PROBLEM: A local bus company named “BAHON PORIBAHAN LIMITED” provides service from khilgaon to Mirpur-14.It starts service at 6 A.M and continues till 10 P.M. At present it has 50 drivers and many of them remain idle most of the time. The owner of the company feels that they have extra drivers and decided to maximize their profit by re-scheduling drivers. A driver can work for 8 hours and can remain idle for next 4 hours. Drivers start their job at 6 AM,10 AM,2 PM, 6 PM or 10 PM. Drivers will be paid depending on the number of trips completed. The number of buses varies from hour to hour ,so does the number of passengers. An OR team has been formed to optimize their number of drivers. From a recent survey, OR team has found the following data: Time Zone 6 AM---------------------10 AM 10 AM------------------- 2 PM 2 PM --------------------6 PM 6 PM---------------------10 PM Number of buses needed 14 09 12 07

We assume that requirement of buses is the same every day. The problem is to determine how many drivers to schedule at each starting time to cover the requirement for buses.

OPTIMIZATION: Variable Definition: x(t)=Number of drivers scheduled at time t; t= 6 ,10,14,18,22 This problem is for infinite number of days and x(t) is the number used every day at time t. Objective: So the main objective is…. Min z=x(6)+x(10)+x(14)+x(18) Constraint: Now we need constraints. For the rime interval 2 PM to 6 PM,drivers starting at time 10 AM and 2 PM cover the need from time 2 PM to 6 PM.

x (6) ≥ 14 x(6)+x(10) ≥ 9 x (10) + x (14) ≥ 12 x (14)+ x (18) ≥7 x (6) x (10) x (14) x (18) ≥ 0

...Scilab Datasheet
Optimization in Scilab
Scilab provides a high-level matrix language and allows to define complex mathematical models and to easily connect to existing libraries. That is why optimization is an important and practical topic in Scilab, which provides tools to solve linear and nonlinear optimization problems by a large collection of tools.
Overview of the industrial-grade solvers available in Scilab and the type ofoptimization problems which can be solved by Scilab.
Objective Linear Bounds y Equality l Inequalities l Problem size m m l l y Nonlinear s Gradient needed y n Solver linpro quapro qld qpsolve optim neldermead optim_ga fminsearch optim_sa lsqrsolve leastsq optim/"nd" optim_moga semidef lmisolve
Quadratic
y
l
l
s Nonlinear Least Squares Min-Max Multi-Obj. Semi-Def. y y l*
n
l m s l* l* l l
optional y n n
For the constraint columns, the letter "l" means linear, the letter "n" means nonlinear and "l*" means linear constraints in spectral sense. For the problem size column, the letters "s", "m" and "l" respectively mean small, medium and large.
Focus on nonlinear optimization
w The optim function solves optimization problems with nonlinear objectives, with or without bound constraints on the unknowns. The quasi-Newton method optim/"qn" uses a Broyden-Fletcher-Goldfarb-Shanno formula to update the approximate Hessian matrix. The...

...Answers are hi-lighted yellow.
Company A's nationally advertised brand is Brand X. Contribution to profit with Brand X is $40 per case.
Company A's re-proportioned formula is sold under a private label Brand Y. Contribution to profit with Brand Y is $30 per case.
Company A's objective is to maximize the total contribution to profit.
Three constraints limit the number of cases of Brand X and Brand Y that can be produced.
Constraint 1: The available units of nutrient C (n) is 30.
Constraint 2: The available units of flavor additive (f) is 72.
Constraint 3: The available units of color additive (c) is 90.
Material units per case of Brand X and Brand Y:
Product
Brand X Brand Y Formula for a case of Brand X = 4n+12f+6c
Nutrient C: 4 4 Formula for a case of Brand Y = 4n+6f+15c
Flavor Additive: 12 6
Color Additive: 6 15
Objective Function:
Max 40X + 30Y = total profit contribution
Constraint 1: Units of nutrient C used < units of nutrient C available
Units of nutrient C used = 4X + 4Y
30 units of nutrient C are available so the mathematical equation of constraint 1 is
4X + 4Y < 30
Constraint 2: Units of flavor additive used < units of flavor additive available
Units of flavor additive used = 12X + 6Y
72 units of flavor additive are available so the mathematical equation of constraint 2 is
12X + 6Y < 72
Constraint 3: Units of color additive used < units of color additive available
Units of color additive used = 6X + 15Y90 units of color additive are...

...Data Depth and Optimization
Komei Fukuda
fukuda@ifor.math.ethz.ch
Vera Rosta
rosta@renyi.hu
In this short article, we consider the notion of data depth which generalizes the median to higher dimensions. Our main objective is to present a snapshot of the data depth,
several closely related notions, associated optimization problems and algorithms. In particular, we brieﬂy touch on our recent approaches to compute the data depth using linear
and integeroptimization programming. Although the problem is NP-hard, there are ways
to compute nontrivial lower and upper bounds of the depth.
The notion of data depth has been studied independently in statistics, discrete geometry, political science and optimization. The motivation and the necessity in statistics to
generalize the median and the rank is very natural, as the mean is not considered to be a
robust measure of central location. It is enough to place one outlier to change the mean.
In contrast, the median in one dimension is very robust as half of the observations need
to be changed to corrupt the value of the median.
In nonparametric statistics, several data depth measures were introduced as multivariate generalizations of ranks to complement classical multivariate analysis, ﬁrst by Tukey
(1975), then followed by Oja (1983), Liu (1990), Donoho and Gasko (1992), Singh (1992),
Rousseeuw and Hubert (1999) among others. These measures, though seemingly...

...a corporation faces today is whether optimization, simulation, or a hybrid model (combination of optimization and simulation) is a better option to pursue.
In this paper, we fundamentally distinguish the two modeling approaches – Supply Chain Optimization vs. Supply Chain Simulation, and the scenarios where the each option should be employed.
Overview
Optimization focuses on finding the optimal solution from millions of possible alternatives while meeting the given constraints of the supply chain. Optimization utilizes mixed integer programming (MIP) or linear programming (LP) to obtain the optimal solution. Optimization models are used for network optimization, allocation management (refinery and terminals), route optimization (retail logistics) and vendor-managed inventory (retail network management).
Simulation identifies the impact of different variables on an organization’s entire supply chain. It answers the fundamental question – what will happen to the cost and service levels associated with a Supply Chain if an ‘X’ factor is manipulated The tool however does not drive to an optimal solution. Simulation also enables a user to visualize real world behavior of an “optimal solution” derived from the optimization. Simulation models are best suited for decision analysis, diagnostic evaluation, and project planning.
Supply Chain...

...MATH-2640 MATH-264001
This question paper consists of 3 printed pages, each of which is identiﬁed by the reference MATH-2640 Only approved basic scientiﬁc calculators may be used.
c UNIVERSITY OF LEEDS Examination for the Module MATH-2640 (January 2003)
Introduction to Optimisation
Time allowed: 2 hours Attempt four questions. All questions carry equal marks. In all questions, you may assume that all functions f (x1 , . . . , xn ) under consideration are sufﬁciently ∂2f ∂2f continuous to satisfy Young’s theorem: fxi xj = fxj xi or ∂xi ∂xj = ∂xj ∂xi . The following abbreviations, consistent with those used in the course, are used throughout for commonly occurring optimisation terminology: LPM – leading principal minor; PM – (non-leading) principal minor; CQ – constraint qualiﬁcation; FOC – ﬁrst-order conditions; NDCQ – non-degenerate constraint qualiﬁcation; CSC – complementary slackness condition; NNC – non-negativity constraint.
Q1 (a) You are given that the formula for the total differential at the point x0 of a function f of n variables x1 , . . . , xn is
1 δf (x0 ) = δx· f (x0 ) + 2 (δx)T H(x0 )(δx) + O |δx|3 ,
where x = (x1 , . . . , xn )T , the Hessian of f at x0 .
∂ ∂ ≡ ( ∂x1 , . . . , ∂xn )T is the n-dimensional gradient operator and H(x0 ) is
(i) Deﬁne: the total differential in terms of f , x0 and δx; the Hessian matrix H in terms of f and x0 ; the kth LPM of the Hessian H. (ii) What is meant by saying that x∗ is a stationary point of f ?...

...Management of High Tech Services (HTS) would like to develop a model that will help allocate its technician's time between service calls to regular contract customers and new customers. A maximum of 80 hours of technician time is available over the two-week period. To satisfy cash flow requirements, at least $800 in revenue must be generated through technician time during the two-week period. Technician time for regular customers generates $25 per hour. However, technician time for new customers only generates an average of $8 per hour because in many cases a new customer contact does not provide billable services. To ensure that new customer contacts are being maintained, the technician time spent on new customer contacts must be at least 60% of the time spent on regular customer contacts. Given these revenue and policy requirements, HTS would like to determine how to allocate technician time between regular customers and new customers so that the total number of customers contacted during the two-week period will be maximized. A technician can contact on average 1.2 regular customers per hour and 1 new customer per hour. Develop a linear programming model that will enable HTS to allocate technician time between regular customers and new customers. Report the optimal solution.
Management of High Tech Services (HTS) would like to develop a model that will help allocate its technician's time between service calls to regular contract customers and new customers. A...

...Inventory Optimization at Procter & Gamble
As a leader organization, P&G has an impact in the daily lives of billions of people every day. But accomplishing such a feature has very complex planning and delivering method. This article published in the INFORMS Journal on the Practice of Operations Research, decorticates de process which P&G went through while tackling the matter of the inventory throughout the organisation and how to both, minimize costs and improving customer satisfaction (fill rate).
Since the 70’s, P&G has applied inventory management techniques in the organization, always striving to optimize resources and counted with a horizontal planning tool known as horizontal process networks (HPN), in order to coordinate the inventory planning across the various global business units (GBU). Nonetheless, due to globalization, expansion of the organisation throughout 26 product markets with more than 200 brands and overall sales over $76 billion dollars; managing a cost-efficient supply chain became an exponentially increasing challenge for P&G.
In order to address this, the organisation implements a progressive solution. Its first part consists of applying localized optimization models, through spreadsheets. These were generated at single-staged levels and managed by planners in order to optimize inventory, mainly. The second part of the inventory solution for P&G was a multiechelon model that was first implemented in North...

...COST OPTIMIZATION FOR WORKFORCE SCHEDULING
An Operations Research Report
IET4405
Submitted to Gregory Wiles
By
Group 3:
Amanda Bathe
Eminue Eminue
Danielle Harmon
Theophilus Holley
Coltan Palmer
Abigale Vining
April 24, 2013
EXECUTIVE SUMMARY
This report presents a recommendation for a hiring schedule that would allow Davis Instruments to hire temporary employees based on the product demand that varies from month to month for the six months of January to June, at the lowest cost.
Figure 2
Full-time and Temporary Employee Suggested Schedule
Figure 2
Full-time and Temporary Employee Suggested Schedule
Temporary Employee Suggested Schedule
Figure 1
Temporary Employee Suggested Schedule
Figure 1
In order to find the hiring schedule that would result in the least incremental training and salary costs, a linear programming model was formulated and solved. Temporary employees can be hired under three options. Option one allows the temporary employee to work for one month, Option two for two months, and Option three for three months. The training for all temporary employees occurs once, each time they are hired, even if they have worked for Davis before. The recommended hiring schedules for temporary workers is displayed below, next to a recommended hiring schedule for temporary workers assuming Davis first hired 10 full-time positions.
Figure 1 suggests 43 temporary workers for the 112 available positions. The total cost of...