DAKOTA FURNITURE COMPANY

Transcription

1 DAKOTA FURNITURE COMPANY AYMAN H. RAGAB 1. Introduction The Dakota Furniture Company (DFC) manufactures three products, namely desks, tables and chairs. To produce each of the items, three types of resources are required: lumber board feet, finishing hours and carpentry hours. It is required to determine: How much of each item to produce The corresponding resource requirements Following the above requirements, the decision variables should be: (1) Y = y 1,y 2,y 3 T : the amount of products to produce from items 1,...,3. (2) X = x 1,x 2,x 3 T : the required amount of resources 1,...,3. where items 1,2 and 3 and resources 1,2 and 3 refer to desks, tables and chair, and lumber board feet, finishing hours and carpentry hours correspondingly. The cost of unit resource is given by the cost vector: C = c 1,c 2,c 3 = 2, 4, 5.2, while the furniture selling price per unit is given by the revenue vector: R = r 1,r 2,r 3 = 60, 40, 10. The resource requirements per product are expressed by the matrix B below. B = b 1,1 b 1,2 b 1,3 b 2,1 b 2,2 b 2,3 = b 3,1 b 3,2 b 3, It is implied that the demand is realized before the production decision is made, and therefore the amount produced is the same as that sold (since there is no advantage in producing items and not selling them.) 2. The Deterministic Model (DM) The demand is assumed to be deterministic and given by the demand vector D, where D T = d 1,d 2,d 3 = 150, 125, 300. The following is the formulation of the deterministic model. Date: September 15,

2 2 AYMAN H. RAGAB 2.1. Formulation of Deterministic Model. (DM) = max 2x 1 4x 2 2x 3 +60y 1 +40y 2 +10y 3 x 1 +8y 1 +6y 2 +y 3 0 x 2 +2y y y 3 0 x 3 +4y 1 +2y y 3 0 y y y x 1, x 2, x 3, y 1, y 2, y 3, Solution of Deterministic Model. The deterministic model was solved using cplex 8.0. The expected profit is $4, 165 and the values of the variables are as indicated below: X = 1950, 850, T Y = 150, 125, 0 T 2.3. Sensitivity Analysis. It is required to examine the effect of the fluctuation of the demand on the results of the model. It turned to be that any increase or decrease in the demand of the chairs does not affect the model anyway or the other. As for the desks and the tables, any change in the demand is accordingly reflected in the values of y 1 and y 2. Subsequently, the values of y 1 and y 2 are equal to the demand of desks and tables no matter what the demand turns to be. The amount of acquired resources is adjusted following the values of y 1 and y 2. The sensitivity of the model to the other parameters (namely cost, revenue and resource requirements) is not examined since the emphasis of part B of this exercise is solely on the randomness of the demand. 3. The Stochastic Solution The demand of desks, tables and chairs is assumed to be random and following a multivariate distribution. The low demand, D l = 50, 20, 200 T, happens with a probability p l =0.3. The medium demand, D m = 150, 110, 225 T, happens with a probability p m =0.4. The high demand, D h = 250, 250, 500 T, happens with a probability p h = The Expected Value Solution. In the expected value model (EVM), the expected value of the demand is D e = 150, 125, 300 T which is equal to the deterministic demand of (DM). Both The formulation and the solution of the model are therefore identical to those of (DM). It is obvious that: the solution is infeasible when low and medium demands realize the solution is sub-optimal when high demand realizes due to the combined effect of infeasibility and suboptimality, the estimated expected profit will not be achieved in practice

3 DAKOTA FURNITURE COMPANY The Scenario Analysis Solution. Each of the three scenario can be formulated as the (DM) with the appropriate demand vector replacing the right hand side of the constraints. The profit for the low demand scenario is $1124 and the values of the associated variables are listed below: X = 520, 240, 130 T Y = 50, 20, 0 T The profit for the medium demand scenario is $3982 and the values of the associated variables are listed below: X = 1860, 820, 465 T Y = 150, 110, 0 T The profit for the high demand scenario is $7450 and the values of the associated variables are listed below: X = 3500, 1500, 875 T Y = 250, 250, 0 T It is important to note, however, that this analysis doesn t provide a mean to decide on the quantity of resources to acquire before the realization of the demand. In other words, the model s value is in the insight that it provides the decision maker and not in providing the decision maker with an optimal solution The Fat Solution. The fat solution is intended to avoid the shortcomings of both the expected value solution and the scenario analysis by providing the decision maker with a solution that is always feasible. Below is the formulation of the fat model (FM). (FM) = max 2x 1 4x 2 2x 3 +60y 1 +40y 2 +10y 3 x 1 +8y 1 +6y 2 +y 3 0 x 2 +2y y y 3 0 x 3 +4y 1 +2y y 3 0 y 1 50 y 2 20 y y y y y y y x 1, x 2, x 3, y 1, y 2, y 3, 0 Needless to mention, the medium and hight demand sets of constraints above are redundant and can be omitted without loss of information. As a result, the model produces results identical to that of the low demand scenario. Although the solution is always feasible, it is apparently too conservative in the sense that in two thirds of the times DFC is missing the opportunity to produce and sell more products and hence achieve higher profits.

4 4 AYMAN H. RAGAB 3.4. The Two-stage SLP Recourse Model Solution. In the two-stage SLP recourse model (TRM), the amount of items to be produced and the demand are scenario dependent. The model can be formulated as follows. (TRM) = max 2x 1 4x 2 2x 3 + ω p ω [60y 1 ( ω)+40y 2 ( ω) +10y 3 ( ω)] x 1 +8y 1 ( ω)+6y 2 ( ω) +y 3 ( ω) 0 x 2 +2y 1 ( ω)+1.5y 2 ( ω) +0.5y 3 ( ω) 0 x 3 +4y 1 ( ω)+2y 2 ( ω) +1.5y 3 ( ω) 0 y 1 ( ω) d 1 ( ω) y 2 ( ω) d 2 ( ω) y 3 ( ω) d 3 ( ω) x 1, x 2, x 3, y 1 ( ω), y 2 ( ω), y 3 ( ω), 0 The expected profit for the model is $1, 730, and the value of the decision variables are as follows: First-stage variables: X = 1300, 540, 325 T Second-stage variables: for low demand Y l = 50, 20, 200 T for medium demand Y m = 80, 110, 0 T for high demand Y h = 80, 110, 0 T 4. Solution Analysis The deterministic case is suitable when the values of the demand are known with absolute certainty (e.g. engineer to order situation). However, when the values of the demand are merely an approximation for the uncertain demand (e.g. expected value), the solution provided can be misleading. For example, in the Dakota Furnishing Company s (DFC) situation, it turns to be that the expected value solution is infeasible for two of the scenarios. Hence, on the long run, the expected profit is not going to be achieved. Furthermore, the sensitivity analysis sometimes fails to alarm the decision maker to the need for further analysis, as was clearly the case in the DFC situation where the sensitivity analysis concluded that under no circumstance the production of the chairs is beneficial. The scenario analysis solution can provide a good deal of intuition, yet it does not provide a solution. Taking the expected value for the different scenario is just another way to achieve the same quality of results as achieved by the expected value solution. The fat solution resolve the problem of the possible infeasibility of the expected value solution under some scenarios, but the most conservative scenarios have an undue influence over the solution that in the case of the DFC the fat solution was identical to the low demand scenario solution. Finally, the two stage SLP recourse model provides the best results in terms: (1) the solution is always feasible under any scenario (it is more effective than the expected value solution)

5 DAKOTA FURNITURE COMPANY 5 (2) the solution is not unduly driven by the most conservative scenario (it performs considerably better than the fat solution because it takes into consideration reacting differently to different scenarios) (3) the solution is implementable and not informative (unlike the scenario analysis solution)