L.7. Analisis Model Matematis

Pemodelan Sistem (TKI 128)Teknik IndustriUNIJOYO

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Analisis Model 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

• Mathematical Model: [Empirical Model & Theoretical Model]

• Classification of Mathematical Models• Analysis of Mathematical Formulations-I• Model Testing & Sensitivity Analysis


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Mathematical Model• A mathematical model of a system

– A symbolic representation involving a abstract mathematical formulation

• A mathematical formulation– composed of symbols, and makes no sense outside

mathematics– not a model by itself– it is only by relating the mathematical formulation to a

system characterization (i.e. variables and relationships)


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Empirical vs Theoretical • Empirical Model

– The characterization is based on no theory or knowledge.

– The system is viewed as a black box– The mathematical formulation to serve as a dummy

- must be selected on an ad-hoc basis– Models called empirical models

• Theoretical Model– The characterization is done using well established

theory– Models called theoretical models


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Classification of Mathematical Models

• Four categories based on mathematical structure of the underlying formulation– A: Formulations suitable for modelling deterministic

static systems– B: Formulations suitable for modelling deterministic

dynamic systems– C: Formulation suitable for modelling probabilistic

static systems– D: Formulations suitable for modelling stochastic

dynamic systems


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Mathematical Formulations• Deterministic static Formulations

– e.g. Linear Programming• Deterministic dynamic Formulations

– e.g. Dynamic Programming• Probabilistic static Formulations

– e.g. Regression Analysis; Design of experiment • Stochastic dynamic Formulations

– e.g. Markov Process; Renewal Process


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Mathematical Formulations-I

• The math. Formulations suited for modelling deterministic systems– Static Formulations

• involve either algebraic equations or function optimization

– Dynamic Formulations • involve two types of variables -dependent variables

and independent variables, e.g. X(t)


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Analysis of Mathematical Formulations-I

• Step 1:– Discard the association so that only the math.

formulation is retained.• Step 2:

– Carry out an analysis of the formulation using appropriate math techniques

• Step 3:– Re-introduce the discarded association so that the

analysis can be interpreted in term of physical variables of the system characterization to yield model behaviour


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Types of Analysis• Qualitative Analysis

– deals with the study of qualitative aspects of a given mathematical formulation without explicitly solving it

• Quantitative Analysis– concerned with finding the explicit solution which

satisfies the given math formulation.• Analytical methods (the solution can be exact or

approximate)• Computational methods (the solution is only

approximate and depends on the method used)


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Model Testing & Sensitivity Analysis

• Internal Validity (Verification)– Is the model mathematically correct and logically

consistent?– This also involves verifying that each expression is

dimensionally consistent.• External Validity (Validation)

– Is the model a sufficiently valid representation of reality?

• Testing the solution performance– To determine the expected benefits, such as net

profits or net savings


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Analysis of Sensitivity of Solution

• Analysis of Sensitivity of Solution– Sensitivity analysis (evaluate the response

of the best solution to changes in various inputs)

– Error analysis • The input parameters are estimated on the basis of

past data• There is no guarantee that the future will be similar

to the past• There could be an error in input data


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Rules for Testing Validity

• The evaluation of the proposed policy has to be based on observations of actual (or simulated) performance

• The data used for the test should be independent of the data used to derive the best policy

• The test should not just give expected performance, but also some measure of its variability, such the standard deviation


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Sensitivity Analysis: [Purposes]• If the optimal solution is relatively insensitive,

then the decision maker and user can place more confidence in the validity and usefulness of the model

• Sensitivity analysis provides information about the value of additional amounts of each scarce resource (shadow price of the resource)

• Sensitivity analysis is used for exploring how the optimal solution changes as a function of such uncertain data


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Procedure of Error Analysis1.Determine the optimal policy based on the best estimate

values for all input parameters (assume that one of these, say p, is in error).

2.Assume that the value of the input parameter p differs from the correct value, P (p=kP). Find the optimal policy, using the (assumed) correct value P.

3.Compute the actual value of the objective function if the pseudo-optimal policy determined in (1) were implemented.

4.Find the difference between the optimal objective function values obtained from (2) –using the correct value and (3) –using the estimate value.


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CASE OF “LOD”

• q=50.87• The demand actual is not 4140 (but only

2875) – overestimated: 44%• The true optimal: EOQ=42.4;

T(q=42.4)=2442• T(q=50.87)=2482 with D=2875• [T(q)-T(EOQ)]/ T(EOQ)x100% =1,64%


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Keputusan Pembelian Terigu

• Pabrik roti membeli terigu dengan harga $1000 per ton. Kebutuhan terigu relatif konstan selama setahun dengan total permintaan per bulan sebesar 20 ton. Terigu dikirim dari pabrik terigu dengan menggunakan truk dan ongkos per sekali kirim $132, tidak tergantung dari jumlah terigu yang diangkut.

• Uang yang digunakan untuk membeli terigu berasal dari suatu investasi dengan interest sebesar 8% per tahun. Juga, terigu yang disimpan diasuransikan dengan premi 16% yang dihitung berdasarkan nilai rata-rata persediaan per tahun. Manajer pembelian ingin mendapatkan kebijakan pembelian terigu yg lebih baik dari yang terjadi sekarang.


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System Description


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Mathematical Model


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Mathematical Model


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Mathematical Model


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EOQ Model


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Solution


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System Description


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Validation:[Internal]


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Validation:[Internal]


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System/Model Overview

L.7. Analisis Model Matematis

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