l.6. pemodelan matematis
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Pemodelan Matematis
Disampaikan Oleh
M. Imron Mustajib, S.T., M.T.
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Referensi1. Daellenbach, H. G., (1994), “Systems and Decision Making”, John
Wiley & Sons, Chichester-England.
2. Murthy, D.N.P., Page, M.W., and Rodin,E.Y., Mathematical Modelling, Pergamon Press, 1990
3. Simatupang, T.M., (1995), Pemodelan Sistem, Nindita: Klaten
4. Tunas, B. (2007), “Memahami dan Memecahkan Masalah dengan Pendekatan Sistem”, PT Nimas Multima.
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OUTLINE• Introduction• What is a mathematical model? • Why do we build a mathematical model?• How to build a mathematical model?• An illustrative case (Case of LOD) • Formal Approaches for finding the optimal
solution
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INTRODUCTION
• We use the OR/MS Methodology• To capture the relationships between
various elements of the relevant system in a mathematical model and explore its solution.
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What is a mathematical model?• A mathematical model: Express, in quantitative
term, the relationships between various components, as defined in the relevant system for the problem (e.g. using Influence Diagram).
• Terminology:– Decision variables or the alternative courses of
action (controllable inputs)– Performance measure (how well the objectives
are achieved)
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What is a mathematical model?
• Terminology:– Objective function (the performance
measure is expressed as a function of decision variables)
– Uncontrollable inputs: parameters, coefficients, or constants
– Constraints –limit the range of the decision variables
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Relationship Between Input-System-Output
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Why build mathematical models?
• Real-life tests are not possible–Disruptive–Risky–Expensive
• Math Models are easy to manipulate–Quick exploration of the effect of changes in the inputs on the objective functions
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Properties of Good mathematical models
• Simple –simple models are more easily understood by the problem owner
• Complete –should include all significant aspect of the problem situation affecting the measure of effectiveness
• Easy to manipulate –possible to obtain answer from the model
• Adaptive –changes in the structure of the problem situation
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Properties of Good mathematical models
• Easy to communicate with –easy to prepare, update, and change the inputs and get answer quickly
• Appropriate for the situation studied –produces the relevant outputs at the lowest possible cost and in the time frame required
• Produce information that is relevant and appropriate for decision making –has to be useful for decision making
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The Art of Modeling
• A scientific process• More akin to art than science• A few guidelines• Ockham’s Razor:
– “Things should not be multiplied without good reason”.
– The modeler has to be selective in including aspects into a model
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The Art of Modeling
• An iterative process of enhancements –begin with a very simple model and move in an evolutionary fashion toward more elaborate models
• Working out a numerical example –observe how variables of interest behave
• Diagram and Graphs –to see things in the form of graphs or other drawings expressing relationships and patterns.
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Math. Model For The LOD Problem• Simplification
– Constraints (Warehouse space & mixing and filling capacities)
– Two decision variables (cutoff point, L and order size,Q)• First Approximation
– Ignore the constraints– Involve only one decision variable, Q
• Performance measure– Total annual relevant cost (TAC) (per year)– TAC=Annual stock holding cost+Annual set up
cost+Annual handling cost+Annual product values
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Math. Model For The LOD Problem
• Annual stock holding cost– (Average stock level x Unit product value) x
Holding cost/$/year• Annual set up cost
– Setup cost per batch x Annual number of stock replenishments
• Annual handling cost– Product handling cost per unit x annual volume
met from stock• Annual product values
– Unit product value x Annual volume of demand
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Math. Model For The LOD Problem
][][]/[]5.0[)( 1111 vDDhQsDQvrQT +++=
][]/5.0[][][),( 11122 DhQsDQvrDhsNLQT ++++=
)( LQT
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Math. Model – LOD[Second Approximation]
• Two decision variables, L and Q.• Two additional costs
– The annual set up cost for special production run• Annual volume by special prod.runs x Product handling
cost per unit– The annual handling cost for big order
• Production setup per batch x Annual number of special prod.runs
• Total cost = The annual set up cost for special production run + The annual handling cost for big order +Associated annual EOQ cost given L +The annual handling cost for small order.
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Deriving A Solution To The Model
• Enumeration• Search Methods • Algorithmic Solution Methods• Classical Methods of Calculus• Heuristic Solution Methods• Simulation
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Deriving A Solution To The Model
• Enumeration – Number of alternatives of action is relatively small.– Computational effort is relatively minor– Optimal solution is obtained by evaluating the
performance measure for each alternatives.• Search Methods
– e.g. Golden section search• Algorithmic Solution Methods
– A set of logical and mathematical operations performed repeatedly in a specific sequence
– Iteration– Stopping rules.
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Deriving A Solution To The Model
• Classical Methods of Calculus• Heuristic Solution Methods
– Impossible to find the optimal solution with the computational means currently available (intractable)
– If the optimal solution is possible to obtain, but the potential benefit do not justify the computational effort needed.
– Heuristic methods: to find a good solutions or to improve an existing solutions (out put based techniques)
• Simulation– For complex dynamic systems– To identify good policies rather than the optimal one.
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