Pertemuan 3 (LP-Formulation

7/29/2019 Pertemuan 3 (LP-Formulation

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Linear Programming:

Basic Sensitivity Analysis

Samsul Amar

References:

Hillier, F. S., & Lieberman, G. J., 2001, Introduction To Operations

Research, McGraw-Hill, New York. Taha, H. A., 2007, Operations Research: An Introduction, 8th ed.,

Pearson Prentice Hall, Upper Saddle River, NJ 07458.


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The Standard Formulation


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The Wyndor Glass Formulation:


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The Graphical Solution:


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Sensitivity Analysis

To determine which parameters of the model are

most critical (the sensitive parameter) in

determining the solution.

Typically, greater attention is given to performing

sensitivity analysis on the bi and cj parameters than

aijparameters.


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Changing cjParameters


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The solution will not change if:

0 c1/c

2 3/2


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Changing biParameters:

BindingConstraints


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Shadow price (dual price)

The shadow price for resource i (denoted by yi*)

measures the marginal value of this resource, i.e.,

the rate at which Z could be increased by (slightly)

increasing the amount of this resource (bi) beingmade available.


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Quiz

Find the solution of the following problem:

Max z = 40 x1 + 50 x2

Subject to

2500 x1 + 1000 x2 5000250 x1 + 500 x2 1500

x1, x2 0


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Linear Programming:

More Formulation

Samsul Amar

References:

Hillier, F. S., & Lieberman, G. J., 2001, Introduction To Operations

Research, McGraw-Hill, New York. Masruroh, N., 2011, Handout Metode Optimasi, Teknik Industri,

UGM.

Taha, H. A., 2007, Operations Research: An Introduction, 8th ed.,

Pearson Prentice Hall, Upper Saddle River, NJ 07458.


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The Standard Formulation


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Case 1 (Oil Blending Problem)

A refinery produces 3 grades of oil A, B, and C from 4

grades of crude oil 1, 2, 3, and 4.

Data

Crude Quantityavailable

Cost/barrel

1 3000 3

2 2000 6

3 4000 4

4 1000 5


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Case 1 (Cont.)

Blending requirements:

Oil A

Not more than 30% of 1

Not less than 40% of 2


Oil B


Not less than 10% of 2

Oil C



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Case 1 (Cont.)

Selling price A - $5.5

B - $4.5

C - $3.5

Problem: what blends of crude should be used to

make each grade of oil in order to maximize profitsubject to quantities available and blending

requirements?


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Case 2 (Product Mix)

A company has excess production capacity on three

types of machines which can be used to make two

products A and B

Data:

A B Cost/hr

Machining 25 units/hr 40 units/hr 20

Boring 28 units/hr 35 units/hr 14

Polishing 35 units/hr 25 units/hr 17.5

Selling price $5/unit $6/unit

Cost of material $2/unit $3/unit


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Case 2 (Cont.)

Problems:

How should the excess capacity on machines be

used to maximize profit?

Minimize cost while maximize revenue

Maximize utilization of the machinery


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Case 3 (Regional Planning)

A group of 3 farming communities (kibbutz) isplanning agricultural production for the coming year.

The agricultural output is limited by both the amount

of available irrigable land and the quantity of water

allocated for irrigation by the Water. The data are:


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Case 3 (Cont.)The crops suited for this region include sugar beets,

cotton, and sorghum. These crops differ primarily intheir expected net return per acre and their

consumption of water. In addition, the Ministry of

Agriculture has set a maximum quota for the totalacreage that can be devoted to each of these crops as

the following:


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Case 3 (Cont.)Because of the limited water available for irrigation, the

Group will not be able to use all its irrigable land forplanting crops in the upcoming season. To ensure equitybetween the three kibbutzim, it has been agreed thatevery kibbutz will plant the same proportion of its

available irrigable land. For example, if kibbutz 1 plants200 of its available 400 acres, then kibbutz 2 must plant300 of its 600 acres, while kibbutz 3 plants 150 acres of its300 acres. However, any combination of the crops may be

grown at any of the kibbutzim. The problem is to planhow many acres to devote to each crop at the respectivekibbutzim while satisfying the given restrictions. Theobjective is to maximize the total net return to the group

as a whole.


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Case 4 (Diet Problem)Ozark Farms uses at least 800 lb of special feed daily.

The special feed is a mixture of corn and soybeanmeal with the following compositions:

The dietary requirements of the special feed are atleast 30% protein and at most 5% fiber. Ozark Farmswishes to determine the daily minimum-cost feed

mix.


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Case 5 (Loan Policy)Emon Bank is in the process of devising a loan policy

that involves a maximum of $12 million. The followingtable provides the pertinent data about available

types of loans.


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Case 5 (Cont.)

Bad debts are unrecoverable and produce no interest

revenue.

Competition with other financial institutions requires

that the bank allocate at least 40% of the funds to

farm and commercial loans. To assist the housing

industry in the region, home loans must equal at least

50% of the personal, car, and home loans. Emon bank

also has a stated policy of not allowing the overallratio of bad debts on all loans to exceed 4%.

Pertemuan 3 (LP-Formulation

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