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25TH JUNE LINEAR PROGRAMMING
Creation date：2024-05-14 16:50:55
Belonging category：计算机computer science
Tag DIGITAL
LINEAR PROGRAMMING

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888411 OPERATIONS
RESEARCH FOR DIGITAL
INNOVATION
LINEAR PROGRAMMING I
Introduction to Linear Programming
The most common type of application involves the general problem of allocating
limited resources among competing activities in a best possible way
Introduction to Linear Programming
The adjective linear means that all the mathematical functions in this model are
required to be linear functions.
Introduction to Linear Programming
The adjective linear means that all the mathematical functions in this model are
required to be linear functions.
PROTOTYPE EXAMPLE
The WYNDOR GLASS CO. produces high-quality glass products, including windows and
glass doors. It has three plants. Aluminum frames and hardware are made in Plant 1, wood
frames are made in Plant 2, and Plant 3 produces the glass and assembles the products.
Plant 1 Plant 2 Plant 3
PROTOTYPE EXAMPLE
top management has decided to revamp the company’s product line. Unprofitable products are
being discontinued, releasing production capacity to launch two new products having large sales
potential:
Product 1: An 8-foot glass door with aluminum framing
Product 2: A 4 x 6 foot double-hung wood-framed window
Product 1 Product 2
PROTOTYPE EXAMPLE
Product 1 need only plant 1 and 3. Product 2: need only plant 2 and 3.
The marketing division has concluded that the company could sell as much of either
product as could be produced by these plants
Determine what the production rates should be for the two products in order to maximize
their total profit, subject to the restrictions imposed by the limited production capacities
available in the three plants.
PROTOTYPE EXAMPLE
The OR team also identified the data that needed to be gathered:
• Number of hours of production time available per week in each plant for these
new products.
• Number of hours of production time used in each plant for each batch
produced of each new product.
• Profit per batch produced of each new product
Obtaining reasonable estimates of these quantities required enlisting the help of key
personnel in various units of the company
PROTOTYPE EXAMPLE
PROTOTYPE EXAMPLE: Formulation as a Linear Programming Problem
Let assign variables as follows:
The objective is to choose the values of and so as to maximize = +
PROTOTYPE EXAMPLE: Formulation as a Linear Programming Problem
Objective Function
Functional Constraints
Nonnegativity Constraints
https://www.geogebra.org/m/epfhezbp
PROTOTYPE EXAMPLE: Graphical Solution
Find the feasible region. The resulting region of permissible values of
PROTOTYPE EXAMPLE: Graphical Solution
Find slope-intercept form of the objective function.
PROTOTYPE EXAMPLE: Graphical Solution
Move the ruler with fixed slope through the feasible region in the direction of improving Z.
(When the objective is to minimize Z, move the ruler in the direction that decreases Z.)
Stop moving the ruler at the last instant that it still passes through a point in this region.
This point is the desired optimal solution.
PROTOTYPE EXAMPLE: Conclusion
This solution indicates that the Wyndor Glass Co. should produce products 1 and
2 at the rate of 2 batches per week and 6 batches per week, respectively, with a resulting
total profit of $36,000 per week. No other mix of the two products would be so profitable—
according to the model.
Are we finished?
No! this is just the step 2 out of 6 steps
Let’s Try!
https://www.geogebra.org/graphing
Suppose that the profit per batch of product 2 is change to $2,000. What is the optimal
produce rate for product 1 and product 2 to gain maximize profit?
The Weakness of Graphical method in Linear programming
It can be used to solve any linear programming problem with two decision variables. With
considerable difficulty, it is possible to extend the method to three decision variables but
not more than three.
The algorithm for this method is take too much resource from computer.
The Linear Programming Model
The Linear Programming Model
Let assign variables as follows:
The Linear Programming Model: A Standard Form of the Model
Objective Function
Functional Constraints
Nonnegativity Constraints
The Linear Programming Model: A Standard Form of the Model
The Linear Programming Model: Other Forms
How we use Graphical method?
What is the weakness of graphical method?
Recap questions:
Terminology for Solutions of the Model
A solution: any specification of values for the decision variables (1, 2, … , ), regardless of
whether it is a desirable or even an allowable choice.
A feasible solution is a solution for which all the constraints are satisfied.
An infeasible solution is a solution for which at least one constraint is violated.
Terminology for Solutions of the Model
An optimal solution is a feasible solution that has the most favorable value of
the objective function.
It is possible to have no feasible solution or have more than one optimal solution.
Terminology for Solutions of the Model
Is it possible to have feasible region but have no optimal solution?
Terminology for Solutions of the Model
A corner-point feasible (CPF) solution is a solution that lies at a corner of the
feasible region.
Terminology for Solutions of the Model
Relationship between optimal solutions and CPF solutions: Consider any linear
programming problem with feasible solutions and a bounded feasible region. The problem
must possess CPF solutions and at least one optimal solution.
Furthermore, the best CPF solution must be an optimal solution. Thus, if a problem has
exactly one optimal solution, it must be a CPF solution. If the problem has multiple
optimal solutions, at least two must be CPF solutions.
Assumptions of Linear Programming
We need to see why the OR team for the Wyndor Glass Co. concluded that a linear
programming formulation provided a satisfactory representation of the problem.
• Proportionality
• Additivity
• Divisibility
• Certainty
Assumptions of Linear Programming: Proportionality
Proportionality assumption: The contribution of each activity to the value of
the objective function is proportional to the level of the activity .
Similarly, the contribution of each activity to the left-hand side of each functional
constraint is proportional to the level of the activity .
Assumptions of Linear Programming: Proportionality
-1
Assumptions of Linear Programming: Proportionality
Assumptions of Linear Programming: Proportionality
-1
Assumptions of Linear Programming: Proportionality
This violation of proportionality might occur because of economies of scale that can
sometimes be achieved at higher levels of production.
Assumptions of Linear Programming: Proportionality
-1
Assumptions of Linear Programming: Proportionality
It might be possible to sell product 1 at the rate of 1 per week with no advertising, whereas
attaining sales to sustain a production rate of 2 might require a moderate amount of advertising.
Assumptions of Linear Programming: Additivity
Additivity assumption: Every function in a linear programming model (whether
the objective function or the function on the left-hand side of a functional constraint)
is the sum of the individual contributions of the respective activities.
Assumptions of Linear Programming: Additivity
This case would arise if the two products were complementary in some way that increases profit.
Assumptions of Linear Programming: Additivity
This case would arise if the two products were competitive in some way that decreased their
joint profit.
Assumptions of Linear Programming: Additivity
Assumptions of Linear Programming: Additivity
extra time is wasted switching the production processes back
and forth between the two products.
Assumptions of Linear Programming: Additivity
Because each product goes through a sequence of production operations, individual production
facilities normally dedicated to that product would incur occasional idle periods. During these
otherwise idle periods, these facilities can be used by the other product.
Assumptions of Linear Programming: Divisibility
Divisibility assumption: Decision variables in a linear programming model are
allowed to have any values, including noninteger values, that satisfy the functional
and nonnegativity constraints. Since each decision variable represents the level of some
activity, it is being assumed that the activities can be run at fractional levels.
In certain situations, the divisibility assumption does not hold because some of or all
the decision variables must be restricted to integer values.
Assumptions of Linear Programming: Certainty
Certainty assumption: The value assigned to each parameter of a linear programming
model is assumed to be a known constant.
The Assumptions in Perspective
• It is very common in real applications of linear programming that almost none of the four
assumptions hold completely.
• This is especially true for the certainty assumption, so sensitivity analysis normally is a must to
compensate for the violation of this assumption.
• However, it is important for the OR team to examine the four assumptions for the problem
under study and to analyze just how large the disparities are.
• If any of the assumptions are violated in a major way, then use other models.
• For some applications, the powerful linear programming approach is used for the initial analysis,
and then a more complicated model is used to refine this analysis.
Additional Examples: Radiation therapy
Mary has just been diagnosed as having a cancer at a fairly advanced stage. Specifically,
she has a large malignant tumor in the bladder area.
Mary is to receive the most advanced medical care
available to give her every possible chance for survival.
This care will include extensive radiation therapy
The two decision variables 1 and 2 represent the dose (in kilorads) at the entry point for beam 1
and beam 2, respectively. Because the total dosage reaching the healthy anatomy is to be
minimized, let Z denote this quantity.
Additional Examples: Radiation therapy
Additional Examples: Radiation therapy
Additional Examples: Radiation therapy
Additional Examples: Regional Planning
The SOUTHERN CONFEDERATION OF KIBBUTZIM currently is planning agricultural production
for the coming year.
Sugar beets Cotton Sorghum
Additional Examples: Regional Planning
Additional Examples: Regional Planning
Additional Examples: Regional Planning
What are assumptions of linear programming?
Why, in some case, we still use linear programming model
even though many assumptions are violated?
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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