HW1 Processing and Applications | Assignments Family and Consumer Science

IE 370, Spring 2026

Homework 1

You have to show how you arrive at solutions. A correct answer, without correct substantiation,

does not count as a solution.

Problem 1. You plan to go on a month-long hiking trip, mostly in the forest. You know that you

will need a bug repellant. But how much depends on the weather and other factors. From previous

experience you know the probability distribution of the number B of bug repellant cans that you

will need. It is as follows:

i|0 1 2 3 4 5 6 7 8 9 10

πi=P(B=i)|.02 .03 .05 .1.2.2.1.1.1.05 .05

You already have 2 cans. You can order more from Amazon, with the cost of $1.20 per can plus

$10 shipping and handling per the entire order. If you run out of bug repellant while on the trip,

you can buy it at the local stores, but there they cost $5 per can. (If you have some leftovers after

you return, you just keep them; cannot sell and do not need to get rid of.)

1.1 (25 points) If you want to minimize the expected cost, i.e. expected amount of money you

spend for the repellant, should you order additional cans from Amazon? If so, how many? What

is your expected total cost if you make the best decision?

1.2 (25 points) Consider the decision you made in 1.1. What is the probability that your total

cost will be strictly less than $10? What is the variance of cost that you will incur?

Problem 2. (50 points)

The problem setting is the one in the note “Multi-period inventory problem”, with setup cost

cf= 50. (We considered exactly same model in class.) Same parameters. Recall that the random

demand on each day is independent of the demands on other days, and has the same probability

distribution: it takes values 0,1,2,3,with equal probabilities 1/4.

But now consider the three period problem. Namely, suppose on Tuesday evening (i.e.

before Wednesday) we have 2 items on hand. How many items (if any at all) should you order

to arrive before Wednesday to minimize the expected total cost in the next three days (Wednesday,

Thursday and Friday)? What is the optimal expected total cost? Remember, you have a chance to

order items not only on Tuesday evening, but also on Wednesday evening and Thursday evening.

Remark 2.1. In class we essentially derived the general rule (Bellman equation for this particular

problem), which allows one to recursively compute functions vn+1(x) and yn+1(x) for period n+ 1,

via function vn(x) for period n:

vn+1(x) = min

y≥x

EC(x, y) + vn(y−D)+,

yn+1(x) = arg min

y≥x

EC(x, y) + vn(y−D)+,

where arg minyF(y) means a value of ywhich minimizes F(y).

Remark 2.2. It is known (from a general theory we are not going into) that function

F(y) = EC(x, y) + vn(y−D)+

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IE 370, Spring 2026

Homework 1

You have to show how you arrive at solutions. A correct answer, without correct substantiation, does not count as a solution.

Problem 1. You plan to go on a month-long hiking trip, mostly in the forest. You know that you will need a bug repellant. But how much depends on the weather and other factors. From previous experience you know the probability distribution of the number B of bug repellant cans that you will need. It is as follows:

i | 0 1 2 3 4 5 6 7 8 9 10 πi = P(B = i) |. 02. 03. 05. 1. 2. 2. 1. 1. 1. 05. 05

You already have 2 cans. You can order more from Amazon, with the cost of $1.20 per can plus $10 shipping and handling per the entire order. If you run out of bug repellant while on the trip, you can buy it at the local stores, but there they cost $5 per can. (If you have some leftovers after you return, you just keep them; cannot sell and do not need to get rid of.) 1.1 (25 points) If you want to minimize the expected cost, i.e. expected amount of money you spend for the repellant, should you order additional cans from Amazon? If so, how many? What is your expected total cost if you make the best decision? 1.2 (25 points) Consider the decision you made in 1.1. What is the probability that your total cost will be strictly less than $10? What is the variance of cost that you will incur?

Problem 2. (50 points) The problem setting is the one in the note “Multi-period inventory problem”, with setup cost cf = 50. (We considered exactly same model in class.) Same parameters. Recall that the random demand on each day is independent of the demands on other days, and has the same probability distribution: it takes values 0, 1 , 2 , 3 , with equal probabilities 1/4. But now consider the three period problem. Namely, suppose on Tuesday evening (i.e. before Wednesday) we have 2 items on hand. How many items (if any at all) should you order to arrive before Wednesday to minimize the expected total cost in the next three days (Wednesday, Thursday and Friday)? What is the optimal expected total cost? Remember, you have a chance to order items not only on Tuesday evening, but also on Wednesday evening and Thursday evening. Remark 2.1. In class we essentially derived the general rule (Bellman equation for this particular problem), which allows one to recursively compute functions vn+1(x) and yn+1(x) for period n + 1, via function vn(x) for period n:

vn+1(x) = min y≥x

E

C(x, y) + vn(y − D)+

yn+1(x) = arg min y≥x

E

C(x, y) + vn(y − D)+

where arg miny F (y) means a value of y which minimizes F (y). Remark 2.2. It is known (from a general theory we are not going into) that function

F (y) = E

C(x, y) + vn(y − D)+

is convex for y ≥ x + 1. It is not necessarily convex for y ≥ x, when Cf > 0. What this means, in particular, is that the cases y = x (order nothing) and y ≥ x + 1 (order something) do need to be evaluated separately. In the latter case (y ≥ x + 1), the convexity may be helpful when you compute the minimum of F (y). Remark 2.3. In your calculations you can assume that it never makes sense to have more than 5 items, i.e. y ≤ 5.

HW1 Processing and Applications, Assignments of Family and Consumer Science

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