“Operations scheduling optimization through mathematical programming in a high                 mix-low volume manufacturin...
1. AbstractConsider a manufacturing environment with more than 500 multilevel shippable products,characterized by a high m...
consequences of the interruption. The planning and scheduling current process used by this Plant isbriefly described in Fi...
The characteristic considerations of a Flow Shop scheduling problem are:           1) Each of every machine is available c...
The plant in which this project is developed has 10 clients, represented by literals A, B, C, … J; thesecustomers demand a...
CLIENT         N° OF SHIPPABLE           N° PURCHASED         N° MANUFACTURED PARTS                  PRODUCTS             ...
8.  Number of productive processes used by each specific                                                               sub...
The following chart shows the graphical representation of the Model Flow, it starts with the customerdemand that triggers ...
1) Initial product inventory             2) Preparation Time (setup), it´s specific to each part number and it´s of extrem...
Note:       These variables appear in the F.O. with a negative                                                            ...
4) Balance of production per day constraints.                                                                  X i, j     ...
Where:                               C1k= Selling price of the k-th product, k= 1,…, 56                               C2i=...
4.9. Model Flowchart:          CUSTOMER DEMAND AT A SHIPPABLE PART NUMBER LEVEL                                           ...
4.11. Model ResultsTable 2 shows the numeric results of the optimization model. Not only OTD was drastically improved,but ...
This model efficiently solves the problem of programming complex manufacturing scenarioscharacterized by the high mix of p...
7. Bibliography1. HILLER F. y LIEBERMAN G. J, Introduction to Operations Research, McGraw Hill Companies, Inc.,    2005.2....
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Article final ivan f

  1. 1. “Operations scheduling optimization through mathematical programming in a high mix-low volume manufacturing environment” Author: Ivan F. Rodriguez El Paso, TX| 79912 Phone +1-915-999-6845 mr.ivan.rodriguez@gmail.com 1
  2. 2. 1. AbstractConsider a manufacturing environment with more than 500 multilevel shippable products,characterized by a high mix and low volume demand profile. This paper presents a multivariablelinear programming model that efficiently addresses the challenging master scheduling process, itoptimizes the objective function by providing a viable solution to produce the assemblies requestedby the customer while maximizing the profit to the Plant; transformation cost, inventory cost,delayed deliveries/expediting cost, overtime cost and setup cost are all involved to generate thebest possible production plan where customer demand; capacity constraints; and MOQ are eachand every one considered too. 2. Key WordsProfit, multilevel structure, high mix low volume manufacturing configuration, MOQ, batch andqueue, continuous flow, lean manufacturing, Overall Equipment Efficiency 3. IntroductionThis paper is developed using a metal fabrication plant located in Guadalajara, Mexico, it wasselected to present a reality that is probably applicable to many of other companies in this country,first, it´s facing a lot of operation and financial issues mostly related to the way the plan isstructured and the way plant has faced the growth; second able, several difficulties to effectivelyand quickly adapt their production schedule process to the dynamic and always challenging marketneeds. This particular plant has been losing money for the last 4 quarters; among some others, theway the plant is setup (batch and queue flow, functional layout, vertical organizational structurewith a lot of layers in between, etc.) and the way production plan is defined, are severely affectingplant ability to address this issue on a sustainable manner. The combination of unplanned growthand lack of vision to adapt the plant´ master scheduling system, are the main reasons why this plantis struggling to survive and keep stakeholders satisfied.Fundamental industrial engineering principles and mathematic optimization are combined to createa flexible production plan that supported on a continuous flow operation that will significantlyimprove the way this plant works making its operational and financial metrics a lot better and moreimportantly, predictable and sustainable.The proposal for an effective solution is critical, is a matter of survival, reality that probably notdiffer much of hundreds of businesses in Mexico that succumb to the violent and continuousassaults of competition, that don´t cease and redefine itself every minute, not giving space topassivity, nor to partial and short term solutions. Mexico has in its people, the biggest if itsstrengths to, once forever, over pass the historical reactive and fully dependant behavior. 4. Basic conceptsEfficiency is usually expressed as the percentage of time that the resources are used productively.The unproductive utilization is usually the result of activities related to equipments start over(setup), stoppages (downtime), reconfiguration of machines by change of products (mix cost),maintenance, times and interruptions by breakage and/or equipment failures. All of them with theexception of the machinery breakages can be anticipated, so their impact to the overall equipmentefficiency through out a good planning system. Equipment failures are obviously, not scheduledevents and its consequences to the plant, severely affect its performance. First, it creates significantlosses in terms of produced units, giving a low absorption of resources; and secondly, in the case aprolonged breakages, a new production schedule must be generated immediately to minimize the 2
  3. 3. consequences of the interruption. The planning and scheduling current process used by this Plant isbriefly described in Figure 1. Strategic Planning Tactical Planning Decisions Production Plan Feedback Production System Figure 1 – Planning ProcessIndeed, the planning process is divided into several functional components, with multiple supportplans. The typical planning model is based on forecasts, master production plan, materials plan,production plan and operations scheduling as detailed in Figure 2. A problem of operationsprogramming on a Flow Shop environment is a problem in which n tasks must be processed by a setof m different machines and tasks must have the same processing or technological stream flow inthe respective machines/equipments. Market Forecast Production Master Planning Production Plan Materials Materials Plan requirement Production Production Plan Scheduling Operations Operations Planning Schedule Figure 2 - Typical planning model 3
  4. 4. The characteristic considerations of a Flow Shop scheduling problem are: 1) Each of every machine is available continuously and without interruptions. 2) Each of every machine can process one task at a time. 3) Each of every task can only be processed by a specific machine at a time. 4) Tasks processing times are perfectly defined and they are fixed. 5) Tasks have the same likelihood to be scheduled. 6) Setup times are included on the processing times. 7) Once an operation starts on any given machine, it cannot be interrupted.According to theory that studies the mathematical complexity, this problem is classified as NP-Complete. An example of the sequencing online problem is shown in Figure 3, where in a linear flow layout,there are 3 tasks that must be processed in a set of 4 machines, each task has the same order oftechnological stream through the machines, i.e., each of the tasks must first be processed onmachine 1, then in the 2 machine, and so successively until it´s reached the 4 machine. Scripts arecalled technically permutation scripts. The task processing time j in the machine i is denoted by pijand total time of processing or execution is called Cmax (Makespan). Execution Time Figure 3 – Online sequencing of three tasks in four machines 4
  5. 5. The plant in which this project is developed has 10 clients, represented by literals A, B, C, … J; thesecustomers demand a total of 500 Products called shippable products, they are comprised by a totalof 2.263 purchased parts and 1.083 manufactured ones. Table 1 shows the details of clients with itsrespective shippable products, manufactured and purchased parts; this data is presented as of July2009. For the purpose of this paper, one of the ten customers has been selected to develop andrun the mathematical model.Demand is received from customers on a weekly basis; and it is put under a standard process called“Supply Commit” described in Figure 4. Production Plan is generated from it, it details theassemblies and subassemblies that will be manufactured per day, the sequence and the machineryrequired. This plan will then provide the elements to create the Shipment Plan which will be used bythe program managers to provide customers with a day by day delivery schedule. Figure 5 showshow the raw data was collected and organized on an Excel Spreadsheet to later make themathematical programming using Premium Solver Optimization Tool. Figure 4 – Supply Demand ProcessIn this case, purchased parts are administered through a denominated program VMI (VendorManaged Inventory) its lead time is less than a 1 day, hence, they are not considered as a problemin the model, which is absolutely focused on the make parts. 5
  6. 6. CLIENT N° OF SHIPPABLE N° PURCHASED N° MANUFACTURED PARTS PRODUCTS PARTS A 56 221 267 B 29 464 168 C 40 332 161 D 20 303 230 E 90 69 34 F 120 760 83 G 20 217 112 H 40 120 27 I 50 210 78 J 100 620 41 Table 1 - Active Clients, Shippable Products, Make Parts and Purchased parts Figure 5 – Data structuring for mathematical model developmentWhere: 1. Part Number of shippable product (Top Level) 2. Part Number of subassembly 3. Level in the structure (A, B, C or D) 4. Factor use of the subassembly in the shippable product 5. Operations and processes required to make the product 6. Number of available machines by process type 7. Number of productive processes non used by each specific subassembly 6
  7. 7. 8. Number of productive processes used by each specific subassembly 9. Number of total productive processes in the plant (this is used to confirm that the total each part is effectively programmed accurately) 10. Minimum order quantity after Lean transformation (MOQ)4.1. Generic Process Flow DIAGRAMA GENERAL DE PROCESO RECIBO Recibo de INCOMMING Operación Material Inspección Almacen INSPECCION QUALITY CALIDAD NO PASA Simbologia INSPECTION NO PASS MATERIAL PASA REJECTED RECHAZADO MATERIAL PASS PUNCHING PUNZONADO CUTTING CORTE STAMPING QUALITY INSPECCION PRENSAS INSPECTIONCALIDAD NO PASS PASS PASA ALMACEN NO PASA WAREHOUSE QUALITY INSPECCION INSPECTION CALIDAD NO PASS NO PASA PULIDO Modificación de Program POLISHING Programa Modification PASA PASS DOBLEZ DOBLEZ BENDING BENDING NO NO PASA PASS INSPECCION NO PASS CALIDAD QUALITY NO PASA QUALITY INSPECTION INSPECCION INSPECTION CALIDAD INSERSION Y HARDWARE AND WELDING PASS AREA DE SOLDADURA PASA AREA DE SCRAP AREA SCRAP SCRAP SCRAP AREA PASS PASA PULIDO NO PASS NO PASA INSPECCION QUALITY CALIDAD INSPECTION TREN DE SERIGRAFIA LAVADO WASHING TRAIN SILK SCREEN NO PASS SECADO DRYING INSPECCION CALIDAD ENMASCARILLADO MASKING QUALITY INSPECTION PAINTING PINTURA POLVO LIQUIDA POWDER PAINT WET PAINT HORNO DE CURADO CURE OVEN UNMASKING DESENMASCARILLADO QUALITY INSPECTION INSPECCION CALIDAD NO PASA NO PASS PASA PASS ENSAMBLE EMPAQUE PACKING ASSEMBLY QUALITY INSPECCION NO PASS NO PASA CALIDAD PASA INSPECTION PASS ALMACEN WAREHOUSE Figure 6 – Metal Fabrication Generic Processes Flow4.2. Graphical representation of the conceptual model 7
  8. 8. The following chart shows the graphical representation of the Model Flow, it starts with the customerdemand that triggers the Job Order generation through the Plant ERP System (it is made at a Father PartNumber Level), then, these jobs are imploded using the same ERP at a Subassembly level, on this step ofphysical inventory located in the plant is discounted from the gross requirement, giving the Net Quantitythat needs to be produced. In the current master production scheduling process that prevail on thisPlant, the planner manually starts programming the available capacity, this process lasts between 1 to 2working days; as a result of this process, each machine is programmed with a sequence of parts, ifsomething changes in the customer demand, the current “planning system” is not flexible nor fasterenough to update the machinery planning accordingly. Figure 7 shows the current flow followed to planand manufacture multilevel products. Job order created at a Shippable P/N Level BOM Implosion is run through the ERP System to get Sub-Assemblies Sub-assemblies are organized using Demand the process sequence, setup and They are processed sequentially of processing time loaded in the ERP Level “D”, to the “C”, of “C” to the “B” and of the “B” to the “A” [Shippable P/N] Figure 7 – Graphical representation of the Conceptual Model (Multilevel Structure)4.3. Model Objective: Generate the optimal manufacturing sequence that maximizes plant´ profits while ensuring customer demand is reached using available capacity (equipment, machinery and manpower). Gross profit is the result of the arithmetical difference between Revenue (product of the volume of each shippable product and its respective selling price); minus the incurred costs. There are 5 types of costs considered in the model: (1) Transformation Cost (it includes raw material, labor and overhead); (2) Inventory Cost; (3) Delayed deliveries cost; (4) Overtime Cost; and (5) Setup Cost.4.4. Input data: 8
  9. 9. 1) Initial product inventory 2) Preparation Time (setup), it´s specific to each part number and it´s of extreme importance for the correct resource planning, it has a direct effect in the efficiency performance. 3) Processing Time by workstation 4) Process Efficiency, average of 12 months historical performance [90%] 5) Manufacturing sequence, process progression 6) Product structure level by level (Indexed BOM Multilevel) 7) Factor Use (number of times that each subassembly is used in the shippable product) 8) Minimum Order Quantity (MOQ), the plant does not have to produce an inferior number to the MOQ for each product i, since fixed cost will not be recovered; therefore, in the case where the demand of a product is less than MOQ, the demand is replaced by the corresponding MOQ. 9) Standard Gross Profit Margin (SGPM)4.5. Structural Variables: Xi, j Production of i-th product in j-th day of the week i=1, 2,… 267 j=1, 2,… 7 Ii, j Initial inventory of i-th product in j-th day of the week i= 1, 2,… 267 j=1, 2,… 8* * Ii,8 correspond to the inventory of i product in day 8, i.e. the final inventory at the end of the week Yk Unfulfilled demand of k-th product, where k identifies the shippable products k= 1, 2,…, 56 Dj, dj Non-negative variables used to level load production per day so capacity resources are best utilized, it indeed express the difference of production time between the day j+1 and the day j. This arithmetical difference can be positive, negative or zero. In order to be able to use them in a model of linear programming like the one created on this paper, they are defined as the difference of two nonnegative variables. In the presentation of constraints the followed logic is detailed. j= 1, 2,… 7 9
  10. 10. Note: These variables appear in the F.O. with a negative coefficient that penalizes the profit and by as much the model force them to take values from zero or very small ones.4.6. Constraints: 1) Capacity Constraint, amount of equipment and its respective schedules on watch (as it is in Figure 8) where TP is Time of Processing of i-th product in the resource of limited capacity respective m. Subassembly is raised at level. TPiXi, j < Capacitym [N° of constraints 10*7] i= 1, 2,… 267 j= 1, 2,… 7 m= 1.2,…, 10; (m=resources, for example, bending) In the model this is how the mathematical programming is made using Premium Solver Platform: Figure 8 – Mathematical Model Programming 2) Balance Constraint, this constraint ensures that the initial inventory plus the production minus the final inventory is always zero, this allows keeping the mathematical equality between “entrances” and “exits”. Formulas are defined part number by part number, considering its level with respect to shippable product; this constraint has been created to control multilevel structures in which the dependency of the subassembly and the shippable product is required along with the factor use. It is a constraint that practically, defines the manufacturing sequence at the level of each subassembly in the structure, consequently, it´s defined at a subassembly level. Ii, j + Xi, j - FUi*Xi, j - Ii, j+1 = 0; i is subassembly of i [N° of constraints 267*7] i= 1, 2,… 267 j=1, 2,… 7 3) Demand Constraint: The final inventory of the week (Ik, 8) plus unfulfilled demand must be greater or equal to the demand of the client. This constraint is defined at a Shippable Product level, it´s actually the only constraint that works with shippable products. Ik, 8 + Yk > Demand k [N° of Constraints 56*1] k= 1, 2,…, 56; k represents shippable products, “top level” or father products 10
  11. 11. 4) Balance of production per day constraints. X i, j X 1, j 1 D j d j i i [N° of Constraints 4 to 7] This way: Dj > 0 X i, j X i, j 1 Then i i dj = 0 And: Dj = 0 X i, j X i, j 1 Then i i dj > 0 In addition: Dj = 0 X i, j X i, j 1 Then i i dj = 04.7. Objective Function: For the purpose of this paper, objective function is defined using week 28 data from the Plant´ ERP; considers (July 19 to July 25, 2009), in which customer demand is for 16 shippable products. This demand must be satisfied in the available working days, i.e. Monday, Tuesday, Wednesday and Thursday, this is, j=1,…, 4; if this time is not sufficient, Fridays, Saturdays and Sundays, might be used, j=5,…, 7; but incurring on a overtime fee (OT) since the plant will need to get this skilled personnel to work those days. 56 56 267 7 267 56 267 7 267 7 3MAX Z C1k * Dk - C1k * Yk - C2i * Xi, j - C3i * Ii,8 - Ii,1 - C4k * Yk - C5i * Xi, j - 175 * C6i * Zi, j - 100 * (Dji dj) k 1 k 1 i 1 j 1 i 1 k 1 i 1 j 5 i 1 j 1 j 1 11
  12. 12. Where: C1k= Selling price of the k-th product, k= 1,…, 56 C2i= Transformation Cost of the i-th product, i=1,…, 267 C3i= Inventory Cost of the product i-th, i=1,…, 267 C4k= Delayed Deliveries Cost of the k-th product, k=1,…, 56 C5i= Overtime Cost of the i-th product, i=1,…, 267 C6i= Setup Cost of the i-th product, i=1,…, 267Logic followed by the model to make the demand fulfillment calculation: Initial inventory + Total Production + Unfulfilled Demand > Demand Numerical Representation: Ik, 8 = 100  Final inventory of k-th product Dk = 80  Demand of k-th product Ik, 8 - Dk=20  The Revenue could never be greater than product between demand (Dk) times the respective selling price (C1i), i.e. the model does not restrict the production by above of the demand of the client, nevertheless, this does not imply that the client will buy it, for that reason, the Revenue considers the following constraints: If Ik, 8 < Dk  Revenue V= Ik, 8 * C1k If Ik, 8 > Dk  Revenue V= Dk * C1k4.8. Optimization Method: Linear Programming Optimization implies to determine the production schedule, what toproduce, when and how many so the maximum profit is obtained considering the technical and humanlimitations.Model of Optimization Deliverables: 1. Production sequence at a Subassembly Level that minimizes the makespan (produce the greater gross profit margin) and maximizes the number of shippable products sent to the customer right on time and right on quantity. 2. Shippable products fulfilled (volumetric and mix on time delivery) 3. Gross Profit reached with proposed production schedule that creates the optimal manufacturing sequence (considering Revenue, transformation, inventory, overtime and delayed deliveries costs) 12
  13. 13. 4.9. Model Flowchart: CUSTOMER DEMAND AT A SHIPPABLE PART NUMBER LEVEL SUBASSEMBLY DEMAND ON HAND INVENTORY IMPLOSION PROCESSING TIMES OPTIMIZER ENGINE CREATES DIFFERENT PRODUCTION SEQUENCES TO MAXIMIZE THE PROFIT SETUP TIMES PROCESS FLOW FEEDBACK OPTIMAL RESULT PRODUCTION SEQUENCE CONTRIBUTION MARGIN Figure 9 – Optimization Model Flow4.10. Model Presentation Figure 10 shows the Model specifications on the Premium Solver Platform: Figure 10 – Screenshot of the Model Specifications in Premium Solve 13
  14. 14. 4.11. Model ResultsTable 2 shows the numeric results of the optimization model. Not only OTD was drastically improved,but the overall productivity of the plant, lower inventory levels, lower transformations cost, zeroovertime cost, and significant gross profit improvement.Table 2 Summary of Results (Comparison between the traditional planning model vs. optimized model) 5. Conclusions A mathematical programming model has been created that generates the optimal productionplan that maximizes the profits of the plant, organizing the manufacturing floor by defining thesubassemblies that must be produced, how many of them and the specific day of the week.While this paper shows the application of a linear programming model on one of the 10 customers thatthis metal fabrication plant has, it can be straightforwardly escalated to the total number of customersand shippable parts of the plant, capacity resources and client demands are to be updated in the modelso the “complete master plan” is generated in less than 15 min.This is a “quantum leap” step in the way the production scheduling task has historically been performedin the plant, not only assigned manpower was reduced by 85%, but, more importantly, the accuracy, thepredictability and the speed to create a scheduling production plan, has been improved. Above andbeyond, with this model, different manufacturing scenarios can now be produced in terms of minutes,so new customer requirements can be “added” to the existing plan to see its effect to the OEE (OverallEquipment Efficiency), Revenue and Gross Profit. This is real value to the decision makers of this plantthat will help them to improve the time and the quality of their decisions in terms of what needs to beproduced, how many prices by part number, the sequence and the specific day of the week. All relevantinformation such as setup time, manufacturing sequence, available capacity (equipment, machinery,manpower, processing time), MOQ, operation costs, and customer demand are all playing anactive/direct role in the objective function to produce the best possible manufacturing plan. 14
  15. 15. This model efficiently solves the problem of programming complex manufacturing scenarioscharacterized by the high mix of products and the low volumes; HM-LV-HC (High Mix - Low Volume -High Complexity), which is, indeed, the new reality that the Manufacturing Industry in Mexico has beenfacing for at least a lustrum and this model will continue.What is next?To further expand the application of this model, the planner would need: (1) Update model parameters, i.e. the number of available machines by process, costs, selling prices and MOQs (2) Select the project that he/she wants to program (3) Select the week of the year that he/she wants to program (4) Load the Initial Inventory of each subassembly; with this, the model will run and produce the master production plan to reach maximum level of gross profit. DATA ENTRY PROJECT/CUSTOMER TO PLAN WEEK OF YEAR TO PLAN INITIAL INVENTORY BY SUBASSEMBLY PROGRAMMING PARAMETERS IN THE MODEL Figure – Sequence of updates to be made by the planner to apply this model 6. Appendix6.1. About the Optimizer Software The model is loaded in a spreadsheet in Microsoft Excel and uses Premium Solver to solve the mathematical model. Premium Solver is Frontline Systems basic upgrade for the standard Excel Solver. Its 100% upwardly compatible from the standard Solver -- which Frontline developed for Microsoft -- with the capacity to solve much larger problems -- up to 2,000 variables -- at speeds anywhere from three to 100 times faster than the standard Solver. Good application for problems like the one presented on this paper with bigger limits than the ones of the standard Solver (200 decision variables). Premium Solver is dramatically enhanced in Version 9.0, with an all-new user interface that uses Excels Ribbon and Task Pane for easier, faster model definition; new charting features to visualize linear and nonlinear functions; and new features to easily solve multiple, parameterized optimizations. You can easily upgrade Premium Solver to Risk Solver Premium for new simulation power, Premium Solver Platform for greater optimization power, or our "super-product" Risk Solver Platform. 15
  16. 16. 7. Bibliography1. HILLER F. y LIEBERMAN G. J, Introduction to Operations Research, McGraw Hill Companies, Inc., 2005.2. REEVES Colin. A generis Algorithm for Flowshop Sequencing. Pergamon. Computers Ops Res. Vol. 22 No 1, pp 5-13, 1995 Great Britain.3. MOCELLIN. Joao Vitor, BUZZO Walter Rogerio. Programacao da Producao em sistemas Flow- Shop utilizando um Método Heurístico Híbrido Algoritmo Genético- Simulated Annealing. Revista Gestao & producao. Brasil. VII. No 3, December 2000. 16

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