" This is a multiperiod production problem scheduling problem. Plastic Molding manufacturer is planning the production storage and marketing of a product whose demand and selling price vary seasonally.The company is in the process of developing a manufacturing strategy. According to the formulated strategy, the company wants to manufcature the product in the seasons during which production costs are low, and then store and sell product when selling price is high. The manufacturer has obtained the following estimates of the demand, cost, capacity, and expected selling price of the product for each period.

*Let Pj = Product made (in tons) during period j.*

*Ij= product left in storage at the end of period j*

*Dj= product demanded during period j*

*Beginning inventory = 500 tons*

*Desired ending inventory = 300 tons*

*(inventory ) storage cost = $2 per ton form one period to the next*

**Period ----Production cost ----Production capacity---- Demand---- Price**

**1 ---------20------------------ 1500---------------------1100 -------180**

**2 ---------25------------------ 2000--------------------- 1500 -------180**

**3 ---------30 ------------------2200 --------------------1800 --------250**

**4 --------40------------------- 3000 --------------------1600 --------270**

**5 -------50 --------------------2700 --------------------2300 --------300**

**6 -------60 --------------------2500-------------------- 2500 --------320**

__Formulate the linear programming problem that provides the optimal production schedule and maximizes the total profit__

*And also the second problem:*BH Canada designs and manufacturers high-grade palstic based products for industrial, commercial, and residential use. The company is considering leasing a local warehouse to fullfill its needs for aditional storage spaces over the next 4 months

**Month --Additional storage space**

**1 -------5000**

**2 -------3500**

**3 -------6500**

**4 -------4000**

A local warehouse company has offered short term leasing options that differ in terms of the length of lease

**Option ---Length of lease--- Cost per square metre**

**1 ---------1 month ----------1.50**

**2--------- 2 months ---------2.10**

**3 ---------3 months--------- 2.85**

**4--------- 4 months ----------3.40**

Given that the monthly reguirements for additional storage spaces are variable, the company considering acquiring lease options that differ in terms of length and space size at the beginning of each month

Formulate a linear programming model that can be used by the company to determine the minimum cost for additional storage space

Thanks in advance