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MMT 9: Mathematical Modeling

Title Name	IGNOU MMT 9 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 9
Subject Name	Mathematical Modeling
Year	2026
Session	-
Language	English Medium
Assignment Code	MMT 9/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT009 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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MMT 9 2025 - English

Assignment

Course Code: MMT-009

Assignment Code: MMT-009/TMA/2025

Maximum Mark 100

1. a) Companies located on the banks of a river, dumping their chemicals waste into river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeds certain specified concentration limits. State, giving reasons, the type of modeling you will use to find a policy for discharging the waste to ensure that the concentration level never exceeds the specified limits. Also state four essentials and two non-essentials for the problem.

b) Consider the following data

x	2	3	4	5	6
y	8.3	16.5	30.2	65.2	125.6

Use a best fit line to estimate the value of y when x = 4.5.

2. a) Five securities have the following expected returns

A=20%, B=15%, C=25%, D=22%, E=18%. Calculate the expected returns for a portfolio consisting of all five securities under the following conditions

i) The portfolio weights are of equal percentage in each

ii) The portfolio weights are 32% in A and remaining are equally divided among other four securities.

b) Let P=(w₁,w₂) be a portfolio of two securities. If variance of P is minimum then find the value of w₁and w₂ in the following situations.

i) ρ12 = −1

ii)σ1 = σ2

iii)ρ12 = − ,5.0 σ1 = 5.1 and σ2 = .5.2

3. a) Assume that the return distribution on the two securities X and Y be as given below:

	Market 1	Market 2	Market 3
Probability	2.0	5.0	3.0
Security X	− 20%	18%	40%
Security Y	−10%	20%	15%

which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

b) In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.

4. A model for insect populations leads to the difference equations

$N_{k+1}=frac{lambda N_k}{1+aN_k}$

where λ and a are positive constants.

i) Write the equation in the form $N_{k+1}=N_k+R(N_k)N_k$ , and hence identify the growth rate.

ii) Plot the graph of R(N_k) as a function of N₂

iii) Express the intrinsic growth rater and the carrying capacity K, for this model, in terms of the parameters, a and λ.

iv) Find the steady-state solution of this model and analyse the solution.

5. Do the stability analysis of the following competing species system of equations with diffusion and advection

$egin{align*}frac{partial N_1}{partial t}&=a_1 N_1-b_1 N_1 N_2+D_1frac{partial^2 N_1}{partial x^2}-V_1frac{partial N_1}{partial x}\frac{partial N_2}{partial t}&=-d_1 N_2+c_1 N_1 N_2+D_2frac{partial^2 N_2}{partial x^2}-V_2frac{partial N_2}{partial x},quad 0leq xleq Lend{align*}$

where Vand V, are advection velocities in x direction of the two populations with densities N, and N, respectively. a, is the growth rate, b, is the predation rate, d, is the death rate, C, is the conversion rate. D, and D, are diffusion coefficients. The initial and boundary conditions are:

$N_i(x,0)=f_i(x)>0,quad 0leq xleq L,quad i=1,2,dots$

$N_i=overline{N} ext{at}x=0 ext{and}x=Lquadforall t,quad i=1,2,dots$

where $overline{N_{i}$ are the equilibrium solutions of the given system of equations. Interpret the solution obtained and also write the limitations of the model.

6. The population dynamics of a species is governed by the discrete model

$x_{n+1}=x_nexpleft[rleft(1-frac{x_n}{K} ight) ight]$

7. Do the stability analysis of the following model formulated to study the effect of toxicant on one competing species where the environment toxicant concentration is being taken to change w.r.t. time.

$egin{align*}frac{dN_1}{dt}&=r_1 N_1-alpha_1 N_1 N_2-d_1 C_0 N_1\frac{dN_2}{dt}&=r_2 N_2-alpha_2 N_1 N_2\frac{dC_0}{dt}&=k_1 P-g_1 C_0-m_1 C_0\frac{dP}{dt}&=Q-hP-kPN_1+gC_0N_2end{align*}$

along with the initial conditions.

$N_1(0)=N_{10},quad N_2(0)=N_{20},quad C(0)=0,quad P(0)=P_0>0$

Here,

N₁ (t)=Density of prey population

N₂(t)=Density of predator population

C_o(t) = Concentration of the toxicant in the individual of the prey population

P=Constant environmental toxicant concentration.

a,, a, are the predation rates, r,r, are the growth rates or birth rates, d, is the death rate due to C, m, is the depuration rate, Q, h, k, g are positive rate constants.

8. The owner of a readymade garments store sells two types of shirts: Zee-shirts and Button-down shirts. He makes a profit of Rs. 5 and Rs. 10 per shirt on Zee-shirts and Button-down shirt, respectively. He has two tailors, A and B at his disposal to stitch the shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively. Both these shirts are to be stitched by both the tailors. Tailors A and B spend 2 hours and 5 hours, respectively in stitching one Zee-shirts, and 4 hours and 3 hours, respectively in stitching a Button-down shirt. How many shirts of both types should be stitched in order to maximize daily profit?

a) Formulate and solve this problem as an LP problem.

b) If the optimal solution is not integer-valued, use Gomory technique to derive the optimal integer solution.

9. a) A company has three factories that supply to three markets. The transportation costs from each factory to each market are given in the table. Capacities of the factories and market requirements are shown. Find the minimum transportation cost.

	M₁	M₂	M₃	a_i
F₁	2	1	3	20
F₂	1	2	3	30
F₃	2	1	2	10
b_j	10	10	20	40/60

b) For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, p_o, = 0.056, calculate

i) The average time a customer is in the system

ii) The average number of customers in the system

iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?

10. a) Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter- arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?

b) A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

MMT 9 2026 - English

Assignment (MMT – 009)

Course Code: MMT-009
Assignment Code: MMT-009/TMA/2026
Maximum Marks: 100

1. a) A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 6 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.

b) Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?

Portfolio	Expected return	Standard deviation
W	10%	25%
X	5%	7%
Y	17%	37%
Z	12%	13%

2. a) Let G(t) be the amount of the glucose in the bloodstream of a patient at time t. Assume that the glucose is infused into the bloodstream at a constant rate of $g / \text{min}$ . At the same time, the glucose is converted and removed from the bloodstream at a rate proportional to the amount of the glucose present. If at $t = 0, G = 70\text{g}$ then

i) formulate the model.

ii) find g(t) at any time t.

iii) discuss the long term behavior of the model.

b) A tumour is developing from the organ of a human body with concentration $5.2 \times 10^9$ with growth and decay control parameters 7.2 and 2.7 respectively. In how many days the size of the tumor will be twice?

3. a) Return distributions of the two securities are given below:

Return		Probabilities
X	Y	`P_xj=P_yj=P_j`
0.20	0.15	0.30
0.15	0.08	0.25
0.10	0.05	0.15
0.11	0.09	0.25

Find which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

b) Let $\text{P} = (w_1, w_2)$ be a portfolio of two securities X and Y. Find the values of w₁ and w₂ in the following situations:

i) $\rho_{xy} = -1$ and P is risk free.

ii) $\sigma_x = \sigma_y$ and variance P is minimum.

iii) Variance P is minimum and $\rho_{xy} = -0.6, \sigma_x = 3$ and $\sigma_y = 5$ .

4. a) Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:

Customer Orders (in thousands)	Add-delete items (in thousands)	CPU time (in hours)
123.5	2.108	141.5
146.1	9.213	168.9
133.9	1.905	154.8
128.5	0.815	146.5
151.5	1.061	172.8
136.2	8.603	160.1
92.0	1.125	108.5

i) Find a linear regression equation that best fit the data.

ii) Estimate the error variance for the regression model obtained in i) above.

b) The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:

Husband’s age (years)	Wife’s age (years)	Husband’s age (years)	Wife’s age (years)
32	27	51	50
25	30	48	46
36	34	37	36
72	65	50	42
37	37	51	46
36	38	36	35

i) Draw a scatter plot of the data

ii) Write two important characteristics of the data that emerge from the scatter plot.

iii) Fit a linear regression model to the data and interpret the result in terms of the comparative change in the age of husband and wife.

iv) Calculate the standard error of regression and the coefficient of determination for the data.

5. Consider a discrete model given by

$$N_{t+1} = \frac{r N_t}{1 + b N_{t-1}^2} = f(N_t), r > 1.$
Investigate the linear stability about the positive steady state N* by setting $N_t = N^* + n_t$ . Show that n_t satisfies the equation
$$n_{t+1} - n_t + 2(r - 1)r^{-1}n_{t-1} = 0.$

Hence show that $r = 2$ is a bifurcation value and that as $r \rightarrow 2$ the steady state bifurcates to a periodic solution of period 6.

6. a) The population dynamics of a species is governed by the discrete model

$$x_{n+1} = x_n \exp \left[ r \left( 1 - \frac{x_n}{K} \right) \right],$

where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population for $r = 2 + \varepsilon$ , where $0 < \varepsilon \ll 1$ . Since a species becomes extinct if $x_n \leq 1$ for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity $k < r \exp[1 + e^{r-1} - 2r]$ .

b) Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.
$$\frac{dN_1}{dt} = r_0 N_1 - r_1 C_0 N_1 - b N_1 N_2$
$$\frac{dN_2}{dt} = -d_0 N_2 - d_1 V_0 N_2 + \beta_0 b N_1 N_2$
$$\frac{dC_0}{dt} = k_1 P - g_1 C_0 - m_1 C_0$
$$\frac{dV_0}{dt} = k_2 P - g_2 V_0 - m_2 V_0$
$$\frac{dP}{dt} = Q - hP - kP(N_1 + N_2) + gC_0N_1 + \ell V_0 N_2 \text{.}$

Where all the variables and constants are same as defined in the system (32)-(35) except for the following

$Q = \text{constant input rate}$

$h = \text{decay rate}$

$P(t) = \text{environmental toxicant concentration}$

$k = \text{ingestion rate of toxicant by the populations}$

$g, \ell = \text{return rate of toxicant in the environment after the death of the populations,}$
$\text{assuming that toxicant is non-degradable}$

$Q > 0, h, k, g, \ell \text{ are positive constants.}$

7. Do the stability analysis of the following competing species system of equations with diffusion and advection

$$\frac{\partial N_1}{\partial t} = a_1 N_1 - b_1 N_1 N_2 + D_1 \frac{\partial^2 N_1}{\partial x^2} - v_1 \frac{\partial N_1}{\partial x}$

$$\frac{\partial N_2}{\partial t} = -d_1 N_2 + C_1 N_1 N_2 + D_2 \frac{\partial^2 N_2}{\partial x^2} - v_2 \frac{\partial N_2}{\partial x}, \quad 0 \le x \le L$

where V₁ and V₂ are advection velocities in x direction of the two populations with densities N₁ and N₂ respectively. a₁ is the growth rate, b₁ is the predation rate, d₁ is the death rate, C₁ is the conversion rate. D₁ and D₂ are diffusion coefficients. The initial and boundary conditions are:

$N_i(x, 0) = f_i(x) > 0, \quad 0 \le x \le L, \quad i = 1, 2.$

$N_i = \overline{N}_i$ at $x = 0$ and $x = L \forall t, \quad i = 1, 2.$

where $\overline{N}_i$ are the equilibrium solutions of the given system of equations.

Interpret the solution obtained and also write the limitations of the model.

8. a) Maximize $Z = 5x_1 + x_2 + 3x_3$ , subject to the constraints
$-x_1 + 2x_2 + x_3 \le 4, 4x_2 - 3x_3 \le 2, x_1 - 3x_2 + 2x_3 \le 3$ and $(x_1, x_2, x_3) \ge 0$ and are integers.

b) Ships arrive at a port at the rate of one in every 6 hours with exponential distribution of inter-arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 12 hours. If the average delay of ships waiting for berths is to be kept below 15 hours, how many berths should be provided at the port?

c) A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 2 every 10 minutes and service time follows exponential distribution with a mean of 15 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

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