2. Do the LP case (Son, Ltd.) on pg. 851 of the textbook. First do the LP formulation of the problem i.e. all the 4-steps of the LP formulation for just Question 1 (no need to do questions 2 and 3). Then solve the problem using LINGO program. Submit both the formulation (all 4-steps) and computer printout solution. Computer printout should include the highlighted optimal solution. Son, Ltd., manufactures a variety of chemical products used by photo-processors.

"Son was recently bought out by a conglomerate, and managers of the two organizations have been working together to improve the efficiency of Sons operations. Managers have been asked to adhere to weekly operating budgets and to develop operating plans using quantitative methods whenever possible.

The manager of one department has been given a weekly operating budget of $11,980 for production of three chemical products, which for convenience shall be referred to as Q, R, and W. The budget is intended to pay for direct labor and materials.

Processing requirements for the three products, on a per-unit basis, are shown in the table.

The company has a contractual obligation for 85 units of product R per week.

Material A costs $4 per pound, as does material B ($4). Labor costs $8 an hour.

Product Q sells for $122 a unit, product R sells for $115 a unit, and product W sells for $76 a unit.

The manager is considering a number of different proposals regarding the quantity of each product to produce. The manager is primarily interested in maximizing contribution. Moreover, the manager wants to know how much labor will be needed, as well as the amount of each material to purchase.

Questions Prepare a report that addresses the following issues: 1.The optimal quantities of products and the necessary quantities of labor and materials.

POTENTIAL WORKED THROUGH SOLUTION:

Step 1: Define the decision variables:

Let q = number of units of Product Q to produce.

Let r = number of units of Product R to produce.

Let w = number of units of Product W to produce.

Let A = number of pounds of materials used for each product under A.

Let B = number of pounds of materials used for each product under B.

Let L = number of labor hours used for each product under L.

Step 2: Define the objective function:

The objective is to maximize the contribution. The contribution is calculated as the revenue minus the cost.

Revenue:

Revenue from Product Q = $122

Revenue from Product R = $115

Revenue from Product W = $76

Cost:

Cost of Labor = $8

Cost of Material A = $4

Cost of Material B = $4

Unit Requirements for each product

Q = 5L + 2A + 1B

R = 4L + 2A + 0B

W = 2L + .5A + 2B

Cost from Product Q = 5(8) + 2(4) + 1(4) = 40+8+4 = $52 per unit

Cost from Product R = 4(8) + 2(4) + 0(4) = 32+8+0 = $40 per unit

Cost from Product W = 2(8) + .5(4) + 2(4) = 16+2+8 = $26 per unit

Profit Margin

Q = $122 - $52 = $70 per unit

R = $115 - $40 = $75 per unit

W = $76 - $26 = $50 per unit

Objective Function:

Maximum profit = 122*q + 115*r + 76*w

Maximum operating budget = (4*(2*q + 2*r + .5*w) + 4 *(1*q + 0*r + 2*w) + 8*(5*q + 4*r + 2*w) = 52*Q + 40*R + 26*W = $11980

Maximize Contribution = (122*q + 115*r + 76*w) - (52*Q + 40*R + 26*W 11980)

Step 3: Define the constraints:

Weekly operating budget constraint: 52*Q + 40*R + 26*W <= 11980

Contractual obligation for Product R: r = 85

Non-negativity constraint: q >= 0, r >= 0, w >= 0

Step 4: LP formulation:

Maximize Contribution = (122 * q + 115 * r + 76 * w) - (52*Q + 40*R + 26*W)

Subject to: 52*Q + 40*R + 26*W <= 11980 r = 85 q >= 0, r >= 0, w >= 0

Lingo

MAX=122 * q + 115 * r + 76 * w;

52*Q + 40*R + 26*W <= 11980;

r = 85;

What am I missing? Can you check my work and ensure this is correct?