Assignment 1: Linear Programming Applications in Operations Management

Problem Statement:

An electronics manufacturing company specializes in producing custom-designed electronic devices using highly skilled labor. The company has received an order for two types of devices: Device A and Device B. For each unit of Device A sold, the company earns a profit of $3500, while each unit of Device B generates $5300 in profit. The production manager is evaluating how many units of each device should be manufactured during the next production cycle. The company has 96,000 labor hours available in its assembly and testing facility for the upcoming month. Manufacturing a single unit of Device A requires 12 labor hours, while Device B demands 21 labor hours. Due to the recent shutdown of a major supplier of rare electronic components, there is now a shortage of a critical component required in both devices. The company has managed to obtain only 20,000 units of this component for use next month. Producing a unit of Device A requires 4 units of this component, while Device B needs 2 units per unit. Additionally, due to softened market demand, the company will not produce more than 3,500 units of Device B in the upcoming period, as they do not expect to sell more than that amount.

Problem Statement:

Two products, Assembly A and Assembly B, are being manufactured in an assembly shop in Galway. Each unit of Assembly A is manufactured after completing three assembly steps, assembly step 1 (for three hours), assembly step 2 (for two hours), and a final assembly step (for 1.5 hours). Each unit of Assembly B is manufactured after completing the same three assembly steps: assembly step 1 (for two hours), assembly step 2 (for one hour), and the final assembly step (for 0.5 hours). For the upcoming production period, the available production capacity (in time units) for each assembly step is as follows: assembly step 1: 240 hrs, assembly step 2: 210 hrs, and the final assembly step: 120 hrs. The estimated profit generation per unit sold for Assembly A is $22, and Assembly B is $15.

Problem Statement:

A Linear Programming model has the following constraint sets:

-10X1 + 20X2 ≤ 500

-20X1 + 10X2 ≤ 500

Problem Statement:

A humanitarian organization is organizing an emergency airlift to deliver vital supplies (denoted as Relief Packages A, B, and C) to a disaster-stricken region. The objective is to maximize the impact value of the supplies delivered by efficiently using their cargo aircraft, which is divided into multiple compartments with strict weight and volume restrictions for safety and balance. Relief Packages A, B, and C have 10, 12, and 17 tons available,respectively.

The estimated impact value per ton of each package is 700, 725, and 685 units, reflecting how crucial each type is to the affected population. Each package also varies in size:

• Package A: 2,000 cu. ft. per ton
• Package B: 3,500 cu. ft. per ton
• Package C: 3,000 cu. ft. per ton

The aircraft has a maximum carrying capacity of 32 tons, divided into several compartments, each with its own weight and volume limits. For balance and structural integrity, the total weight must be distributed across compartments according to fixed ratios. Table 1 tabulates the shipment distributions and their requirements.

Determine the optimal tonnage and placement of each relief package into the aircraft compartments to maximize the overall impact value delivered to the region, while respecting all volume, weight, and balance constraints. Formulate a Linear Programming (LP) model (present the decision variables, objective function, and constraints) and solve it using Microsoft Solver.

Problem Statement:

Table 2 displays data collected from a medical device manufacturing facility that assembles three types of devices—D1, D2, and D3—following a defined assembly sequence across four production units. The production output rates, measured in units per hour, for each unit are listed in Table 2. For example, D1 requires processing only in Production Units U1 and U2, with Unit U1 capable of assembling 20 D1 units per hour. The company owner is aiming to maximize profits by determining the optimal mix of devices to manufacture in the upcoming month.

The company has the following constraints:

• Available working hours per production unit: 150 (U1), 160 (U2), 130 (U3), 100 (U4).
• Production Units U1 and U2 must be scheduled for at least 100 hours each.
• A minimum monthly demand of 1,000 units for Device D1 must be met.
• Only 4,200 units of a specialized component are available for the coming month. Each unit of D1 requires one of these components, while each unit of D3 requires three.

The main goal is to ensure that they meet their production goals and satisfy customer demand in the most profitable manner.