Question

contract to produce 10 items of x per week for a particular customer. • Part (a). Formulate the problem of deciding how much to produce per week as a linear program. • Part (b). Draw the feasible region of your LP formulation. • Part (c). Find the optimal solution of your LP model using graphical solution method. 2. Consider a factory that produces tables. The demand for the tables is known for the coming four months. The manager wants the make a planning of the production quantity in each period. There are two types of costs: cost for each produced table (the production cost) and cost for holding tables in inventory at the at the end of each period (the inventory holding cost). At the start of period 1 there are 5 tables in inventory. The demand and costs per month are given in the following table: Table 2: Monthly demands and costs month 2 3 demand 15 20 30 production cost per table 7 3 holding cost per table 1 2 3 5 4 25 6 2 Furthermore, the factory has a limited production capacity and at most 35 tables can be produced per month. Finally, tables in inventory get damaged during the storage operations and, as a result, 10 percent of the ending inventory is lost in each period. The manager wants to make a production planning at minimum cost such that all demand needs to be satisfied and such that the above infor- mation is taken into account. Formulate this problem as a linear programming (LP) problem. Define the decision variables and provide a clear explanation of the constraints of your model. (Hint: You do not need to take into account the fact that an integer number of tables should be produced).

(I just need Question 2)Please show the steps clearly

Answer 1

LP model is formulated as under:

Decision variables:

Let Pi be the number of tables produced in month i

Vi be the number of tables held in inventory at the end of month i

Objective function:

Min 7P1+3P2+5P3+6P4+1V1+2V2+3V3+2V4   (total production cost and holding cost)

Constraints:

P1-V1+V0 = 15   (production - ending inventory + opening inventory = demand of month 1)

P2-V2+0.9V1 = 20   (production - ending inventory + 90% of opening inventory = demand of month 2)

P3-V3+0.9V2 = 30   (production - ending inventory + 90% of opening inventory = demand of month 3)

P4-V4+0.9V3 = 25   (production - ending inventory + 90% of opening inventory = demand of month 4)

P1, P2, P3, P4 <= 35   (production capacity of each month)

V0 = 5   (beginning inventory of month 1 or ending inventory of month 0)

Pi, Vi >= 0