OPTIMAL BATCHING AND SHIPMENT CONTROL IN A SINGLE-STAGE SUPPLY CHAIN SYSTEM

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1 Abstract OPIMAL BACHING AN SHIPMEN CONROL IN A SINGLE-SAGE SUPPLY CHAIN SYSEM Shaojun Wang epartment of Industrial & Engineering echnology Southeast Missouri State University Cape Girardeau, MO 6370, USA Bhaba R Sarker epartment of Industrial & Manufacturing Systems Engineering Louisiana State University Baton Rouge, LA , USA his research studies the single-stage supply chain system operated under JI technique. A kanban mechanism is employed to assist in linking the production processes in a supply chain system. he number of kanbans, batch size, and the number of batches are determined optimally. Based on this information, kanban operation between two adjacent plants can be worked out for controlling the supply chain system. he model of supply chain system is a mixed integer nonlinear programming (MINLP) which is solved optimally by a branch-and-bound method. he total cost of supply chain system is minimized through kanban mechanism. eywords: anban, supply chain, inventory control, and mixed-integer nonlinear programming.. Introduction A supply chain (SC) system is usually composed of a series of organizations and independent companies. he main operational activities of the supply chain system include: () Raw materials (the suppliers): sales forecasting, inventory planning, and purchasing, transportation between suppliers and manufactures, () Work-in-process (manufactures): processing, inventory management inside the manufactures, (3) Finished goods (the customers or retailers): warehousing, finished goods inventory management, customer service, and transportation among the wholesalers, retailers, and customers. Usually, a manufacturing system is composed of a series of plants (or shops, or, workstations). he manufacturing facilities may be located at the same city, or at different cities or in different states. hus, there may be significant material flows (the work-in-process formed by the semi-products) between two consecutive manufacturing facilities (workstations, shops, plants). he plant has different meanings in different context and so does the kanban (see able ). he reference points of the kanban travel, in this research, are assumed as plants. Also, kanban can be considered as an AGV (Automated Guided Vehic le), cart, tote, truck, ship, train, etc., depending on the situations. ransportation Between Workstations or work-centers Workstations or work-centers Plants or companies Plants or companies able. he general meanings of the plant and kanban. Relative Plant anban ransporters Form of anban Locations AGV, fork-lifter, totes, anban or electronic Intra-house Shops trolley message anban or electronic Inter-house plant Fork-lifter, totes, trolley Intra- or Inter-cities Inter-states Plant or company Plant or company Companies Countries Companies rucks, train, ship rucks, trains, ships, aircargo rucks, trains, ships, aircargo message Paperwork, or electronic message Paperwork, or electronic message Paperwork, or electronic message

2 he role played by the kanban in production control is to tie the different manufacturing processes together and improve the operations in the production process. he improvements in reduction of inventory, wasted labor and customer service are usually accomplished through kanban operations. he material flow and information flow between two adjacent plants compose a kanban stage. If a supply chain system consists of only two plants, it is called single-stage supply chain system (SSSCS). It is very difficult to develop a tractable generalized mathematical model of the kanban controlled supply chain system incorporating all its salient features such as demand pattern, lead time, information (kanban) delivery time/cost, setup time/cost, production capacity, and batch size. Sarker and Balan [] and Wang [] determined the number of kanbans between two workstations for both single-stage and multi-stage kanban systems. In their models, the demand rate was assumed as linear over each of the three phases (inception, maturation and declination) of a product s life cycle. Wang and Wang [3] developed a Markovian model that calculates the steady-state probability distribution for the expected number of kanbans required. Askin et al. [4] minimized the sum of inventory holding and backorder cost in their research. Steady-state results are derived for an M/G/ system for the cases of a few and many part types. Mascolo et al. [5] developed a general-purpose analytical method for performance evaluation of multistage kanban controlled production systems. In Nori and Sarker [6] models, the total cost was expressed as a function of the number of kanbans, the shortage cost of materials, and holding cost of containers. Although numerous models have been developed to describe the supply chain system, the studies published did not consider the supply/retailer, number of kanbans, and kanban operations systematically and quantitatively. he goal of this research is to effectively control a supply chain system by kanban mechanism to achieve the purpose of JI philosophy. he supply chain system considered in this research consists of three parts: the suppliers, manufacturers, and retailers. In the supply chain systems, the deliveries of raw material from the suppliers, the workin-process (WIP) in production stage, and the finished goods to retailers are all broken into batches and are controlled by the kanbans. When the batch sizes and the number of batches are obtained, the number of kanbans can be determined by the number of batches by scheduling.. Problem description he function of the kanban is best explained through the single-stage supply chain system depicted in Figure. Production is first triggered by the demand at the final plant (the final processing). In a kanban operation, first, a withdrawal kanban attached to a loaded container in a succeeding plant is detached from the container and put into the kanban post (W) when the first part from the container is to be used. Second, the withdrawal kanbans in the post are collected at a fixed or non-fixed interval and brought to the preceding plant by the transportation vehicle. he withdrawal kanban indicates such information as the quantity of parts to be filled in a container, the preceding and succeeding plants involved with the part, the collection interval, etc. he withdrawal kanban is then attached to the container in a store at the preceding plant in place of the production ordering kanban permitting the worker at the preceding plant to produce the required amount of parts; that is, the detached production-ordering kanban triggers the production of the preceding plant. he containers filled with parts together with the withdrawal kanban are brought, in turn, to the succeeding plant by the vehicle. his kanban cycle realizes smooth, timely, and unwasted flow of parts between two adjacent plants. - 5 Raw material anban Stage 3 4 Finished goods P W Work-in-process Figure. A single-stage kanban supply chain system.

3 he problems for these kanban controlled supply chain systems are: first, the number of batches in each stage that is to be shipped by kanbans should be determined. According to the number of batches, considering the delivering time and load/unload time, the number of kanbans needed to transport the batches is determined. Second, the ordering policy to the suppliers at the first stage and delivering policy to the retailers at last stage are to be decided, respectively. Finally, the behavior of each stage should be linked together. hat is, when modeling the system, the relationship between the plants should be considered. All these problems should be solved based on the objective of minimizing the supply chain s total operation cost. he total cost of this supply chain system consists of three parts: the cost in raw material stage, the cost in WIP stage, and the cost of finished goods stage. 3. he Single-Stage anban Model he notation used in the model is: Parameters: p i H r H w H f A r A s A w A f C r C w C f C Variables: r w f n o m n s Production rate of plant i (i =, ), units/year; he demand rate, units/year; Holding cost of raw material inventory, dollar/unit/year; Holding cost of work-in-process (WIP) inventory, dollar/unit/year; Holding cost of finished goods inventory, dollar/unit/year; Setup (ordering) cost, dollar/setup (order); Setup (manufacturing) cost, dollar/batch; Setup (shipping) cost, dollar/ship (setup); Setup (shipping) cost at plant, dollar/setup (shipment); Cost of raw material inventory, dollars/year; Cost of WIP inventory, dollars/year; Cost of finished goods inventory, dollars/year; otal cost of a supply chain system, dollars/year. otal quantity of finished goods produced over a period, units/year; Order quantity, units/order; WIP shipping quantity, units/shipment; finished goods shipping quantity, units/shipment; Number of shipments; Numb er of order of raw material inventory placed; Number of shipments placed during the production uptime; Number of shipments placed at the final plant (finished goods); Number of shipments placed during the uptime of the final plant. hree cost components, the cost of raw materials at the first plant, the cost of WIP between two plants, and the cost of finished goods at the second plant, are calculated as follows. 3.. Cost of Raw Material Inventory For the raw material inventory, it is assumed that the rate of demand for the products is nothing but the production rate of the first plant, p. he order arrives in lots on time when an order is placed. So, the input rate (replenishment) is considered as infinite. In this supply chain system, instead of EO arriving at one time, the company orders raw material in batches, i.e., the EO is divided into a number of batches, n o. When the production starts, the shipment (one batch) is set at a fixed interval during one period. his inventory model is referred to as a classical economic batch-size model where the total raw material cost, C R, is given by Ar r C R = + Hr () r 3.. Cost of Work-In-Process Inventory he production uptime of plant is u. uring that time, m batches are shipped to the succeeding plant. he production of plant stops when the quantity of parts produced by the plant is enough for the demand of plant, but during the downtime, the demand of plant consumes the stocks. Plant consumes the raw material and produces the semi-finished products that build the WIP inventory. As the stock level in this plant reaches to lot size, w, the parts are carried by containers to the succeeding plant. uring the uptime, plant continuously manufactures the parts. When it is the time to deliver, w amount of parts are loaded onto the container with the kanban and

4 transported to plant. At this time, the stock level does not drop to zero. hen, WIP inventory increases again in that level as production continues. Upon the next delivering time comes, the w amount of parts is transported to Plant, and so on. he products in the time period u are equal to. he average inventory of the WIP is given by: I avg = [ ] [ ( )] + [ ( )] 0 pdt w m w m ( + ( m))( m) (+ m )( m ) = + w [ ] = w ( m + )] () where is the number of shipments of WIP and m is the number of shipments placed during the uptime. he cost of WIP inventory, C w, is comprised of setup cost, kanban delivering cost, and the inventory holding cost. Hence, the WIP cost is given by w C w = As + Aw + H w ( m + ) (3) w 3.3. Cost of Finished Goods Inventory he throughput of the plant forms the inventory model of finished goods. he total stock in the stage increases at a rate of p. he finished goods are shipped to the buyers or to the warehouse. Because the demand rate is constant, the optimal policy of this inventory is determined as a fixed lot size. Each shipment is made in a fixed interval. It is observed that the behaviors of finished goods inventory is the same as the WIP inventory over the cycle time. herefore, Equation (3) can be directly used for calculating finished goods average inventory: f C f = = As + Af + H f ( n s + ) (4) f 3.4. otal Cost of Single-Stage Supply Chain System hus, the total cost of SSSCS can be obtained by summarizing the cost expressions for raw material, manufacturing stage, and finished goods stage: C = C r + C w + C f A = ( r Aw A f + + ) + ( As + As) + [ H r r + H rr + H ww( m + ) + H f f ( n s + ) ] (5) r w f Substituting the relations = n o r = w = n f, m = / p, and s = n / p into Equation (5), yields: H H r H w f C s (n o,, n, ) = ( noar + Aw + naf + As+ As ) H w( ) + H f ( ) no n p p where n o,, and n are all integers and are greater than, is a real (positive) number variable. hus, Equation (6) is a mixed integer nonlinear programming (MINLP). he solution method for MINLP problem will be addressed and a numerical example will be given to illustrate the solution method. 4. A Branch And Bound Algorithm for Solving MINLP MINLP problem is hard to solve optimally. In this research, the branch and bound algorithm are adopted to solve the problems. he branch and bound algorithm for MINLP problems is based on the same idea as it is for solving the mixed integer linear programming (MILP). First, the nonlinear programming (NLP) relaxation of the original problem is solved. he NLP relaxation of the original problem is obtained by ignoring the integer restrictions. If the solution satisfies the integer constraints, it is the optimal solution of MINLP, then the procedure stops. If not, the solution of the relaxed problem provides a lower bound (minimization problems) to the optimal solution. hen, the original problem is separated into two sets by adding additional constraints one at a time, and the resulting NLP relaxed subsets are solved one by one. When an integer solution is found, it provides an upper bound to the optimal solution of MINLP. All nodes that exceed this bound are fathomed or dropped from further consideration. he (6)

5 search procedure is continued in this way until all the nodes are fathomed. he algorithm is presented in the subsection below. In Equation (6), let x, x, and x 3 replace n o,, and n, respectively, i.e., x = (x, x, x 3, ). Also let Z replace C. herefore, Equation (6) is rewritten as H H r H w f Z(x, x, x 3, ) = ( Ar x Aw x + Af x3 + As + As ) x x x3 + H w / p + H f / p x, x, x 3 and integer, > 0. + ( ) ( ) (7) For the NLP relaxation of the original problem (7), the optimum total cost of a single-stage supply chain system defined by the relaxed MINLP problem is given by: where = A s + As and ( A H + A H + A H + αβ ) Z ( x*) = r r w w f f (8) β = H / p p + H f / α, ( ) ( ) x = = w H r, x Ar Hw = H A w, x3 = w ( As + As) α = f ( / p ) + H ( / p ) β, and x* = (x *, x *, x 3 *, *) is given by H f Af, (9) Example: A kanban controlled single-stage supply chain system. For a single-stage supply chain system with two plants, the demand rate, production rates p and p, setup costs (kanban shipping cost) A s and A s, manufacturing setup costs A r, A w and A f, and holding costs H r, H w and H f are given in able. In this problem, for the MINLP, determine the number of shipments and batch size in raw material stage, WIP, and finished good stage, and determine the part quantity produced in one cycle period. able. he parameters of a SSSCS. units/year P units/year 5000 p = 5500 p = 5600 Setup cost (shipping) dollar/batch A s = 300 A s = 50 Setup cost dollars/setup A r = 0 A w = 0 A f = 00 Holding cost dollars/unit/year H r = 45 H w = 30 H f = 35 Substituting the values in able into Equation (7), yields: Min Z(x,x,x 3, ) = (x + 0 x + x3 + 55) x x x3 n o,, n and are integers. (0) his MINLP problem is solved by the B&B algorithm. From Equation (8), the optimum total cost for relaxation problem is given by: ( Ar H r + Aw H w + A f H + αβ ) = (5000 ) ( 45(0 ) + 30(0 ) + 35(00 ) (0.833 ) ) C ( x*) = f = 00( ) = $4,90 Step : Solve the NLP relaxation of the original problem, set the result as the lower bound. From equation (9), the solution of NLP relaxation is given by:

6 x = 5.89, x = 4.6, x 3 = 5.45, = 9, Z = $4,90 Step : As all x i s are not integers, the process continues. he upper bound is set to Z U ; Step 3: o form the subsets, the additional constraints, x 5, x 6, x 4, x 5, x 3 5, x 3 6, are added into the Equation (0) by one at a time. hus, six subsets are formed at this level. hese six subsets are solved one by one. After that, the feasibility of the solutions is checked and the feasible solutions form the nodes. At the same time, the integer solutions are checked. If integer solution is found, then update Z L by setting Z L equal to the integer solution. Step 4: As none of these six subsets is an integer solution, the search has to continue at a most promising node. So, the Z values of these six subsets are compared. It is observed that the node with Z value of $4,9 is the smallest. hen, Z U, which is $4,90, is updated by Z L (Z L = $4,9). Step 5: Find the most promising node for further fathom. he search continues at the node. Four children of node are checked. he most promising node for further fathom at this step is node 8. hen, at the node, an integer solution is found which is X = (x, x, x 3, ) = (6, 5, 6, 966), z = 4,94. It is observed the node 5 at the top level is less than z = 4,94. hus, the node 5 needs to be fathomed. Step 6: Repeat Steps 4. By following the B&B algorithm, finally the optimal result is obtained at: X * = (x, x, x 3, ) = (6, 5, 6, 966), z * = $4,94. he integers 6, 5, and 6 are the number of batches in raw material stage, work-in-process stage, and finished good stage, respectively. he value of 966 is the total quantity in one period. With these data, the batch size in each stage can be determined by dividing by corresponding x. herefore, the number of kanbans for delivering the materials in each stage can be obtained by scheduling the batches of that stage. 5. Conclusions Usually, the number of batches in each stage that are obtained from the MINLP model is not equal to the number of kanbans that are used to deliver the batches. In kanban controlled supply chain system, determining the number of batches in each stage is very critical to achieve smooth material movement. Although the MINLP can be solved by B&B algorithm, for the large size instances, the heuristic is needed. he most important conclusions found in this research are consolidated as () Single -stage supply chain problem can be solved by a kanban mechanism, () Single -stage supply chain problem forms a MINLP, (3) B&B produces a good solution to MINLPs, (4) he quantity of parts in each stage over a period ties individual stages, and (5) he number and size of containers determines the operations schedules. References. Sarker, B. R. and Balan, C. V., 996, Operations planning for kanbans between two adjacent workstations, Computers & Industrial Engineering, 3(-), -4.. Wang, S., 00, Control of Supply Chain Systems by anban Mechanism, Ph issertation, epartment of Industrial and Manufacturing Systems Engineering, Louisiana State University, March 0, Baton Rouge, LA. 3. Wang, H., and Wang, H. P., 99, Optimum number of kanbans between two adjacent workstations in a JI system, International Journal of Production Economics, (), Askin, R. G., Mitwasi, M. G., and Goldberg, J. B., 993, etermining the number of kanbans in multiitem just-in-time systems, IIE ransactions, 5(), Mascolo, M.., Frein, Y. and allery, Y., 996, An analytical method for performance evaluation of kanban controlled production systems, Operations Research, 44(), Nori, V. S. and Sarkar, B. R., 998, Optimum number of kanbans between two adjacent stations, Production Planning & Control, 9(),