Suppose the cost structure with respect to the net inventory level is similar to that in the HW problem on EOQ models

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

ORIE 4120 Spring 2022

Dr. David A. Goldberg

Cornell

Homework 2, 40 points

Q1 (12 points). In this problem, we will model a multi-stage inventory problem as a linear program. In particular, suppose you run a small business, selling a single item.

First, we will model the single-stage problem as a linear program.

Thus consider the following simple setup.

At the start of today (day 1), you have 0 inventory.

At the start of today, you select a (possibly fractional) amount of inventory o1 to order.

You receive the o1 units instantaneously.

Suppose it costs you c1 dollars per unit of inventory you order.

You know the demand for your product that arrives today will be D1.

Suppose all demand for the day arrives at a single moment in the middle of the day.

Suppose that (as in the EOQ model with backlogging), your net inventory level can go negative. In particular, if D1 > o1, then your net inventory level at the end of the day is negative. Of course, it may also be positive.

Let I1 denote your net inventory level at the end of Day 1.

a. (1 point) Express I1 explicitly in terms of o1 and D1

Suppose the cost structure with respect to the net inventory level is similar to that in the HW problem (in HW 1) on EOQ models, although here without any notion of infinitesimals or integrals. In particular, suppose that at the end of the day you incur a one-time inventory cost equal to h1×max(0, I1)+b1×max(0, −I1).

b. (1 point) Express the total cost incurred on day 1, including ordering costs and inventory costs, explicitly in terms of o1, c1, D1, b1, h1.

c. (4 points) Consider the following linear program

min c1 × o1 + h1 × I+ 1 + b1 × I− 1

s.t.

I1 = o1 − D1 I+ 1 ≥ I1 I+ 1 ≥ 0

I− 1 ≥ −I1 I− 1 ≥ 0 o1 ≥ 0

i. (2 points) Argue that in any feasible solution, I+ 1 must be at least max(0, I1) and I− 1 must be at least max(0, −I1)

ii. (2 points) Argue that in any optimal solution, one would never set I+ 1 to be strictly greater than max(0, I1); and similarly one would never set I− 1 to be strictly greater than max(0, −I1). Conclude that in any optimal solution, it will hold that I+ 1 = max(0, I1) and I− 1 = max(0, −I1).

d. (1 point) Based on your answers to a.,b., and c., argue that the value for o1 in the solution to the linear program from part c. gives the optimal amount to order in the stated inventory system.

Consider now a multi-period version of the problem considered in a. - d. In particular, suppose now you are optimizing an inventory system with similar dynamics over T days.

Attachments:

Related Questions

. The fundamental operations of create, read, update, and delete (CRUD) in either Python or Java

CS 340 Milestone One Guidelines and Rubric  Overview: For this assignment, you will implement the fundamental operations of create, read, update,

. Develop a program to emulate a purchase transaction at a retail store. This  program will have two classes, a LineItem class and a Transaction class

Retail Transaction Programming Project  Project Requirements:  Develop a program to emulate a purchase transaction at a retail store. This

. The following program contains five errors. Identify the errors and fix them

7COM1028   Secure Systems Programming   Referral Coursework: Secure

. Accepts the following from a user: Item Name Item Quantity Item Price Allows the user to create a file to store the sales receipt contents

Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip

. The final project will encompass developing a web service using a software stack and implementing an industry-standard interface. Regardless of whether you choose to pursue application development goals as a pure developer or as a software engineer

CS 340 Final Project Guidelines and Rubric  Overview The final project will encompass developing a web service using a software stack and impleme