston, Birmingham, UK. Abstract: We address an imortant roblem of a manufacturing system. The system rocures raw materials from outside suliers in a lot and rocesses them to roduce finished goods. It rooses an ordering olicy for raw materials to meet the requirements of a roduction facility. In return, this facility has to deliver finished roducts demanded by external buyers at fixed time intervals. First, a general cost model is develoed considering both raw materials and finished roducts. Then this model is used to develo a simulated annealing aroach to determining an otimal ordering olicy for rocurement of raw materials and also for the manufacturing batch size to minimize the total cost for meeting customer demands in time. The solutions obtained were comared with those of traditional aroaches. Numerical examles are resented. Keywords: Inventory, rocurement, eriodic delivery, otimum order quantity, heuristic, simulated annealing. 1. INTRODUCTION We discuss a manufacturing system where the manufacturer uses raw material, received from an outside sulier, so to roduce a finished roduct. Traditionally the economic lot size for raw material urchase and manufacturing batch size are determined searately. However, when the raw material is used in roduction, its ordering quantities are deendent on the batch quantity of the roduct. Therefore, it is undesirable to searate the roblem of economic urchase of raw materials from economic batch quantity. As a result, one should determine the otimum roduction

2 46 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing batch size of a roduct and the ordering quantities of associated raw materials together. This could be done by treating roduction and urchasing as comonents of an integrated system, minimizing the total cost of the system. Consider the raw material inventory system of an organization. Since the amount of raw material used for a roduct is given, the amount of raw material to be used in a batch of known quantity is also known. The raw materials are consumed at a given rate only during the roduction u-time, and not throughout the whole cycle time. The ordering olicy of a raw material can either be based on its economic ordering quantity (EOQ) or on the requirement of raw material in a lot of economic roduction quantity (EPQ). The latter reflects the deendent relationshi between raw material requirement and roduction quantity. The relationshi between a roduct and its raw material has been the basic consideration for the model develoed in this aer. The roblem is also considered from the oint of view of the benefit to the manufacturing firm. The roosed systematic aroach suorts the fact that the otimum urchasing-roduction olicy should be determined by considering an integrated system, rather than by considering several indeendent systems [1,, 3]. It would be ossible to consider either a continuous or eriodic roduct suly olicy. In this research we consider a fixed, known quantity suly at a fixed interval of time. However, the suly of raw material will be taken in a lot where the lot size is a decision variable. The roblem considered in this aer has the following attributes: a finite roduction-rate environment uses raw materials from outside suliers, only the roduct lot sizing and its associated raw material suly quantities are under consideration, the suly olicy for the roduct is to deliver an equal quantity at fixed intervals, and the roduct cannot be delivered until the whole lot is finished and quality certification is comlete. The above situation exists in many industries. The objective is to determine the otimum batch size of the roduct and the ordering quantities of raw materials minimizing the overall system cost. To simlify the model, we consider that a single raw material urchase rovides stock for several roduction runs. In the literature, there are different models for deendent lot sizing. The suly atterns of raw material considered in those models are (a) Lot for lot and (b) Multile lot for lot. The multile lot for lot is further classified into (i) single raw material urchase for several roduction runs and (ii) several raw material urchases for a single roduction run. The delivery olicy of the finished roduct is classified into (a) continuous and (b) eriodic suly. In eriodic suly, a lot can either be sulied in a single shiment or in several equal shiments. Sarker et al. [1,, 3] develoed a model oerating under continuous suly at a constant rate for both raw material delivery olicies. In the lot-for-lot system, the ordering quantity of raw material is assumed to be equal to the raw material required for one roduction run. The raw material that is relenished at the beginning of a finished roduct inventory cycle will be fully consumed at the end of the roduction run. It is assumed that the length of a roduction run is always less than the finished

3 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing 47 roduct inventory cycle time. In multile lot for lot, it is assumed that the ordering quantity of raw material is n times the quantity required for one roduction run, where n is an integer. An ordering olicy for raw materials to meet the requirements of a roduction facility under a fixed quantity, eriodic delivery olicy has been develoed by Sarker and Parija [4], Jamal and Sarker [5], Golhar and Sarker [6] and Sarker and Golhar [7]. They considered that the manufacturer is allowed to lace only one order for raw material er finished roduct inventory cycle. In this case, a fixed quantity of finished goods (say x units) is to be delivered to the customer at the end of every L units of time (fixed interval). This delivery attern forces inventory build-u in a saw-tooth fashion during the roduction u-time. The on-hand inventory deletes sharly at regular intervals during the roduction down time until the end of the cycle. The latter one forms a stair case attern. Recently, Sarker and Parija [8] develoed another model of several urchases of raw material for a single roduction run under a eriodic delivery olicy. In this aer, we consider a roblem for determining the ordering olicy for raw materials to meet the requirements of a roduction facility under a fixed quantity, eriodic delivery olicy. In this case the roduct cannot be delivered until the whole lot is roduced because of the quality assurance requirement. We analyze a system where a single raw material urchase is utilized for several roduction runs. This system is economic when the ordering cost of raw material urchase is much higher than the setu cost of a roduction run. Figure 1 shows the finished roduct inventory system. The roduct is roduced at a constant rate during the roduction u-time eriod t. The delivery of finished roduct starts as soon as the level of accumulated finish roduct inventory reaches the manufacturing batch quantity, Q. A fixed quantity of finished roduct ( x ) is delivered to the customers at the end of every L units of time. This forms a staircase attern during the roduction down time of the cycle. To fulfill the demand on time, there may be an overlaing of two consecutive finished roduct inventory cycles. The duration or amount of overlaing deends on the roduction rate, demand rate and other relevant cost data. The shaded area in Figure 1 shows the overlaing of inventory deletion of a finished roduct inventory cycle with the inventory accumulation of the next cycle. Figure resents the raw material inventory level of the system. At the beginning, the raw material of quantity nrq is relenished for the urose of roviding stock for n roduction runs. The raw material is deleted at a constant rate during the roduction u-time of the finished roduct inventory cycle. The level of raw material inventory remains constant during the roduction down-time. In this aer, the cost factors considered are ordering/setu, holding and material costs. We have the total cost equation of the system with resect to the roduction quantity and the equation as an unconstrained otimization roblem. Two heuristics have been roosed in this aer to solve this otimization roblem. The first is a simle heuristic, which is much simler than other existing methods. The solution obtained using this aroach may be subotimal because of the deendent relationshi and the nature of the solution method. The second method is based on simulated annealing (SA) [9,10,11], which is a owerful stochastic search algorithm that can be alied to comlex and nonconvex otimization roblems. The uroses of

4 48 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing designing simulated annealing are (i) to exlore its use in solving batch-sizing roblems and (ii) to comare its solution to those obtained by other methods. Following this general introduction, the aer resents a brief exosition of simulated annealing. Then the mathematical formulation of the coordinated olicies is develoed. In the following section, the solution aroaches are rovided. Subsequently, numerical examles and analysis are given. The conclusions are rovided in the final section. T Q Inventory Level x L x x f i e g k j t s Time Figure 1: Finished roduct inventory level Q i Inventory Level 0 T T.. nt Figure : Raw material inventory system

5 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing 49. SIMULATED ANNEALING Simulated annealing (SA) [9,10,11] is a owerful stochastic search method alicable to a wide range of roblems for which little rior knowledge is available. It can roduce high-quality solutions for hard combinatorial otimization [1]. The basic idea of SA comes from condensed matter hysics. It is well-known that in condensed matter hysics a good way to find minimum energy states, called ground states, of comlex systems, such as solids, is to use the annealing technique, in which the system (solid) is first heated to some high temerature and then slowly cooled down. The system (solid) will reach a ground state if the cooling rate around the freezing oint of the system is sufficiently slow. This rocess can be simulated on comuters via abstract models, such as systems of interacting articles with many degrees of freedom. At each ste of the simulation, a new state of the system is generated from the current state by giving a random dislacement to a randomly selected article. The new state will be acceted as the current one if the energy of the new state is no greater than that of the current state; otherwise, it will only be acceted with robability e Enew _ state Ecurrent _ state T where E stands for the energy of the system and T is the temerature. This ste can be reeated as many times as necessary with a slow decrease of temerature in order to find a minimum energy state. Of course, only finite stes are taken in ractical simulations. This simulation rocedure was roosed by N. Metroolis et al. [9] and is called the Metroolis rocedure [10]. SA has been alied to numerous roblems in oerations research and industrial engineering [18-5], such as cell formation [18], scheduling with resource constraints [19], machine conditioning [0], scheduling with multi-level roduct structure [1], lot sizing [3-4], and guillotine cutting [5]. A good survey of SA alications can be found in []. However, none of these aers discusses the roblem considered in this aer. It is well-known that SA works well for some roblems, but not for all. It is interesting to investigate whether SA can be alied effectively to the manufacturing batch-sizing roblem described in this aer and whether SA is robust (i.e., not sensitive to different arameter settings) for this roblem. 3. MATHEMATICAL FORMULATION A general cost model is develoed considering the major oints of both the sulier of raw materials and the buyer of finished roducts. This model will be used to determine an otimal ordering olicy for the rocurement of raw materials, and the manufacturing batch size by minimizing the total cost for meeting equal shiments of the finished roduct, at fixed intervals, to the buyers. In order to find an economic order quantity (EOQ) for the raw materials and an economic roduction quantity (EPQ) for the roduction run, it is customary to consider the cost comonents: (i) raw material inventory carrying cost; (ii) finished

6 50 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing goods inventory carrying cost; (iii) raw material ordering cost; and (iv) manufacturing setu cost. To develo the model, the assumtions have been made: (i) the roduction rate is finite and constant; (ii) the roduction caacity is greater than the demand; (iii) no shortages are ermitted; (iv) the time horizon is infinite; and (v) a fixed quantity of the roduct is delivered after a fixed interval of time. The notation used in develoing the cost functions is shown below: A i = ordering cost of raw material A = setu cost for a roduct ($/setu) D i = demand of raw material for the roduct in a year, Di = rd D = demand rate of a roduct, units er year H i = annual inventory holding cost for raw material, $/unit/year H = annual inventory holding cost, $/unit/year L = time between successive shiments x / D = m = number of full shiments during the cycle time = T/ L = Q / x n = number of roduction run to use one lot of raw material P = roduction rate, units er year (here, P > D ) PR i = rice of raw material Q i = ordering quantity of raw material = nrq Q i = otimum ordering quantity of raw material Q = roduction lot size r = amount/quantity of raw material required in roducing one unit of a roduct t = roduction u-time in years in a cycle of length T T = cycle time measured in years Q / D, where T > t = x = shiment quantity to customer at a regular interval (units/shiment) 3.1. Finished Product Inventory Since the roduct cannot be delivered to the customers until the whole lot is comleted and quality certification is ready, there is a continuous build-u of finished roduct inventory, at the rate equal to roduction, during the roduction u-time of a given lot (Figure 1: line ef ). The delivery of the finished roduct is ermitted during the roduction down time of that lot only. In order to fulfill the demand in time, the roduction of a new lot may be started before finishing the delivery of the revious lot. From Figure 1, We can write: t = Q / P and Q/ D = ml, and the finished roduct inventory in a cycle = Area ( efg + ijk )

7 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing 51 ( m- 1) m Area ( ijk) = x( m 1) L + x( m ) L + + xl + xl = x L. So, the finished roduct inventory in a cycle 1 ( m - 1) = Q t+ x ml Hence, the average finished roduct inventory in a cycle 1 x( m 1) Q t+ ml = T 1 t m 1mL = Q + x T T 1 D mx x = Q + P 1 = Q D + 1 x (1) P 3.. Raw Material Inventory The raw material inventory is shown in Figure. In this case, the inventory for raw material in a cycle of nt eriods is = [ Qit+ QiT+ Qit+ QiT+ Qit+ + Qi( n 1) T+ Qit ] n 1 ( n 1) = Qit+ Q it For raw material the average inventory er cycle 1 ( n 1) Qit+ QiT = nt 1 D = rq + n 1 () P Then the total Cost Function for a year can be reresented as follows: Total cost of the system, D A Q i x TC1 = A + + ( kk) H Q n (3)

8 5 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing D where, = = + 1 D kk H rhi n 1 CC nr H i, P P D where, = D 1 CC H rhi P P Since, Q = mx, then TC 1 can be rearranged as follows: D Ai mx x TC = A + + ( kk) H mx n (4) TC is a nonlinear function with integer variable m and n. The behavior of this function is discussed in the next section. Our objective is to determine the otimal m and n while minimizing the total cost. 4. SOLUTION APPROACH The total cost of the system can be exressed either as a function of continuous variable Q and an integer variable n (equation 3) or as a function of the integer variables m and (equation 4). TC and TC are no differentiable since they contain n 1 integer variables. As such, a closed form solution for it can be shown that TC1 is a iecewise convex function of Q cannot be obtained. However, Q. Sarker and Parija [4] examined and lotted a similar function. Efficient algorithms may be alied to solve this roblem by using a discrete otimization technique. An algorithm has been roosed by Moinzadeh and Aggarwal [13] to obtain a global minimum for such a function. Although it may not be efficient, a simle rocedure is develoed here to obtain an otimal or near-otimal solution in this aer. We consider TC to determine the otimal solutions. The reasons for choosing TC, instead of TC 1, are: (i) the range of Q is too large as comared to m, (ii) Q is a deendent variable of m, (iii) the solution may not guarantee an integer value of (iv) in reality, Q is integer too. m, and With a search algorithm, the number of iterations will be much lower with a function of m and n than that of Q and. The function TC can be lotted n connecting the values of TC for integer n and m. For any given value of m, the connected function of n looks like a convex function. This similarity to a convex function is also true for the connected function of m, for any given value of n. In the following section, we develo two aroaches to otimizing the function TC.

9 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing Heuristic Based on Traditional Otimization The roerties of the function motivated us to develo a conventional otimization based aroach. This aroach is well acceted in the OR/MS literature in regard to finding the near otimal solution (Silver, Pyke and Peterson [14] and Joglekar and Tharthare [15]). Relaxing the requirement that m and n are integer and allowing them to be continuous, and then differentiating TC with resect to m and equating to zero, we get m = A + A ) ni x ( kk) i D (5) Substituting m in Equation (4) we get the annual total cost as i D + ( ) A A D Ai x kk x = + ) + ( ) n x TC A kk H x n i ( kk) D A + A x n Ai x = D A + [( CC) + nrhi] H (6) n Differentiating the modified TC (equation 6) with resect to n and equating to zero, we get Ai( CC) n = A rh i (7) Both m and n should be integers. However the values obtained from equation (5) and (7) may not be integer, as we solved the relaxed function. In such a case, the neighbouring integer oint (values of m and n ) is sorted which incurs the minimum cost. The comlete heuristic algorithm is resented below: Algorithm: finding batch size. Ste 0. Initialize and store D, P, A, Ai, H, Hi, r and x. Ste 1. Comute m using (equation 5). n Ste. Comute and TC using (7) and (6) resectively. If both m and n are integers, then calculate Otherwise go to Ste 3. Q and go to Ste 6.

10 54 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing n Ste 3. If m is integer and is not, then comute TC using Choose the Otherwise go to Ste 4. n that gives minimum TC. Calculate m Ste 4. If n is integer and is not, then comute TC using m n= n and n. Q and go to Ste 6. m= m and m. Choose the that gives minimum TC. Calculate Q and go to Ste 6. Otherwise go to Ste 5. Ste 5. If both n and m are non-integers, then comute TC for all four combinations of n and m : Otion 1: m n Otion : m n Otion 3: m n Otion 4: m n Ste 6. Sto Choose n and m that gives the minimum TC. Calculate Q and go to Ste 6. Numerical Examle A numerical examle is rovided to show the alicability of the model develoed in this research. The data for the examle are as follows: 6 D = 4 10 units 6 P = 5 10 units A = $ er setu A i = $ 3000, er order H = $ 10. er unit er year H i = $ 100. er unit er year x = 1, 000 units. Solution: using traditional otimization based heuristic Let r = 100. From Equation (5), the number of full shiments during the cycle time m = From Equation (7), the number of lot size n = From Equation (6), the total cost TC = $ 18, To find the integer value of m and n, we try for m = 14 and 15 and n = 10 and 11

11 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing 55 m n Total Cost ($) Remarks , ,37.80 Highest Lowest total cost , , In between In between So the number of full shiments during the cycle time is m = 14, the number of lots is n = 11 and the lot size = 14, 000 units. 4.. Simulated Annealing Aroach The classical SA (CSA) [10] starts with an initial configuration generated at random. At each ste, it selects the next solution Y from the neighbourhood N of the current solution X. The next solution will be acceted as the current one if its cost is no greater than that of the current solution; otherwise, it will only be acceted with robability C Y C X e T Where reresents the cost (to be minimized) of solution Y and is the cost of C Y solution X. This rocedure is reeated with a slow decrease of the control arameter T, called temerature, until a sufficiently good solution has been found. CSA can be summarized by the following algorithm: select initial solution X at random; select initial temerature T ; REPEAT REPEAT randomly select Y from N X with uniform distribution; IF C C Y X THEN accet Y as the new solution ELSE accet Y as the new solution with robability e UNTIL Âinner-loo sto criterion' is satisfied decrease temerature T UNTIL Âouter-loo sto criterion' is satisfied C Y C X T One of the major tasks in develoing an SA-based aroach is to define a suitable neighborhood function, i.e., to define a suitable generation function that guarantees every feasible solution in the search sace can be reached. In this aer, the neighbors of a given solution X = ( m, n ) are {( m+ i, n+ j) i, j = 1, 1} in our algorithm. The accetance robability used in our algorithm is the same as that defined in the above CSA. C X X

12 56 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing The initial solution was generated uniformly at random within a user-secified range, e.g., for m and n between 1 and The initial temerature was generated using the following rocedure: (1) Generate 100 solutions at random; () Find the solutions with the minimum cost and with the maximum cost; (3) The initial temerature was set at twice the difference between the maximum and minimum cost. The rogram we develoed allows its user to adjust the initial temerature for each run as well. Two cooling schedules were tested in our algorithms. Both erformed well, i.e., enabled the algorithm to find the global otimum consistently. The first cooling schedule is T = ρ T, where ρ is a arameter that can be adjusted by the user. The otimal setting of ρ is highly roblem-deendent. Different roblems require different values. The common ractice to find a near-otimal ρ is to start with a relatively smaller value, such as 0.85, and than gradually increase it to a large value close to 1.0, such as This trial-and-error rocess can be very tedious. A better alternative is to design a robust SA algorithm which is not sensitive to the arameter setting. In other words, the erformance of the SA algorithm changes little when a different value of ρ is used. The SA algorithm used in this aer is very robust in this sense. There is no need to tune ρ as long as it is within a reasonable range, i.e., 0.85 to 0.99, which is the range used by many SA users and researcher. To show the robustness of the SA algorithm, we ran the SA algorithm for ρ = to 0.99 and obtained the global otimum in all cases using the given numerical examle. The second cooling schedule used in our algorithm is T T =, where β is 1 + β T a arameter that can be adjusted by the user. Similar to the case of ρ, the otimal value of β deends on the roblem. There is no universally otimal β which is best for all roblems. A trial-and-error rocess for finding a good β value has to be used in ractice. However, if an SA algorithm is robust, there will be little need for a lengthy trial-and-error rocess. Robustness is one of the criteria in designing our SA algorithm. To test how robust our algorithm is, we ran it with β = 0.01, 0.05, 0.1, 0.5, 1.0,.0, 3.0 and 4.0 in our exerimental studies. Thirty runs were conducted for each arameter setting. The SA algorithm was able to find the global otimum at m= 14, n=11 consistently for all runs. None of our runs took more than a few seconds. These exeriments have shown that our SA algorithm worked well with a wide range of arameter settings. There is no need to have a time-consuming arameter-tuning rocess. A default value of β = 1. 0 can be used for our roblem. The inner loo sto criterion used in our algorithm was determined by the number of iterations for which the temerature T was ket the same. It can be set by the user easily. The outer loo sto criterion was determined by two factors. One was the maximum number of iterations set by the user. The other was the lowest temerature. That is, the SA algorithm stos when its temerature is below this value. Further imrovement to our SA algorithm was made after the above success. We modified the range of i and j in the neighbourhood definition. The neighbourhood of X = ( m, n ) now becomes {( m+ i, n+ j) i, j in range [ range, + range]}, where i and

13 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing 57 j are generated uniformly at random within [ range, + range], and range is a userdefined arameter that deends on the initial ranges of m and n. This imrovement enables our algorithm to converge much faster, since larger jums are made ossible through this neighbourhood. Our revious work has shown the benefit of having a large neighbourhood size [16]. The otimal solution can be found in less than a second. 5. ANALYSIS AND DISCUSSION It is clear from revious studies that the joint roduct batch size and raw material ordering olicy is more cost-effective than the searate olicy [1,, 3, 17]. In the joint batch sizing roblem, we consider that the raw material of a single urchase will rovide stock to several roduction runs. This is true when an ordering cost of raw material urchase is much higher comared to the setu cost of a roduction run. In the examle roblem, if the ordering cost is $5 or less, it is not economic to use single raw material urchase for more than one roduction run. The behavior of the function encouraged us to develo a simle, calculusbased, heuristic to solve the roblem. This aroach is generally well acceted in the OR/MS literature. However, there is no guarantee that such a heuristic will roduce a global otimum. So we designed an SA algorithm. Although there is no guarantee that SA will find an exact global otimum in finite time, the solution obtained by our SA algorithm for the examle roblem has been found consistently excellent. In order to evaluate the quality of solutions, an enumeration algorithm was used to search for the exact global otimum (several hours on a PC) for the examle roblem. This result confirmed that the solution found consistently by SA and the simle heuristic method is indeed the global otimal solution. 6. CONCLUSIONS A model for a two-stage batch environment has been roosed in this aer. This model takes a finite-rate of roduction into account and jointly determines batchsizes for the roduct and order-sizes for the associated raw materials. To avoid comlexity in formulation, the variable m and n are assumed to be integers for all cases. However, these are relaxed in the solution aroach that is erformed. As a result, it is not exected that the solution will be globally otimal. This method can be considered as a simle heuristic to solve the stated roblem with little comutational effort and an accetable level of solution quality. To imrove the quality of the solution, we develoed an SA algorithm, which can roduce high-quality solutions reliably and efficiently. The SA algorithm does not require the cost function to be differentiable or even continuous. It can deal with functions with multile local otima. A distinct feature of our SA algorithm is its robustness against different arameter settings. The algorithm worked very well under a wide range of arameter values. There was no need to tune arameters such as ρ and β using a lengthy trial-and-error rocess. The exerimental results using an examle roblem show that the SA algorithm would be a very useful tool in attacking other batch-sizing roblems.

14 58 R. Sarker, X. Yao / Simulated Annealing and Joint Manufacturing Batch-Sizing The model develoed in this aer can be alied in real life situations where (i) a finite roduction-rate environment uses raw materials taken from outside suliers, (ii) the suly olicy is to deliver equal quantities at fixed intervals, and (iii) the roduct cannot be delivered until the whole lot is finished and quality certification is ready. This situation may arise in the chemical and harmaceutical industries under certain conditions. Acknowledgement: The authors are grateful to Xin Yao's students in his Modern Heuristic Techniques class for running some of the SA exeriments. They include David McRae, Nick Hicks, Nathan Bracken, Kim Tyler, and Uan Suarchai. REFERENCES [1] Sarker, R.A., Karim, A.N.M., and Azad, S., "Integrated inventory system: cases of roductrawmaterials and roducer-wholesalers", 37th Annual Convention of the Institution of Engineers Bangladesh, Rajshahi, Bangladesh, [] Sarker, R.A., Karim, A.N.M., and Azad, S., "Two cases of integrated inventory", Journal of the Institution of Engineers, Bangladesh, 1 (4) (1995) [3] Sarker, R.A., Karim, A.N.M., and Haque, A.F.M.A., "An otimal batch size for a roduction system oerating under a continuous suly/demand", International Journal of Industrial Engineering, (3) (1995) [4] Sarker, B. R., and Parija, G.R., "An otimal batch size for a roduction system oerating under a fixed-quantity, eriodic delivery olicy", Journal of the Oerational Research Society, 45 (8) (1994) [5] Jamal, A.M.M., and Sarker, B.R., "An otimal batch size for a roduction system oerating under a just-in-time delivery system", International Journal of Production Economics, 3 () (1993) [6] Golhar, D.Y., and Sarker, B. R., "Economic manufacturing quantity in a just-in-time deliver system", International Journal of Production Research, 30 (5) (199) [7] Sarker, B. R., and Golhar, D.Y., "A rely to 'A note to "Economic manufacturing quantity in a just-in-time delivery system"'", International Journal of Production Research, 31 (11) (1993) [8] Sarker, B. R., and Parija, G.R., "Otimal batch size and raw material ordering olicy for a roduction system with a fixed-interval, lumy demand delivery system", Euroean Journal of Oerational Research, 89 (1996) [9] Metroolis, N, Rosenbluth, A., Rosenbluth, M., and Teller, E., "Equations of state calculations by fast comuting machines", Journal of Chemical Physics, 1 (1953) [10] Kirkatrick, S., Gelatt, C. D., and Vecchi, M.P. "Otimization by simulated annealing", Science, 0 (1983) [11] Yao, X., "A new simulated annealing algorithm", International Journal of Comuter Mathematics, 56 (1995) [1] Yao, X., "Call routing by simulated annealing", International Journal of Electronics, 79 (4) (1995) [13] Moinzadeh, K., and Aggarwal, P., "Order exedition in multi-level roduction inventory system", Paer resented at TIMS/ORSA Joint National Meeting, Las Vegas, NV, USA, May 7-9, 1990.

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